Superstrings, Geometry, Topology, and C*-algebras

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American Mathematical SocietyProvidence, Rhode Island

PURE MATHEMATICSProceedings of Symposia in

Volume 81

Superstrings, Geometry, Topology, and C*-algebras

Robert S. DoranGreg FriedmanJonathan Rosenberg Editors

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NSF-CBMS REGIONAL CONFERENCE ONMATHEMATICS ON TOPOLOGY, C∗-ALGEBRAS,

AND STRING DUALITY,HELD AT TEXAS CHRISTIAN UNIVERSITY,FORT WORTH, TEXAS, MAY 18–22, 2009

with support from the National Science Foundation,Grant DMS-0735233

2000 Mathematics Subject Classification. Primary 81–06, 55–06, 46–06, 46L87, 81T30.

Any opinions, findings, and conclusions or recommendations expressed in this materialare those of the authors and do not necessarily reflect the views of the National ScienceFoundation.

Library of Congress Cataloging-in-Publication Data

NSF-CBMS Conference on Topology, C∗-algebras, and String Duality (2009 : Texas ChristianUniversity)

Superstrings, geometry, topology, and C∗-algebras : NSF-CBMS Conference on Topology, C∗-algebras, and String Duality, May 18–22, 2009, Texas Christian University, Fort Worth, Texas /Robert S. Doran, Greg Friedman, Jonathan Rosenberg, editors.

p. cm. — (Proceedings of symposia in pure mathematics ; v. 81)Includes bibliographical references and index.ISBN 978-0-8218-4887-6 (alk. paper)1. Algebraic topology—Congresses. 2. Quantum theory—Congresses. 3. Functional analysis—

Congresses. I. Doran, Robert S., 1937– II. Friedman, Greg, 1973– III. Rosenberg, J.(Jonathan),1951– IV. Conference Board of the Mathematical Sciences. V. National Science Foundation(U.S.) VI. Title.

QA612.N74 2009530.12—dc22

2010027233

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10 9 8 7 6 5 4 3 2 1 15 14 13 12 11 10

Contents

Preface vii

Conference Attendees xi

Conference Speakers xiii

IntroductionJonathan Rosenberg 1

Functoriality of Rieffel’s Generalised Fixed-Point Algebras for Proper ActionsAstrid an Huef, Iain Raeburn, and Dana P. Williams 9

Twists of K-theory and TMFMatthew Ando, Andrew J. Blumberg, and David Gepner 27

Division Algebras and Supersymmetry IJohn C. Baez and John Huerta 65

K-homology and D-branesPaul Baum 81

Riemann-Roch and Index Formulae in Twisted K-theoryAlan L. Carey and Bai-Ling Wang 95

Noncommutative Principal Torus Bundles via Parametrised Strict DeformationQuantization

Keith C. Hannabuss and Varghese Mathai 133

A Survey of Noncommutative Yang-Mills Theory for Quantum HeisenbergManifolds

Sooran Kang 149

From Rational Homotopy to K-Theory for Continuous Trace AlgebrasJohn R. Klein, Claude L. Schochet, and Samuel B. Smith 165

Distances between Matrix Algebras that Converge to Coadjoint OrbitsMarc A. Rieffel 173

Geometric and Topological Structures Related to M-branesHisham Sati 181

Landau-Ginzburg Models, Gerbes, and Kuznetsov’s Homological ProjectiveDuality

Eric Sharpe 237

v

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Preface

The Conference Board of the Mathematical Sciences (CBMS) hosted a regionalconference, funded by the National Science Foundation, during the week of May18–22, 2009, entitled Topology, C*-algebras, and String Duality at Texas ChristianUniversity in Fort Worth, Texas. The principal lecturer was Jonathan Rosenbergof the University of Maryland, whose conference lectures have been published inVolume 111 of the CBMS’s Regional Conference Series in Mathematics. In additionto Professor Rosenberg’s lectures, the conference featured talks by fifteen otherspeakers on topics related to his lectures and the general theme of the conference.The purpose of this volume is to collect the contributions of these speakers and otherparticipants. All papers have been carefully refereed and will not appear elsewhere.At first sight these papers, which are highly interdisciplinary, may appear unrelated.To provide direction and historical context for the reader, a technical introductiondescribing how the various papers fit together in a natural way has been writtenby Professor Rosenberg. It appears as the first article in the volume.

The editors express their sincere gratitude and thanks to the speakers for theirbeautiful talks and their willingness to spend many hours writing them up so thatthe results would be available to the larger scientific community. In addition, we ac-knowledge the hard work and help of the referees. We thank the Conference Boardof the Mathematical Sciences and the National Science Foundation for their sup-port via NSF Grant DMS-0735233. We thank Sergei Gelfand, Christine Thivierge,and the dedicated staff at the American Mathematical Society for their efforts inpublishing these proceedings. Finally, we thank Texas Christian University and allthe participants who helped ensure a wonderfully successful conference.

Robert S. DoranGreg Friedman

Jonathan Rosenberg

vii

1.GregFriedman

2.JonathanRosenberg

3.RobertDoran

4.Shilin

Yu

5.Matthew

Ando6.HuichiHuang

7.VargheseMathai8.EftonPark

9.Ruth

Gornet

10.RishniRatnam

11.StefanMendez-D

iez12.SooranKang

13.Ped

ram

Hekmati

14.Jacques

Distler

15.PhuChung

16.SeunghunHong17.ValentinDeaconu

18.DorinDumitrascu

19.AlanCarey

20.Reb

ecca

Chen

21.Marc

A.Rieffel

22.Letty

Reza

23.Mart

Abel

24.Hisham

Sati

25.JonSjogren

26.EricSharpe

27.Ken

Richardson

28.Joh

nSkukalek

29.MichaelTseng

30.JodyTrout31.Peter

Bouwknegt

32.Wan

gQingy

un

33.Nigel

Higson

34.Dan

ielFreed

35.MarkTomforde

36.MagnusGoff

eng37.ClaudeSchochet

38.Bruce

Doran

39.JohnHuerta

40.JacobShotw

ell

41.DanielPape

42.James

West43.JonathanBlock

44.LoredanaCiurdariu

45.LorenSpice

46.AnnaSpice

47.DanaW

illiams

48.BraxtonCollier

Notpictured:Pau

lBaum,Alexander

A.Katz,Snigdhayan

Mahanta,Scott

Nollet

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Conference Attendees

Mart AbelUniversity of Tartu

Matthew AndoUniversity of Illinois

Paul BaumPennsylvania State University

Jonathan BlockPennsylvania State University

Peter BouwknegtAustralian National University

Alan CareyAustralian National University

Rebecca ChenUniversity of Houston

Phu ChungUniversity at Buffalo

Loredana CiurdariuUniversity Politechnic of Timisoara

Braxton CollierUniversity of Texas at Austin

Valentin DeaconuUniversity of Nevada, Reno

Jacques DistlerUniversity of Texas at Austin

Robert DoranTexas Christian University

Bruce DoranAccenture

Dorin DumitrascuNorthern Arizona University

Daniel FreedUniversity of Texas at Austin

Greg FriedmanTexas Christian University

Magnus GoffengChalmers University of Technology andUniversity of Gothenburg

Ruth GornetUniversity of Texas at Arlington

Pedram HekmatiRoyal Institute of Technology

Nigel HigsonPennsylvania State University

Seunghun HongPennsylvania State University

Huichi HuangUniversity at Buffalo

John HuertaUniversity of California, Riverside

Sooran KangUniversity of Colorado at Boulder

Alexander A. KatzSt. John’s University

Snigdhayan MahantaJohns Hopkins University

Varghese MathaiUniversity of Adelaide

Stefan Mendez-DiezUniversity of Maryland

Scott NolletTexas Christian University

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xii PARTICIPANTS

Daniel PapeMathematisches Intitut Gottingen

Efton ParkTexas Christian University

Wang QingyunWashington University in St. Louis

Rishni RatnamAustralian National University

Letty RezaUniversity of Houston

Ken RichardsonTexas Christian University

Marc A. RieffelUniversity of California, Berkeley

Jonathan RosenbergUniversity of Maryland

Hisham SatiYale University

Claude SchochetWayne State University

Eric SharpeVirginia Tech

Jacob ShotwellArizona State University

Jon SjogrenAir Force Office of Scientific Research

John SkukalekPennsylvania State University

Anna SpiceUniversity of Michigan

Loren SpiceUniversity of Michigan

Mark TomfordeUniversity of Houston

Jody TroutDartmouth College

Michael TsengPennsylvania State University

James WestUniversity of Houston

Dana WilliamsDartmouth College

Shilin YuPennsylvania State University

Conference Speakers

Jonathan RosenbergTopology, C∗-algebras, and StringDuality

Matthew AndoTwisted Generalized Cohomology andTwisted Elliptic Cohomology

Paul BaumEquivariant K Homology

Jonathan BlockHomological Mirror Symmetry andNoncommutative Geometry

Peter BouwknegtThe Geometry Behind Non-geometricFluxes

Alan CareyTwisted Geometric Cycles

Jacques DistlerGeometry and Topology of Orientifolds I

Dan FreedGeometry and Topology of Orientifolds II

Nigel HigsonThe Baum-Connes Conjecture andParametrization of Representations

Sooran KangThe Yang-Mills Functional andLaplace’s Equation on QuantumHeisenberg manifolds

Varghese MathaiThe Index of Projective Families ofElliptic Operators

Marc RieffelVector Bundles for “Matrix AlgebrasConverge to the Sphere”

Hisham SatiFivebrane Structures in String Theoryand M-theory

Claude SchochetAn Update on the Unitary Group

Eric SharpeGLSMs, Gerbes, and Kuznetsov’sHomological Projective Duality

Dana WilliamsProper Actions on C*-algebras

xiii

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Proceedings of Symposia in Pure Mathematics

Introduction

Jonathan Rosenberg

Abstract. The papers in this volume are the outgrowth of an NSF-CBMSRegional Conference in the Mathematical Sciences, May 18–22, 2009, orga-nized by Robert Doran and Greg Friedman at Texas Christian University.This introduction explains the scientific rationale for the conference and someof the common themes in the papers.

During the week of May 18–22, 2009, Robert Doran and Greg Friedman orga-nized a wonderfully successful NSF-CBMS Regional Research Conference at TexasChristian University. I was the primary lecturer, and my lectures have now beenpublished in [29]. However, Doran and Friedman also invited many other mathe-maticians and physicists to speak on topics related to my lectures. The papers inthis volume are the outgrowth of their talks.

The subject of my lectures, and the general theme of the conference, was highlyinterdisciplinary, and had to do with the confluence of superstring theory, algebraictopology, and C∗-algebras. While with “20/20 hindsight” it seems clear that thesesubjects fit together in a natural way, the connections between them developedalmost by accident.

Part of the history of these connections is explained in the introductions to [11]and [17]. The authors of [11] begin as follows:

Until recently the interplay between physics and mathematics fol-lowed a familiar pattern: physics provides problems and mathe-matics provides solutions to these problems. Of course at timesthis relationship has led to the development of new mathematics.. . . But physicists did not traditionally attack problems of puremathematics.

This situation has drastically changed during the last 15 years.Physicists have formulated a number of striking conjectures (suchas the existence of mirror symmetry) . . . . The basis of the physi-cists’ intuition is their belief that underlying quantum field theory

2010 Mathematics Subject Classification. Primary 81-06; Secondary 55-06, 46-06, 46L87,81T30.

Key words and phrases. string theory, supersymmetry, D-brane, C∗-algebra, crossed product,topological K-theory, twisted K-theory, classifying space, noncommutative geometry, Landau-Ginzburg theory, Yang-Mills theory.

Partially supported by NSF grant DMS-0805003.

c©0000 (copyright holder)

1

Proceedings of Symposia in Pure MathematicsVolume 81, 2010

c©2010 American Mathematical Society

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2 JONATHAN ROSENBERG

and string theory is a (as yet undiscovered) self-consistent mathe-matical framework.

Of course this was written over 10 years ago. In the last 10 years, the sameprinciple has been borne out time and again. As far as the subject matter of thisvolume is concerned, there are a few key developments from the last 35 years thatone can point to, that played an essential role:

(1) The Baum-Douglas [2, 3] and Kasparov [19, 20] approaches to (respec-tively) topological and analytic K-homology, and the realization thatthese theories are naturally isomorphic.

(2) The “Second Superstring Revolution” around 1995. Geometric objects,known as D-branes, were shown to play a fundamental role in string the-ory, and as time went on, it was realized that they naturally carry vectorbundles and topological charges (see for example [23, 31, 22, 32]), liv-ing in K-theory or K-homology (or still more complicated generalizedhomology theories).

(3) The development of Connes’ theory of “noncommutative differential ge-ometry,” epitomized by the book [9], and the gradual acceptance of non-commutative geometry as a natural tool in quantum field theories.

(4) The invention of “twisted K-theory,” and the realization that it has anatural realization in terms of continuous trace C∗-algebras (see [28, 1,18]).

My own interest in combining string duality with topology and noncommutativegeometry followed a rather circuitous route. A classical theorem of Grothendieckand Serre [15] computed the Brauer group BrC(X) for X a finite complex, andfound that it is isomorphic to the torsion subgroup of H3(X,Z). In the 1970’s, PhilGreen [14] worked out a more general theory of the Brauer group of C0(X), for Xa locally compact Hausdorff space. Green had the idea to drop all technical con-ditions on X and to allow continuous-trace algebras with infinite fiber dimension,not just classical Azumaya algebras, so as to get an isomorphism of the Brauergroup BrC0(X) with all of H3(X,Z), not just with its torsion subgroup. (WhenX is a finite complex, it doesn’t matter what cohomology theory one uses, but forgeneral locally compact spaces, Cech cohomology is appropriate here.) Now it sohappens that Donovan and Karoubi [12] had used classical Azumaya algebras todefine twisted K-theory with torsion twistings, so Green’s idea of using more gen-eral continuous-trace algebras to replace Azumaya algebras made possible definingtwisted K-theory of X with arbitrary twistings fromH3(X,Z). In [27, §6] I pointedthis out and explained how to generalize the Atiyah-Hirzebruch spectral sequenceto make this twisted K-theory somewhat computable. But for the most part, theidea just sat around for a while since nobody had any immediate use for it.

A number of years later, Raeburn and I [24] happened to study crossed productsof continuous trace algebras by smooth actions, and we discovered the followinginteresting “reciprocity law” [24, Theorem 4.12]:

Theorem 1. Let p : X → Z be a principal T-bundle, where T = R/Z is thecircle group. Also assume X and Z are second-countable, locally compact Haus-dorff, with finite homotopy type. Let H ∈ H3(X,Z) and let A = CT (X,H) be thecorresponding stable continuous-trace algebra with Dixmier-Douady class H. Thenthe free action of T = R/Z on X lifts (in a unique way, up to exterior equivalence)

2

INTRODUCTION 3

to an action of R on A inducing the given action of R/Z on A = X. The crossedproduct AR is again a stable continuous-trace algebra A# = CT (X#, H#), withp# : X# → Z again a principal T-bundle. Furthermore, the characteristic classesof p and p# are related to the Dixmier-Douady classes H and H# by

p!(H) = [p#], (p#)!(H#) = [p],

where p! and (p#)! are the Gysin maps of the circle bundles.

At the time, Raeburn and I regarded this entirely as a curiosity, and we certainlydidn’t expect any physical applications. A bit later [28], I continued my studiesof continuous-trace algebras and twisted K-theory, but I still didn’t expect anyphysical applications.

Much to my surprise, I discovered many years later that my studies of continu-ous-trace algebras and twisted K-theory were starting to show up in the physicsliterature in papers such as [7] and [4]. In fact, twisted K-theory seemed to beexactly the mathematical framework needed to studying D-brane charges in stringtheory. Not only that, but the “reciprocity law” of [24] for continuous-trace algebrasassociated to circle bundles also showed up in physics, as the recipe for topologychange and H-flux change in T-duality [5, 6]. Since that time, there has been afruitful continuing interaction between the subjects of string theory, topology, andC∗-algebras, an interaction that led to the organization of the CBMS conference atTCU in 2009.

With this as background, I can now explain how the various papers in thisvolume fit together. The papers of Baum and of Carey and Wang deal with D-brane charges in K-homology and twisted K-homology, a natural continuation ofthe combination of items (1), (2), and (4) on the list of key developments above(page 2). Baum’s paper deals with the extension to the twisted case of the Baum-Douglas approach to topological K-homology. While Baum does not go into theassociated physics, D-branes in type II string theories come with precisely thestructures he is discussing, and thus produce “topological charges” in the twistedK-homology of spacetime. The paper of Carey and Wang goes into more detail onthe same subject, and also discusses a Riemann-Roch theorem in twisted K-theory.Carey and Wang explain how D-brane charges in twisted K-theory arise in bothtype II and type I string theories.

The papers of Ando and Sati deal with roughly the same theme as those ofBaum and Carey-Wang, but in a somewhat generalized context. Ando explains(from the point of view of a stable homotopy theorist) how to construct twistedgeneralized cohomology theories in general, and then specializes to the cases oftwisted K, twisted elliptic cohomology, and twisted TMF. TMF [16], topologicalmodular forms, is a version of elliptic cohomology that seems to play an impor-tant role in M-theory, the “master” theory that gives rise (on reduction from 11dimensions to 10) to the five superstring theories. Sati’s paper concentrates on thephysics side of the same topic, and explains how the physics of M-branes (whichplay the same role in M-theory that the D-branes play in string theory) leads totwisted String and Fivebrane structures. (These are higher-dimensional analoguesof Spin and Spinc structures.) Sati also discusses the kinds of orientation conditionsthat arise for branes in F-theory [30], a 12-dimensional theory that is supposed toreduce to M-theory in certain circumstances.

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4 JONATHAN ROSENBERG

Two of the papers in this volume, by Kang and by Baez and Huerta, dealwith Yang-Mills gauge theory and its connection with noncommutative geometry.The basic Yang-Mills action on a spacetime manifold M is (up to a scalar factor)−∫M

Tr(F ∧ F ), where F is the curvature of a connection on a principal G-bundleover M . Here G is some Lie group which depends on the details of the theory; forexample, in the “standard model” of particle physics it is SU(3) × SU(2) × U(1).Physicists have known for some time [8] that in some circumstances one can makethis action supersymmetric, by adding in a fermionic term of the form (again, upto a constant factor) 〈ψ, /∂ψ〉, where ψ is a spinor field and /∂ is the Dirac operator.However, this only appears to work in three, four, six and ten dimensions. Thepaper of Baez and Huerta gives an explanation for this fact in terms of the factthat division algebras over R only occur in dimensions 1, 2, 4, and 8 (where one hasthe reals, complexes, quaternions, and octonions, respectively). Kang’s paper dealswith noncommutative Yang-Mills in the sense of Connes and Rieffel [10], where thebasic Yang-Mills action becomes −Tr(Θ,Θ), where Θ is the curvature 2-formfor a connection on a finitely generated projective module (the natural analogue ofa vector bundle) over the smooth subalgebra of some C∗-algebra A. Connes andRieffel took A to be Aθ, the irrational rotation algebra generated by two unitariesU and V with UV = e2πiθV U . Kang considers the somewhat more complicatedcase of the “quantum Heisenberg manifold” in the sense of Rieffel [25]; this is adeformation quantization of the algebra of functions on a Heisenberg nilmanifold.

Just to relate the papers of Baez-Huerta and Kang to the rest of the volume, it isperhaps worthwhile to explain how Yang-Mills and super-Yang-Mills are related tostring theory. There are two interconnected ties between the two subjects. On theone hand, as we mentioned already, D-branes naturally carry certain Chan-Patonvector bundles; on these there is a natural Yang-Mills action. In addition, there isa duality, known as the AdS/CFT correspondence, between type IIB string theoryon S5×AdS5 (AdS5 is anti-de Sitter space, a 5-dimensional Lorentz manifold witha metric of constant negative curvature) and 4-dimensional super-Yang-Mills on S4

in the large-N limit [21].The paper of Sharpe deals with Landau-Ginzburg models, a class of mod-

els which were originally constructed to model superconductivity, but which haveturned out to be extremely useful for superstring theory as well. A Landau-Ginzburg model in string theory describes propagation of strings on a noncompactspacetime (always a complex manifold) with a holomorphic superpotential W , of-ten having a degenerate critical point. One of the results explained in Sharpe’spaper is that A-twisted correlation functions in the Landau-Ginzburg model on

X = Tot(E∨ π−→ B), E → B a holomorphic vector bundle, with W = p · π∗s,p a tautological section of π∗E∨ and s a holomorphic section of E , should matchcorrelation functions in the nonlinear sigma model on s = 0. Since the com-plex geometry of the Landau-Ginzburg spacetime is usually quite different fromthe one which the sigma model lives, sometimes one gets interesting relations inenumerative algebraic geometry which are hard to explain directly.

The papers of Hannabuss-Mathai, Reiffel, Klein-Schochet-Smith, and an Huef-Raeburn-Williams all deal with various aspects of C∗-algebraic noncommutativegeometry. Several of them also have ties to quantum physics and to topology. Rief-fel’s paper gives explicit examples of sequences of matrix algebras with dimensionsgoing to ∞ whose “proximity” in a rather precise but technical sense goes to 0.

4

INTRODUCTION 5

This sort of calculation is motivated by the use of “matrix models” to approximatequantum field theories on spaces with complicated geometry.

The paper of Klein-Schochet-Smith computes the rational homotopy type ofthe group U(A) of unitary elements in the Azumaya algebra A of sections of abundle of matrix algebras Mn over a compact space X. This turns out to beindependent of what Azumaya algebra one chooses (so that one might as well takeA = C(X)⊗Mn), basically because the Brauer group of C(X) is torsion, and theauthors are only interested in rational information. This paper also computes themap πj(U(A))⊗Q → Kj(A)⊗Q; this gives explicit information on the stable rangefor rationalized topological K-theory of X. The paper of Hannabuss and Mathaideals with Rieffel’s theory of strict deformation quantization [26] and the theory ofnoncommutative principal bundles due to Echterhoff, Nest, and Oyono-Oyono [13].The main theorem of this paper is that for every such bundle with a suitable smoothstructure A∞(X), there is a principal torus bundle T → X and a correspondingstrict deformation quantization σ of C∞

fibre(Y ) (the continuous functions on Y thatare fibrewise smooth), so that A∞(X) ∼= C∞

fibre(Y )σ.Finally, the paper by an Huef, Raeburn, and Williams talks about functoriality

issues in the theories of C∗-crossed products and fixed-point algebras for properactions. Issues like this come up when one tries to use C∗-algebraic noncommutativegeometry to study the geometry of spacetime in various physical theories.

We hope the diversity of the papers in this volume will give the reader someidea of the breadth and vitality of the current interplay between superstring theory,geometry/topology, and noncommutative geometry.

Acknowledgments

I would like to thank Robert Doran and Greg Friedman again for their excel-lent work in organizing the conference. In addition, all three of us would like tothank the Conference Board of the Mathematical Sciences and the National ScienceFoundation for their financial support. NSF Grant DMS-0735233 supported theconference, and NSF Grant DMS-0602750 supported the entire Regional Confer-ence program. Finally, we would like to thank the American Mathematical Societyfor encouraging the publication of this volume in the Proceedings of Symposia inPure Mathematics series.

References

[1] Michael Atiyah and Graeme Segal, Twisted K-theory, Ukr. Mat. Visn. 1 (2004), no. 3, 287–330; Engl. translation, Ukr. Math. Bull. 1 (2004), no. 3, 291–334, arxiv.org: math/0407054.MR2172633 (2006m:55017)

[2] Paul Baum and Ronald G. Douglas, Index theory, bordism, and K-homology, in Operatoralgebras and K-theory (San Francisco, Calif., 1981), Contemp. Math., vol. 10, Amer. Math.Soc., Providence, RI, 1982, pp. 1–31. MR658506 (83f:58070)

[3] , K-homology and index theory, in Operator algebras and applications, Part I(Kingston, Ont., 1980), Proc. Sympos. Pure Math., vol. 38, Amer. Math. Soc., Providence,RI, 1982, pp. 117–173. MR679698 (84d:58075)

[4] Peter Bouwknegt, Alan L. Carey, Varghese Mathai, Michael K. Murray, and Danny Stevenson,Twisted K-theory and K-theory of bundle gerbes, Comm. Math. Phys. 228 (2002), no. 1,17–45, arxiv.org: hep-th/0106194. MR1911247 (2003g:58049)

[5] Peter Bouwknegt, Jarah Evslin, and Varghese Mathai, T -duality: topology change fromH-flux, Comm. Math. Phys. 249 (2004), no. 2, 383–415, arxiv.org: hep-th/0306062.MR2080959 (2005m:81235)

5

6 JONATHAN ROSENBERG

[6] , Topology and H-flux of T -dual manifolds, Phys. Rev. Lett. 92 (2004), no. 18, 181601,3, arxiv.org: hep-th/0312052. MR2116165 (2006b:81215)

[7] Peter Bouwknegt and Varghese Mathai, B-fields and twisted K-theory, J. High Energy Phys.2000, no. 3, Paper 7, arxiv.org: hep-th/0002023. MR1756434 (2001i:81198)

[8] Lars Brink, John H. Schwarz, and J. Scherk, Supersymmetric Yang-Mills theories, NuclearPhys. B 121 (1977), no. 1, 77–92. MR0446217 (56 #4545)

[9] Alain Connes, Noncommutative geometry, Academic Press Inc., San Diego, CA, 1994, now

available at http://www.alainconnes.org/en/downloads.php. MR1303779 (95j:46063)[10] Alain Connes and Marc A. Rieffel, Yang-Mills for noncommutative two-tori, in Operator

algebras and mathematical physics (Iowa City, Iowa, 1985), 237–266, Contemp. Math., 62,Amer. Math. Soc., Providence, RI, 1987. MR0878383 (88b:58033)

[11] Pierre Deligne, Pavel Etingof, Daniel S. Freed, David Kazhdan, John W. Morgan, andDavid R. Morrison (eds.), Quantum fields and strings: a course for mathematicians. Vol. 1,2, American Mathematical Society, Providence, RI, 1999, Material from the Special Year onQuantum Field Theory held at the Institute for Advanced Study, Princeton, NJ, 1996–1997.MR1701618 (2000e:81010)

[12] P. Donovan and M. Karoubi, Graded Brauer groups and K-theory with local coefficients,

Inst. Hautes Etudes Sci. Publ. Math. No. 38 (1970), 5–25. MR0282363 (43 #8075)[13] Siegfried Echterhoff, Ryszard Nest, and Herve Oyono-Oyono, Principal non-commutative

torus bundles, Proc. Lond. Math. Soc. (3) 99 (2009), no. 1, 1–31, arxiv.org: 0810.0111.MR2520349

[14] Philip Green, The Brauer group of a commutative C∗-algebra, handwritten manuscript, 1978.[15] Alexander Grothendieck, Le groupe de Brauer. I. Algebres d’Azumaya et interpretations

diverses, in Seminaire Bourbaki, Vol. 9, Exp. No. 290, Soc. Math. France, Paris, 1995, pp. 199–219. (Also printed in Dix exposes sur la cohomologie des schemas, Advanced Studies in PureMathematics, Vol. 3, North-Holland Publishing Co., Amsterdam; Masson & Cie, Editeur,Paris 1968.) MR0244269 (39 #5586a), MR1608798

[16] M. J. Hopkins, Algebraic topology and modular forms, in Proceedings of the InternationalCongress of Mathematicians, Vol. I (Beijing, 2002), 291–317, Higher Ed. Press, Beijing, 2002,arxiv.org: math.AT/0212397. MR1989190 (2004g:11032).

[17] Kentaro Hori, Sheldon Katz, Albrecht Klemm, Rahul Pandharipande, Richard Thomas, Cum-run Vafa, Ravi Vakil, and Eric Zaslow, Mirror symmetry, Clay Mathematics Monographs,vol. 1, American Mathematical Society, Providence, RI, 2003, With a preface by C. Vafa.MR2003030 (2004g:14042)

[18] Max Karoubi, Twisted K-theory, old and new, in K-Theory and Noncommutative Geome-try (Valladolid, Spain, 2006), EMS Ser. Congr. Rep., European Math. Soc., Zurich, 2008,pp. 117–149, arxiv.org: math.KT/0701789. MR2513335

[19] G. G. Kasparov, Topological invariants of elliptic operators. I. K-homology, Izv. Akad. NaukSSSR Ser. Mat. 39 (1975), no. 4, 796–838; Engl. translation, Math. USSR-Izv. 9 (1975), no.

4, 751–792 (1976). MR0488027 (58 #7603)[20] , The operator K-functor and extensions of C∗-algebras, Izv. Akad. Nauk SSSR Ser.

Mat. 44 (1980), no. 3, 571–636, 719; Engl. translation, Math. USSR Izv. 16, 513–572 (1981).MR0582160 (81m:58075)

[21] Juan Maldacena, The large N limit of superconformal field theories and supergravity, Adv.Theor. Math. Phys. 2 (1998), no. 2, 231–252, arxiv.org: hep-th/9711200. MR1633016(99e:81204a)

[22] Ruben Minasian and Gregory Moore, K-theory and Ramond-Ramond charge, J. High EnergyPhys. 1997, no. 11, Paper 2, see updated version at arxiv.org: hep-th/9710230. MR1606278(2000a:81190)

[23] Joseph Polchinski, Dirichlet branes and Ramond-Ramond charges, Phys. Rev. Lett. 75(1995), no. 26, 4724–4727, arxiv.org: hep-th/9510017. MR1366179 (96m:81185)

[24] Iain Raeburn and Jonathan Rosenberg, Crossed products of continuous-trace C∗-algebras bysmooth actions, Trans. Amer. Math. Soc. 305 (1988), no. 1, 1–45. MR920145 (89e:46077)

[25] Marc A. Rieffel, Deformation quantization of Heisenberg manifolds. Comm. Math. Phys. 122(1989), no. 4, 531–562. MR1002830 (90e:46060)

[26] , Deformation quantization and operator algebras, in Operator theory: operator alge-bras and applications, Part 1 (Durham, NH, 1988), 411–423, Proc. Sympos. Pure Math., 51,Part 1, Amer. Math. Soc., Providence, RI, 1990. MR1077400 (91h:46120)

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[27] Jonathan Rosenberg, Homological invariants of extensions of C∗-algebras, in Operator alge-bras and applications, Part 1 (Kingston, Ont., 1980), Proc. Sympos. Pure Math., vol. 38,Amer. Math. Soc., Providence, RI, 1982, pp. 35–75. MR679694 (85h:46099)

[28] , Continuous-trace algebras from the bundle theoretic point of view, J. Austral. Math.Soc. Ser. A 47 (1989), no. 3, 368–381. MR1018964 (91d:46090)

[29] , Topology, C∗-algebras, and string duality, CBMS Regional Conference Series inMathematics, vol. 111, American Mathematical Society, Providence, RI, 2009. MR2560910

[30] Cumrun Vafa, Evidence for F -theory, Nuclear Phys. B 469 (1996), no. 3, 403–415, arxiv.org:hep-th/960202. MR1403744 (97g:81059)

[31] Edward Witten, Bound states of strings and p-branes, Nuclear Phys. B 460 (1996), no. 2,335–350, arxiv.org: hep-th/9510135. MR1377168 (97c:81162)

[32] , D-branes and K-theory, J. High Energy Phys. 1998, no. 12, Paper 19, arxiv.org:hep-th/9810188. MR1674715 (2000e:81151)

Department of Mathematics, University of Maryland, College Park, MD 20742-

4015

E-mail address: jmr@math.umd.edu

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Proceedings of Symposia in Pure Mathematics

Functoriality of Rieffel’s Generalised Fixed-PointAlgebras for Proper Actions

Astrid an Huef, Iain Raeburn, and Dana P. Williams

Abstract. We consider two categories of C∗-algebras; in the first, the iso-

morphisms are ordinary isomorphisms, and in the second, the isomorphismsare Morita equivalences. We show how these two categories, and categories ofdynamical systems based on them, crop up in a variety of C∗-algebraic con-

texts. We show that Rieffel’s construction of a fixed-point algebra for a properaction can be made into functors defined on these categories, and that hisMorita equivalence then gives a natural isomorphism between these functorsand crossed-product functors. There are interesting applications to nonabelian

duality for crossed products.

Introduction

Let α be an action of a locally compact group G on a C∗-algebra A. In [38],Rieffel studied a class of proper actions for which there is a Morita equivalencebetween the reduced crossed product Aα,rG and a generalised fixed-point algebraAα sitting inside the multiplier algebra M(A). Rieffel subsequently proved thatα is proper whenever there is a free and proper G-space T and an equivariantembedding ϕ : C0(T ) → M(A) [39, Theorem 5.7]. In [15], inspired by previouswork of Kaliszewski and Quigg [13], it was observed that Rieffel’s hypothesis saysprecisely that ((A,α), ϕ) is an object in a comma category of dynamical systems.It therefore becomes possible to ask questions about the functoriality of Rieffel’sconstruction, and about the naturality of his Morita equivalence.

These questions have been tackled in several recent papers [15, 7, 8], which webelieve contain some very interesting results. In particular, they have substantialapplications to nonabelian duality for C∗-algebraic dynamical systems. However,these papers also contain a confusing array of categories and functors. So our goalhere is to discuss the main categories and explain why people are interested in them.We will then review some of the main results of the papers [13, 15, 7, 8], and tryto explain why we find them interesting.

In all the categories of interest to us, the objects are either C∗-algebras ordynamical systems involving actions or coactions of a fixed group on C∗-algebras.

2000 Mathematics Subject Classification. 46L55.This research was supported by the Australian Research Council and the Edward Shapiro

fund at Dartmouth College.

c©0000 (copyright holder)

1

Proceedings of Symposia in Pure MathematicsVolume 81, 2010

c©2010 American Mathematical Society

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2 AN HUEF, RAEBURN, AND WILLIAMS

But when we decide what morphisms to use, we have to make a choice, and what wechoose depends on what sort of theorems we are interested in. Loosely speaking,we have to decide whether we want the isomorphisms in our category to be theusual isomorphisms of C∗-algebras, or to be Morita equivalences. We think that,once we have made that decision, there is a “correct” way to go forward.

We begin in §1 with a discussion of commutative C∗-algebras; since Moritaequivalence does not preserve commutativity, it is clear that in this case we wantisomorphisms to be the usual isomorphisms. However, even then we have to dosomething a little odd: we want the morphisms from A to B to be homomorphismsϕ : A → M(B). Once we have the right category, we can see that operatoralgebraists have been implicitly working in this category for years. The motivatingexample for Kaliszewski and Quigg was a duality theory for dynamical systems dueto Landstad [19], and our main motivation is, as we said above, to understandRieffel’s proper actions. We discuss Landstad duality in §2. In §3, we discuss itsanalogue for crossed products by coactions, which is due to Quigg [30], and howthis makes contact with Rieffel’s theory of proper actions.

We begin §4 by showing how the search for naturality results leads us to a dif-ferent category C* of C∗-algebras in which the morphisms are based on right-Hilbertbimodules. Categories of this kind have been around much longer, and [2, 3], forexample, contain a detailed discussion of how imprimitivity theorems provide nat-ural isomorphisms between functors with values in C*. In §5, we discuss a theoremfrom [7] which says that Rieffel’s Morita equivalences give a natural isomorphismbetween a crossed-product functor and a fixed-point-algebra functor. This power-ful result implies, for example, that the version in [9] of Mansfield imprimitivityfor arbitrary subgroups is natural. We finish with a brief survey of one of themain results of [8] which uses an approach based on Rieffel’s theory to establishinduction-in-stages for crossed products by coactions.

1. The Category C*nd and Commutative C∗-algebras

In our first course in C∗-algebras, we learned that commutative unital C∗-algebras are basically the same things as compact topological spaces. To make thisformal, we note that the assignment X → C(X) is the object map in a contravari-ant functor C from the category Cpct of compact Hausdorff spaces and continuousfunctions to the category CommC*1 of unital commutative C∗-algebras and unitalhomomorphisms (which for us are always ∗-preserving); the morphism C(f) asso-ciated to a continuous map f : X → Y sends a ∈ C(Y ) to a f ∈ C(X). Thenthe Gelfand-Naimark theorem implies that the functor C is an equivalence of cat-egories. (This result goes back to [25], and we will go into the details of what itmeans in the proof of Theorem 2 below.)

The Gelfand-Naimark theorem for non-unital algebras says that commutativeC∗-algebras are basically the same things as locally compact topological spaces.However, it is not so easy to put this version in a categorical context, and in doingso we run into some important issues which are very relevant to problems involvingcrossed products and nonabelian duality. So we will discuss these issues now asmotivation for our later choices.

There is no doubt what the analogue of the functor C does to objects: ittakes a locally compact Hausdorff space X to the C∗-algebra C0(X) of continuousfunctions a : X → C which vanish at infinity. However, there is a problem with

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PROPER ACTIONS 3

morphisms: composing with a continuous function f : X → Y does not necessarilymap C0(Y ) into C0(X). For example, consider the function f : R → R definedby f(x) = (1 + x2)−1: any function a ∈ C0(R) which is identically 1 on [0, 1]satisfies a f = 1, and hence a f does not vanish at infinity. One way out isto restrict attention to the category in which the morphisms from X to Y arethe proper functions f : X → Y for which inverse images of compact sets arecompact, and then on the C∗-algebra side one has to restrict attention to thehomomorphisms ϕ : A → B such that the products ϕ(a)b span a dense subspaceof B. In [28], Pedersen does exactly this, and calls these proper homomorphisms.It turns out, though, that there is a very satisfactory way to handle arbitrarycontinuous functions between locally compact spaces, in which we allow morphismswhich take values in Cb(X).

A homomorphism ϕ of one C∗-algebra A into the multiplier algebra M(B) ofanother C∗-algebra B is called nondegenerate if

ϕ(A)B := spanϕ(a)b : a ∈ A, b ∈ Bis all of B. (This notation is suggestive: the Cohen factorisation theorem says thateverything in the closed span factors as ϕ(a)b.) We want to think of the nondegen-erate homomorphisms ϕ : A → M(B) as morphisms from A to B. Every nonde-generate homomorphism ϕ extends to a unital homomorphism ϕ : M(A) → M(B)(see [35, Corollary 2.51], for example); the extension has to satisfy ϕ(m)(ϕ(a)b) =ϕ(ma)b, and hence the nondegeneracy implies that there is exactly one such exten-sion, and that it is strictly continuous.

The following fundamental proposition is implicit in a number of earlier works,including [43, 44], [41] and [13, §1].

Proposition 1. There is a category C*nd in which the objects are C∗-algebras,the morphisms from A to B are the nondegenerate homomorphisms from A toM(B), and the composition of ϕ : A → M(B) and ψ : B → M(C) is ψ ϕ := ψ ϕ.The isomorphisms in this category are the usual isomorphisms of C∗-algebras.

Proof. It is easy to check that the composition ψ ϕ : A → M(C) is non-degenerate, and hence defines a morphism in C*nd. Since ψ ϕ is a homomorphism

from M(A) to M(C) which extends ψ ϕ, it must be the unique extension ψ ϕ.Thus if θ : C → M(D) is another nondegenerate homomorphism, we have

θ (ψ ϕ) = θ (ψ ϕ) = θ (ψ ϕ) = (θ ψ) ϕ

= (θ ψ) ϕ = (θ ψ) ϕ = (θ ψ) ϕ,and composition in C*nd is associative. The identity maps idA : A → A, viewed ashomomorphisms into M(A), satisfy idA = idM(A), and hence have the propertiesone requires of the identity morphisms in C*nd. Thus C*nd is a category, as claimed.

For the last assertion, notice first that every isomorphism is trivially nonde-generate, and hence defines a morphism in C*nd, which is an isomorphism because ithas an inverse. Conversely, suppose that ϕ : A → M(B) and ψ : B → M(A) areinverses of each other in C*nd, so that ψ ϕ = idA and ϕ ψ = idB. Using first thenondegeneracy of ψ and then the nondegeneracy of ϕ, we obtain

ϕ(A) = ϕ(ψ(B)A) = ϕ(ψ(B))ϕ(A) = Bϕ(A) = B.

Thus ϕ has range B, and since ψ|B = ψ, we have ψ ϕ = idA. The same argumentsshow that ϕ ψ = idB, so ϕ is an isomorphism in the usual sense.

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4 AN HUEF, RAEBURN, AND WILLIAMS

If f : X → Y is a continuous map between locally compact spaces and a ∈C0(Y ), then a f is a continuous bounded function which defines a multiplier ofC0(X). For every b in the dense subalgebra Cc(X), we can choose a ∈ Cc(Y )such that a = 1 on f(supp b), and then b = (a f)b, so C0(f) : a → a f is a

nondegenerate homomorphism from C0(Y ) to M(C0(X)); the extension C0(f) toCb(X) = M(C0(X)) is again given by composition with f . We now have a functorC0 from the category LCpct of locally compact spaces and continuous maps to thefull subcategory CommC*nd of C*nd whose objects are commutative C∗-algebras. Thisfunctor has the properties we expect:

Theorem 2. The contravariant functor C0 : LCpct → CommC*nd is an equiva-lence of categories.

Proof. To say that C0 is an equivalence means that there is a functor G :CommC*nd → LCpct such that C0 G and G C0 are naturally isomorphic to theidentity functors. To verify that it is an equivalence, though, it suffices to showthat every object in CommC*nd is isomorphic to one of the form C0(X), which isexactly what the Gelfand-Naimark theorem says, and that C0 is a bijection on eachset Mor(X,Y ) of morphisms (see [21, page 91]). Injectivity is easy: since C0(Y )separates points of Y , af = ag for all a ∈ C0(Y ) implies that f(x) = g(x) for allx ∈ X. For surjectivity, we suppose that ϕ : C0(Y ) → C0(X) is a nondegeneratehomomorphism. Then for each x ∈ X, the composition εx ϕ with the evaluationmap is a homomorphism from C0(Y ) to C, and the nondegeneracy of ϕ implies thatεx ϕ is non-zero. Since ε : y → εy is a homeomorphism of Y onto the maximalideal space of C0(Y ), there is a unique f(x) ∈ Y such that εx ϕ = εf(x), and

f = ε−1 ϕ∗ ε is continuous. The equation εx ϕ = εf(x) then says precisely thatϕ = C0(f).

The result in [21, page 91] which we have just used in the proof of Theorem 2is a little unnerving to analysts. (Well, to us, anyway.) Its proof, for example,makes carefree use of the axiom of choice. So it is perhaps reassuring that in thesituation of Theorem 2, there is a relatively concrete inverse functor Δ which takesa commutative C∗-algebra A to its maximal ideal space Δ(A). (We say “relativelyconcrete” here because the axiom of choice is also used in the proof that the Gelfandtransform is an isomorphism.) The argument on page 92 of [21] shows that, oncewe have chosen isomorphisms ηA : A → C(Δ(A)) for every commutative C∗-algebraA, there is exactly one way to extend Δ to a functor in such a way that

η := ηA : A ∈ Obj(CommC*nd)

is a natural isomorphism. If we choose ηA : A → C0(Δ(A)) to be the Gelfandtransform, then the functor Δ takes a morphism ϕ : A → M(B) to the mapΔ(ϕ) : ω → ω ϕ. So we have the following naturality result:

Corollary 3. The Gelfand transforms ηA : A ∈ Obj(CommC*nd) form a natu-ral isomorphism between the identity functor on CommC*nd and the composition C0Δ.

Of course, modulo the existence of the isomorphisms ηA, which is the content ofthe (highly non-trivial) Gelfand-Naimark theorem, this result can be easily proveddirectly: we just need to check that for every morphism ϕ : A → M(B) the following

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PROPER ACTIONS 5

diagram commutes in CommC*nd:

AηA

ϕ

C0(Δ(A))

C0(Δ(ϕ))

B ηB

C0(Δ(B)).

2. Crossed Products and Landstad Duality

Although the category C*nd has only been studied recently, nondegenerate ho-momorphisms have been around for years. For example, the unitary representa-tions U : G → U(H) of a locally compact group G on a Hilbert space H are inone-to-one correspondence with the nondegenerate representations πU of the groupalgebras L1(G) or C∗(G) on H. In this context, “nondegenerate” usually meansthat the elements πU (a)h span a dense subspace of H, but this is equivalent to thenondegeneracy of πU as a homomorphism into B(H) = M(K(H)). More gener-ally, if u : G → UM(B) is a strictly continuous homomorphism into the unitarygroup of a multiplier algebra, then there is a unique nondegenerate homomorphismπu : C∗(G) → M(B), called the integrated form of u, from which we can recover uby composing with a canonical unitary representation kG : G → UM(C∗(G)). Thecomposition here is taken in the spirit of the category C*nd: it is the composition inthe usual sense of the extension of πu to M(C∗(G)) with kG. We say that kG isuniversal for unitary representations of G.

One application of this circle of ideas which will be particularly relevant hereis the existence of the comultiplication δG on C∗(G), which is the integrated formof the unitary representation kG ⊗ kG : G → UM(C∗(G)⊗ C∗(G)). Thus δG is bydefinition a nondegenerate homomorphism of C∗(G) into M(C∗(G)⊗ C∗(G)). Itsother crucial property is coassociativity: (δG ⊗ id) δG = (id⊗ δG) δG, where thecompositions are interpreted as being those in the category C*nd.

Now suppose that α : G → AutA is an action of a locally compact group G on aC∗-algebra. Nondegeneracy is then built into the notion of covariant representationof the system: a covariant representation (π, u) of a dynamical system (A,G, α) in amultiplier algebraM(B) consists of a nondegenerate homomorphism π : A → M(B)and a strictly continuous homomorphism u : G → UM(B) such that π(αt(a)) =utπ(a)u

∗t . The crossed product is then, either by definition [33] or by theorem

[42, 2.34–36], a C∗-algebra A α G which is generated (in a sense made precisein those references) by a universal covariant representation (iA, iG) of (A,G, α) inM(Aα G). Each covariant representation (π, u) in M(B) has an integrated formπ u which is a nondegenerate homomorphism of A α G into M(B) such thatπ = (π u) iA and u = (π u) iG.

The crossed product Aα G carries a dual coaction α, which is the integratedform of iG ⊗ kG : G → UM((A α G) ⊗ C∗(G)). This is another nondegeneratehomomorphism, and the crucial coaction identity (α⊗ id) α = (id⊗ δG) α againhas to be interpreted in the category C*nd. (Makes you wonder how we ever managedwithout C*nd.)

There is another version of the crossed-product construction which can be moresuitable for spatial arguments, and which is particularly important for the issueswe discuss in this paper. For any representation π : A → B(Hπ), there is aregular representation (π, U) of (A,G, α) on L2(G,Hπ) such that (π(a)h)(r) =

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6 AN HUEF, RAEBURN, AND WILLIAMS

π(α−1r (a))(h(r)) and λsh(r) = h(s−1r) for h ∈ L2(G,Hπ). The reduced crossed

product Aα,rG is the quotient of AαG which has the property that every πλfactors through a representation of A α,r G, and then π λ is faithful wheneverπ is [42, §7.2]. The reduced crossed product is also generated by a canonicalcovariant representation (irA, i

rG), and the dual coaction α factors through a coaction

αn : Aα,r G → M((Aα,r G)⊗ C∗(G)) characterised by

(1) αn irA(a) = irA(a)⊗ 1 and αn irG(s) = irG(s)⊗ kG(s).

This coaction is called the normalisation of α, and is in particular normal in thesense that the canonical map jAG of Aα,rG into M((Aα,rG)αG) is injective(see Proposition A.61 of [3]).

Kaliszewski and Quigg’s motivation for working in the category C*nd came fromthe following characterisation of the C∗-algebras which arise as reduced crossedproducts.

Theorem 4 (Landstad, Kaliszewski-Quigg). Suppose that B is a C∗-algebraand G is a locally compact group. Then there is a dynamical system (A,G, α)such that B is isomorphic to A α,r G if and only if there is a morphism π :C∗(G) → M(B) in C*nd and a nondegenerate (see Remark 6 below) normal coactionδ : B → M(B ⊗ C∗(G)) such that

(2) (π ⊗ id) δG = δ π.In [13] the authors say that this result follows from a theorem of Landstad

[19], and it is certainly true that most of the hard work is done by Landstad’sresult. But we think it is worth looking at the proof; those who are not interestedin the subtleties of coactions should probably skip to the end of the proof below.We begin by stating Landstad’s theorem in modern terminology.

Theorem 5 (Landstad, 1979). Suppose that B is a C∗-algebra and G is alocally compact group. Then there is a dynamical system (A,G, α) such that B isisomorphic to Aα,rG if and only if there are a strictly continuous homomorphismu : G → UM(B) and a reduced coaction δ : B → M(B ⊗ C∗

r (G)) such that

(a) δ(us) = us ⊗ λs for s ∈ G, and

(b) δ(B)(1⊗ C∗r (G)) = B ⊗ C∗

r (G).

The “reduced coaction” appearing in Landstad’s theorem is required to haveslightly different properties from the full coactions which we use elsewhere in thispaper, and which are used in [3] and [13], for example. A reduced coaction onB is an injective nondegenerate homomorphism of B into M(B ⊗ C∗

r (G)) ratherthan M(B ⊗ C∗(G)), and it is required to be coassociative with respect to thecomultiplication δrG on C∗

r (G).

Remark 6. Nowadays, the second condition (b) in Theorem 5 is usually ab-sorbed into the assertion that δ is a coaction. Everyone agrees that for δ to be acoaction δ(B)(1⊗C∗

r (G)) must be contained in B⊗C∗r (G), and Landstad described

the requirement of equality as “nondegeneracy”, which in view of our emphasis onC*nd has turned out to be unfortunate terminology. Coactions of amenable or discretegroups are automatically nondegenerate in Landstad’s sense, and dual coactions arealways nondegenerate. We therefore follow modern usage and assume that all coac-tions satisfy (b), or its analogue in the case of full coactions. (So (b) can now bedeleted from Theorem 5 and the word “nondegenerate” from Theorem 4.)

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PROPER ACTIONS 7

Proof of Theorem 4. For B = Aα,r G, we take δ = αn and π = πirG. The

second equation in (1) implies that

(π ⊗ id) δG(kG(s)) = π ⊗ id(kG(s)⊗ kG(s)) = irG(s)⊗ kG(s)

= αn π(kG(s))for all s ∈ G, which implies (2).

Now suppose that there exist π and δ as described. Then we define u := π krG,and consider the reduction δr of δ, which since δ is normal is just δr := (id⊗πλ)δ.Now we compute:

δr(us) = (id⊗ πλ) δ(π(krG(s))) = id⊗ πλ δ π(krG(s))= id⊗ πλ π ⊗ id δG(krG(s)) = π ⊗ πλ(k

rG(s)⊗ kG(s))

= π krG(s)⊗ λs = us ⊗ λs.

Thus u and δr satisfy the hypotheses of Landstad’s theorem (Theorem 5), and wecan deduce from it that B is isomorphic to a reduced crossed product.

Kaliszewski and Quigg then made two further crucial observations. First, theyrecognised that there is a category of coactions associated to C*: the objects inC*coactnd(G) consist of full coactions δ on C∗-algebras B, and the morphismsfrom (B, δ) to (C, ε) are nondegenerate homomorphisms ϕ : B → M(C) such that(ϕ ⊗ id) δ = ε ϕ. Then (2) says that the homomorphism π in Corollary 4 is amorphism in C*coactnd(G) from (C∗(G), δG) to (B, δ). Second, they knew that forevery object a and every subcategory D in a category C there is a comma categorya ↓ D in which objects are morphisms f : a → x in C from a to objects in D,and the morphisms from (x, f) to (y, g) are morphisms h : x → y in D such thath f = g. Thus Landstad’s theorem identifies the reduced crossed products asthe C∗-algebras which can be augmented with a coaction δ and a homomorphismπ to form an object in the comma category (C∗(G), δG) ↓ C*coactnnd(G), whereC*coact

nnd(G) is the full subcategory of normal coactions.

The main results in [13] concern crossed-product functors defined on the cat-egory C*actnd(G) whose objects are dynamical systems (A,G, α) and whose mor-phisms ϕ : (A,α) → (B, β) are nondegenerate homomorphisms ϕ : A → M(B)such that ϕ αs = βs ϕ for s ∈ G (where yet again the composition on the rightis taken in C*nd). The following theorem is Theorem 4.1 of [13].

Theorem 7 (Kaliszewski-Quigg, 2009). There is a functor CPr from C*actnd(G)to the comma category (C∗(G), δG) ↓ C*coactnnd(G) which takes the object (A,α) to(Aα,r G, αr, irG), and this functor is an equivalence of categories.

Landstad’s theorem, in the form of Theorem 4, says that CPr is essentiallysurjective: every object in the comma category is isomorphic to one of the formCPr(A,α) = A α,r G. Thus Theorem 7 can be viewed as an extension of Land-stad’s theorem, and Kaliszewski and Quigg call it “categorical Landstad duality foractions”. They also obtain an analogous result for full crossed products.

3. Proper Actions and Landstad Duality for Coactions

Quigg’s version of Landstad duality for crossed products by coactions [30] isalso easy to formulate in categories based on C*nd. Suppose that δ is a coaction of Gon a C∗-algebra C, and let wG denote the function s → kG(s), viewed as a multiplier

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8 AN HUEF, RAEBURN, AND WILLIAMS

of C0(G,C∗(G)). A covariant representation of (C, δ) in a multiplier algebra M(B)consists of nondegenerate homomorphisms π : C → M(B) and μ : C0(G) → M(B)such that

(π ⊗ id) δ(c) = μ⊗ id(wG)(π(c)⊗ 1)μ⊗ id(wG)∗ for c ∈ C,

where, as should seem usual by now, the composition is interpreted in C*nd. Thecrossed product CδG is generated by a universal covariant representation (jC , jG)in M(C δ G), in the sense that products jC(c)jG(f) span a dense subspace of

C δ G. The crossed product carries a dual action δ such that δs(jC(c)jG(f)) =jC(c)jG(rts(f)), where rt is defined by rts(f)(t) = f(ts). Quigg’s theorem identifiesthe C∗-algebras which are isomorphic to crossed products by coactions.

Theorem 8 (Quigg, 1992). Suppose that G is a locally compact group and Ais a C∗-algebra. There is a system (C, δ) such that A is isomorphic to C δ Gif and only if there are a nondegenerate homomorphism ϕ : C0(G) → M(A) andan action α of G on A such that (A,α, ϕ) is an object in the comma category(C0(G), rt) ↓ C*actnd(G).

When A = C δ G, we can take ϕ := jG and α := δ, and the hard bit is toprove the converse. This is done in [30, Theorem 3.3]. It is then natural to look fora “categorical Landstad duality for coactions” which parallels the results of [13].However, triples (A,α, ϕ) of the sort appearing in Theorem 8 had earlier (that is,before [13]) appeared in important work of Rieffel on proper actions, and it hasproved very worthwhile to follow up this circle of ideas in Rieffel’s context. Toexplain this, we need to digress a little.

If α : G → AutA is an action of a compact abelian group, then informationabout the crossed product can be recovered from the fixed point algebra Aα, and,more generally, from the spectral subspaces

Aα(ω) := a ∈ A : αs(a) = ω(s)a for ω ∈ G.

A fundamental result of Kishimoto and Takai [16, Theorem 2] says that if thespectral subspaces are large in the sense that Aα(ω)∗Aα(ω) is dense in Aα for every

ω ∈ G, then A α G is Morita equivalent to Aα. There is as yet no completelysatisfactory notion of a free action of a group on a C∗-algebra (see [29], for example),but having large spectral subspaces is one example of such a notion.

WhenG is locally compact, the fixed-point algebra is often trivial. For example,if rt is the action of G = Z on R by right translation, then f ∈ C0(R)

rt if and onlyif f is periodic with period 1, which since f vanishes at ∞ forces f to be identicallyzero. However, if the orbit space for an action is nice enough, then the algebra ofcontinuous functions on the orbit space can be used as a substitute for the fixed-point algebra. A right action of a locally compact group G on a locally compactspace T is called proper if the map (x, s) → (x, x ·s) : T ×G → T ×T is proper. Theorbit space T/G for a proper action is always Hausdorff [42, Corollary 3.43], anda classical result of Green [5] says that if the action of G on T is free and proper,then C0(T )rt G is Morita equivalent to C0(T/G) (for this formulation of Green’sresult see [42, Remark 4.12]). We want to think of C0(T/G) as a subalgebra of themultiplier algebra M(C0(G)) = Cb(T ) which is invariant under the extensions rtsof the automorphisms rts.

In the past twenty-five years, many researchers have investigated analoguesof free and proper actions for noncommutative C∗-algebras [34, 38, 4, 24, 10,

16

PROPER ACTIONS 9

39, 11]. Here we are interested in the notion of proper action α : G → AutAintroduced by Rieffel [38]. He assumes that there is an α-invariant subalgebra A0

of A with properties like those of the subalgebra Cc(T ) of C0(T ), and that there isan M(A)α-valued inner product on A0. The completion Z(A,α) of A0 in this innerproduct is a full Hilbert module over a subalgebra Aα of M(A)α, which Rieffel callsthe generalised fixed-point algebra for α. The algebra K(Z(A,α)) of generalisedcompact operators on Z(A,G, α) sits naturally as an ideal E(α) in the reducedcrossed product A α,r G [38, Theorem 1.5]. The action α is saturated whenE(α) is all of the reduced crossed product. Thus when α is proper and saturated,A α,r G is Morita equivalent to Aα. Saturation is a freeness condition: if G actsproperly on T , then rt : G → Aut(C0(T )) is proper with respect to Cc(G), and theaction is saturated if and only if G acts freely [23, §3]. On the face of it, though,Rieffel’s bimodule Z(A,α) and the fixed-point algebra Aα depend on the choice ofsubalgebra A0, and it seems unlikely that Rieffel’s process is functorial.

The connection with our categories lies in a more recent theorem of Rieffel whichidentifies a large family of proper actions for which there is a canonical choice ofthe dense subalgebra A0 [39, Theorem 5.7].

Theorem 9 (Rieffel, 2004). Suppose that a locally compact group G acts freelyand properly on the right of a locally compact space T , and (A,G, α) is a dynamicalsystem such that there is a nondegenerate homomorphism ϕ : C0(T ) → M(A)satisfying ϕ rt = α ϕ (with composition in the sense of C*nd). Then α is properand saturated with respect to the subalgebra A0 = ϕ(Cc(T ))Aϕ(Cc(T )).

Example 10. A closed subgroup H of a locally compact group G acts freelyand properly on G, and hence we can apply Theorem 9 to the pair (T,G) = (G,H)and to the canonical map jG : C0(G) → M(C δ G). In this case, highly nontrivial

results of Mansfield [22] can be used to identify the fixed-point algebra (C δ G)δ

with the crossed product Cδ,r (G/H) by the homogeneous space [9, Remark 3.4].(These crossed products were introduced in [1]; the relationship with the crossedproduct Cδ| (G/H) by the restricted coaction, which makes sense when H is nor-mal, is discussed in [1, Remark 2.2].) Then Theorem 3.1 of [9] shows that Rieffel’s

Morita equivalence between (C δ G) δ,r H and (C δ G)δ extends Mansfield’s

imprimitivity theorem for coactions to arbitary closed subgroups (as opposed tothe amenable normal subgroups in Mansfield’s original theorem [22, Theorem 27]and the normal ones in [12]).

From our categorical point of view, the hypotheses on ϕ in Theorem 9 say pre-cisely that (A,α, ϕ) := ((A,α), ϕ) is an object in the comma category (C0(T ), rt) ↓C*actnd(G). Then Rieffel’s theorem implies that (A,α, ϕ) → Aα is a constructionwhich takes objects in the comma category to objects in the category C*nd. Onenaturally asks: is this construction functorial? More precisely, is there an analo-gous construction on morphisms which makes (A,α, ϕ) → Aα into a functor from(C0(T ), rt) ↓ C*actnd(G) to C*nd?

This question was answered in [15, §2] using a new construction of Rieffel’sgeneralised fixed-point algebra. The crucial ingredient is an averaging process Eof Olesen and Pedersen [26, 27], which was subsequently developed by Quigg in[31, 32] and used extensively in his proof of Theorem 8. This averaging process E

17

10 AN HUEF, RAEBURN, AND WILLIAMS

makes sense on the dense subalgebra A0 = ϕ(Cc(T ))Aϕ(Cc(T )), and satisfies

ϕ(f)E(ϕ(g)aϕ(h)) =

∫G

ϕ(f)αs(ϕ(g)aϕ(h)) ds for f, g, h ∈ Cc(T ) and a ∈ A;

the integral on the right has an unambiguous meaning because properness impliesthat s → f rts(g) has compact support. It is shown in [15, Proposition 2.4] thatthe closure of E(A0) is a C∗-subalgebra of M(A), which we denote by Fix(A,α, ϕ)to emphasise all the data involved in the construction. It is shown in [15, Propo-sition 3.1] that Fix(A,α, ϕ) and Rieffel’s Aα are exactly the same subalgebra ofM(A). If σ : (A,α, ϕ) → (B, β, ψ) is a morphism in the comma category, so thatin particular σ is a nondegenerate homomorphism from A to M(B), then the ex-tension σ maps Fix(A,α, ϕ) into M(Fix(B, β, ψ)), and is nondegenerate. (This isProposition 2.6 of [15]; a gap in the proof of nondegeneracy is filled in Corollary 2.3of [8].)

Theorem 11 (Kaliszewski-Quigg-Raeburn, 2008). Suppose that a locally com-pact group G acts properly on the right of a locally compact space T . Then theassignments (A,α, ϕ) → Fix(A,α, ϕ) and σ → σ|Fix(A,α,ϕ) form a functor from(C0(T ), rt) ↓ C*actnd(G) to C*nd.

To return to the setting of Quigg-Landstad duality, we take (T,G) = (G,G)in this theorem. This gives us a functor Fix from (C0(G), rt) ↓ C*actnd(G) to C*nd.Because the fixed-point algebra Fix(A,α, ϕ) is defined using the same averagingprocess E as Quigg used in [30, §3], Fix(A,α, ϕ) is the same as the algebra Cconstructed by Quigg (unfortunately for us, he called it B). So Quigg proves in[30] that

δA(c) = ϕ⊗ πλ(wG)(c⊗ 1)ϕ⊗ πλ(wG)∗

defines a reduced coaction of G on C = Fix(A,α, ϕ), and that A is isomorphic tothe crossed product C δA G. An examination of the proof of [31, Theorem 4.7]shows that the similar formula

δfA(c) = ϕ⊗ id(wG)(c⊗ 1)ϕ⊗ id(wG)∗

defines the unique full coaction with reduction δA. The argument on page 2960of [15] shows that this construction respects morphisms, so that Fix extends to afunctor FixG from (C0(G), rt) ↓ C*actnd(G) to C*coact

nnd(G). The following very

satisfactory “categorical Landstad duality for coactions” is Corollary 4.3 of [15].

Theorem 12 (Kaliszewski-Quigg-Raeburn, 2008). Let G be a locally com-

pact group. Then (C, δ) → (C δ G, δ, jG) and π → π id form a functor fromC*coact

nnd(G) to (C0(G), rt) ↓ C*actnd(G). This functor is an equivalence of cate-

gories with quasi-inverse FixG.

In fact, this is a much more satisfying theorem than its analogue for actionsbecause we have a specific construction of a quasi-inverse. We would be interestedto see an analogous process for Fixing over coactions.

4. Naturality

Now that we have a functorial version Fix of Rieffel’s generalised fixed-pointalgebra, we remember that the main point of Rieffel’s paper [38] was to constructa Morita equivalence between Aα = Fix(A,α, ϕ) and the reduced crossed product

18

PROPER ACTIONS 11

RCP(A,α, ϕ) := A α,r G. This equivalence is implemented by an (A α,r G) –Fix(A,α, ϕ) imprimitivity bimodule Z(A,α, ϕ). There is another category C* ofC∗-algebras in which the isomorphisms are given by imprimitivity bimodules, soit makes sense to ask whether these isomorphisms are natural. Of course, beforediscussing this problem, we need to be clear about what the category C* is.

If A and B are C∗-algebras, then a right-Hilbert A –B bimodule is a rightHilbert B-module X which is also a left A-module via a nondegenerate homomor-phism of A into the algebra L(X) of bounded adjointable operators on X. (Theseare sometimes called A –B correspondences.) The objects in C* are C∗-algebras,and the morphisms from A to B are the isomorphism classes [X] of full right-Hilbert A –B bimodules. Every nondegenerate homomorphism ϕ : A → M(B)gives a right-Hilbert bimodule: view B as a right Hilbert B-module over itself with〈b1 , b2〉B := b∗1b2, and define the action of A by a · b := ϕ(a)b. We denote theisomorphism class of this bimodule by [ϕ]. In [2], it is shown that [ϕ] = [ψ] if andonly if there exists u ∈ UM(B) such that ψ = (Adu) ϕ, so we are not just addingmore morphisms to C*nd, we are also slightly changing the morphisms we alreadyhave.

If AXB and BYC are right Hilbert bimodules, then we define the compositionusing the internal tensor product: [Y ][X] := [X ⊗B Y ]. The identity morphism1A on A is [AAA] = [idA]. Now we can see why we have had to take isomorphismclasses of bimodules as our morphisms: the bimodule A⊗AX representing [X]1A =[X][idA] is only isomorphic to X. A similar subtlety arises when checking thatcomposition of morphisms is associative. The details are in [2, Proposition 2.4].In [2, Proposition 2.6], it is shown that the isomorphisms from A to B in C* arethe classes [X] in which X is an imprimitivity bimodule, so that X also carries aleft inner product A〈x , y〉 such that A〈x , y〉 · z = x · 〈y , z〉B . Similar results wereobtained independently by Landsman [17, 18] and by Schweizer [40], and a slightlymore general category in which the bimodules are not required to be full as rightHilbert modules was considered in [3].

Theorem 3.2 of [15] says that, for every nondegenerate homomorphism σ : A →M(B), the diagram

(3) Aα,r G[Z(A,α,ϕ)]

[σid]

Fix(A,α, ϕ)

[σ|]

B β,r G

[Z(B,β,ψ)] Fix(B, β, ψ)

commutes in C*, which means that

Z(A,α, ϕ)⊗Fix(A,α,ϕ) Fix(B, β, ψ) and (B β,r G)⊗Bβ,rG Z(B, β, ψ)

are isomorphic as right-Hilbert (Aα,r G) – Fix(B, β, ψ) bimodules. Thus Rieffel’sbimodules (or rather, the morphisms in C* which they determine) implement a nat-ural isomorphism between the functors RCP and Fix from (C0(T ), rt) ↓ C*actnd(G)to C*.

This naturality theorem certainly has interesting applications to nonabelianduality, where it gives naturality for the extension in [9] of Mansfield’s imprimi-tity theorem to closed subgroups (see [15, Theorem 6.2]). However, it is slightly

19

12 AN HUEF, RAEBURN, AND WILLIAMS

unsatisfactory: we were forced to change the target category from C*nd to C* be-cause the bimodules Z do not define morphisms in C*nd, but in the diagram (3) wehave not fully committed to the change. Our goal in [7] was to find versions ofthe same functors defined on a category built from C* — that is, ones in whichthe morphisms are implemented by bimodules — to establish that Rieffel’s Moritaequivalence gives a natural isomorphism between these functors, and to apply theresults to nonabelian duality. We will describe our progress in the next section.

5. Upgrading to C*

Proposition 3.3 of [2] says that for every locally compact group G, there isa category C*act(G) whose objects are dynamical systems (A,α) = (A,G, α) andwhose morphisms are obtained by adding actions to the morphisms of C*. Formally,if (A,α) and (B, β) are objects in C*(G) and AXB is a right-Hilbert bimodule, thenan action of G on a right-Hilbert bimodule X is a strongly continuous homomor-phism u of G into the linear isomorphisms of X such that

us(a · x · b) = αs(a) · us(x) · βs(b) and 〈us(x) , us(y)〉B = βs

(〈x , y〉B

),

and the morphisms in C*act(G) are isomorphism classes of pairs (X,u).Next we consider a free and proper action of G on a locally compact space

T and look for an analogue of the comma category for the system (C0(T ), rt).The objects are easy: to ensure that Fix is defined on objects, we need to insistthat every system (A,α) is equipped with a nondegenerate homomorphism ϕ :C0(T ) → M(A) which is rt –α equivariant. We choose to use the semi-commacategory C*act(G, (C0(T ), rt)) in which the objects are triples (A,α, ϕ), and themorphisms from (A,α, ϕ) to (B, β, ψ) are just the morphisms from (A,α) to (B, β)in C*act(G). In [7, Remark 2.4] we have discussed our reasons for adding the mapsϕ to our objects and then ignoring them in our morphisms, and the discussion belowof how we Fix morphisms should help convince sceptics that this is appropriate.

We know how to Fix objects in the semi-comma category C*act(G, (C0(T ), rt)),and we need to describe how to Fix a morphism [X,u] from (A,α, ϕ) to (B, β, ψ).We begin by factoring the morphism [X] in C* as the composition [K(X)XB][κA] ofthe isomorphism associated to the imprimitivity bimodule K(X)XB with the mor-phism coming from the nondegenerate homomorphism κA : A → M(K(X)) = L(X)describing the left action of A on X (see Proposition 2.27 of [3]). The action u of Gon X gives an action μ of G on K(X) such that μs(Θx,y) = Θus(x),us(y), and thenκA satisfies κA αs = μs κA. So the morphism [(A,α)(X,u)(B,β)] in C*act(G) fac-tors as [(K(X),μ)(X,u)(B,β)][κA]. Now κA is a morphism in (C0(G), rt) ↓ C*actnd(G)from (A,α, ϕ) to (K(X), μ, κA ϕ), and hence by Theorem 11 restricts to a mor-phism κA| from Fix(A,α, ϕ) to Fix(K(X), μ, κA ϕ). We want to define Fix sothat it is a functor, so our definition must satisfy

(4) Fix([X,u]) = Fix([(K(X),μ)(X,u)(B,β)]) Fix([κA]).

Since we don’t want to change the meaning of Fix on morphisms in C*nd, our strategyis to define Fix([κA]) := [κA|], figure out how to Fix imprimitivity bimodules, andthen use (4) to define Fix([X,u]).

So we suppose that (A,α, ϕ) and (B, β, ψ) are objects in the semi-comma cat-egory C*act(G, (C0(T ), rt)), and that [X,u] is an equivariant (A,α) – (B, β) im-primitivity bimodule. We emphasise that, because of our choice of morphisms inC*act(G, (C0(T ), rt)), we do not make any assumption relating the actions of ϕ

20

PROPER ACTIONS 13

and ψ on X. We let X := (x) : x ∈ X be the dual bimodule, and form thelinking algebra

L(X) :=

(A X

X B

),

as in the discussion following [35, Theorem 3.19]. Then

L(u) :=

(α u

(u) β

)and ϕL :=

(ϕ 00 ψ

)

define an action L(u) of G on L(X) and a nondegenerate homomorphism ϕL ofC0(T ) into M(L(X)) which intertwines rt and L(u). Then (L(X), L(u), ϕL) is anobject in C*act(G, (C0(T ), rt)), and (reverting to Rieffel’s notation to simplify theformulas) we can form L(X)L(u) := Fix(L(X), L(u), ϕL). It follows quite easilyfrom the construction of Fix in [15, §2] that the diagonal corners in L(X)L(u) areAα and Bβ, and we define Xu to be the upper right-hand corner, so that

L(X)L(u) =

(Aα Xu

∗ Bβ

);

with the actions and inner products coming from the operations in L(X)L(u), Xu

becomes an Aα –Bβ-imprimitivity bimodule (see [35, Proposition 3.1]). We nowdefine Fix([X,u]) := [Xu], and use (4) to define Fix in general, as described above.

With this definition, Theorem 3.3 of [7] says:

Theorem 13. Suppose that T is a free and proper right G-space. Then theassignments

(A,α, ϕ) → Fix(A,α, ϕ) and [X,u] → Fix([X,u])

form a functor Fix from the semi-comma category C*act(G, (C0(T ), rt)) to C*.

Proving that Fix preserves the composition of morphisms is surprisingly compli-cated, and involves several non-trivial steps. For example, we needed to show that if

(A,α)(X,u)(B,β) and (B,β)Y(C,γ) are imprimitivity bimodules implementing isomor-

phisms in C*act(G, (C0(T ), rt)), then (X⊗B Y )u⊗v is isomorphic to Xu⊗Bβ Y v asAα –Cγ imprimitivity bimodules.

It follows from [3, Theorem 3.7] that RCP is a functor from C*act(G, (C0(T ), rt))to C* which takes a morphism [X,u] to the class of the Combes bimodule [Xu,rG].We can now state the main naturality result, which is Theorem 3.5 of [7].

Theorem 14. Suppose that a locally compact group G acts freely and properlyon a locally compact space T . Then the Morita equivalences Z(A,α, ϕ) implement anatural isomorphism between the functors RCP and Fix from C*act(G, (C0(T ), rt))to C*.

The proof of Theorem 14 relies on factoring morphisms; then Theorem 3.2 of[15] gives the result for the nondegenerate homomorphism, and standard linkingalgebra techniques give the other half.

We saw in Example 10 that Rieffel’s Morita equivalence can be used to gener-alise Mansfield’s imprimitivity theorem to crossed products by homogeneous spaces,and we want to deduce from Theorem 14 that this imprimitivity theorem gives anatural isomorphism. To get the imprimitivity theorem in Example 10, we appliedRieffel’s Theorem 9 to a crossed product C δ G. So the naturality result we seekrelates the compositions of RCP and Fix with a crossed-product functor.

21

14 AN HUEF, RAEBURN, AND WILLIAMS

Suppose as in Example 10 that H is a closed subgroup of a locally compactgroup G. We know from Theorem 2.15 of [3] that there is a category C*coactn(G)whose objects are normal coactions (B, δ), and whose morphisms are isomorphismclasses of suitably equivariant right-Hilbert bimodules. We also know from Theo-rem 3.13 of [3] that there is a functor CP : C*coactn(G) → C*act(H), and addingthe canonical map jG makes CP into a functor with values in the comma cate-gory (C0(G), rt |H) ↓ C*act(H). We show in [7, Proposition 5.5] that there is afunctor RCPG/H which sends (B, δ) to the crossed product B δ,r (G/H) by thehomogeneous space G/H, and that this functor coincides with Fix CP. We saw in

Example 10 that Rieffel’s bimodules Z(Bδ G, δ|H , jG) implement a Morita equiv-alence between (Bδ G)δ|,r H and Bδ,r G/H. Write RCPH for the functor from

C*act(G) to C* sending (C, γ) → C γ|,r H. Then the general naturality resultabove gives the following theorem, which is Theorem 5.6 of [7].

Corollary 15. Let H be a closed subgroup of G. Then Rieffel’s Morita equiv-

alences Z(Gδ G, δ|H , jG) implement a natural isomorphism between the functorsRCPH CP and RCPG/H from C*coactn(G) to C*.

Corollary 15 extends Theorem 4.3 of [3] to non-normal subgroups, and extendsTheorem 6.2 of [15] to categories based on C* rather than ones based on C*nd.

6. Induction-in-stages and Fixing-in-stages

Rieffel’s theory of proper actions seems to be a powerful tool for studyingsystems in the comma or semi-comma category associated to a pair (T,G). Corol-lary 15 is, we think, an impressive first example. As another example, we discuss anapproach to induction-in-stages which works through the same general machinery,and which we carried out in [8].

The original purpose of an imprimitivity theorem was to provide a way ofrecognising induced representations (as in, for example, [20]), and Rieffel’s theory ofMorita equivalence for C∗-algebras was developed to put imprimitivity theorems ina C∗-algebraic context [36, 37]. One can reverse the process: a Morita equivalenceX between a crossed product CαG and another C∗-algebra B gives an inductionprocess X-Ind which takes a representation of B on H to a representation of Con X ⊗B H, and for which there is a ready-made imprimitivity theorem (see, forexample, [6, Proposition 2.1]). The situation is slightly less satisfactory when onehas a reduced crossed product, but one can still construct induced representationsand prove an imprimitivity theorem.

Mansfield’s imprimitivity theorem, as extended to homogeneous spaces in [9],

gives an induction process IndGG/H from Bδ,r(G/H) to BδG which comes with animprimitivity theorem. One then asks whether this induction process has the otherproperties which one would expect. For example, we ask whether we can induct-in-

stages: if we have subgroups H, K and L with H ⊂ K ⊂ L, is IndG/HG/K(Ind

G/KG/L π)

unitarily equivalent to IndG/HG/L π? If the subgroups are normal and amenable, then

the induction processes are those defined by Mansfield [22], and induction-in-stageswas established in [14, Theorem 3.1]. For non-normal subgroups, not much seemsto be known. There are clearly issues: for example, the subgroups H and K haveto be normal in L for the three induction processes to be defined.

We tackled this problem in [8] using our semi-comma category. Suppose that(T,G) is as usual, N is a closed normal subgroup of G, and (A,α, ϕ) is an object

22

PROPER ACTIONS 15

in C*act(G, (C0(T ), rt)). Then N also acts freely and properly on T , so we canform the fixed-point algebra FixN (A,α|N , ϕ). The quotient G/N has a naturalaction αG/N on Aα|N := FixN (A,α|N , ϕ), and the map ϕ induces a homomorphismϕN : C0(T/N) → M(Aα|N ) such that (Aα|N , αG/N , ϕN ) is an object in the semi-comma category C*act(G/N, (C0(T/N), rt)). We prove in [8] that FixN extendsto a functor

FixG/NN : C*act(G,C0(T ), rt) → C*act(G/N, (C0(T/N), rt)),

and that the functors FixG/N FixG/NN and FixG are naturally isomorphic (see [8,

Theorem 4.5]). The first difficulty in the proof is showing that the functor FixN hasan equivariant version: because the functor Fix is defined using the factorisationof morphisms, we have to track carefully through the constructions in [7] to makesure that they all respect the actions of G/N .

Applying this result on “fixing-in-stages” with (T,G) = (L/H,K/H), gives thefollowing version of induction-in-stages, which is Theorem 7.3 of [8].

Theorem 16. Suppose that δ is a normal coaction of G on B, and that H,K and L are closed subgroups of G such that H ⊂ K ⊂ L and both H and K arenormal in L. Then for every representation π of B δ,r (G/L), the representation

IndG/HG/K(Ind

G/KG/L π) is unitarily equivalent to Ind

G/HG/L π.

Obviously this is not the last word on the subject, and the normality hypotheseson subgroups are irritating. However, Mansfield’s induction process is notoriouslyhard to work with, and it seems remarkable that one can prove very much at allabout an induction process which is substantially more general than his. We thinkthat Rieffel’s theory of proper actions is proving to be a remarkably malleable andpowerful tool.

References

[1] S. Echterhoff, S. Kaliszewski and I. Raeburn, Crossed products by dual coactions of groupsand homogeneous spaces, J. Operator Theory 39 (1998), 151–176.

[2] S. Echterhoff, S. Kaliszewski, J. Quigg and I. Raeburn, Naturality and induced representa-tions, Bull. Austral. Math. Soc. 61 (2000), 415–438.

[3] , A categorical approach to imprimitivity theorems for C∗-dynamical systems, Mem.Amer. Math. Soc. 180 (2006), no. 850, viii+169.

[4] R. Exel, Morita-Rieffel equivalence and spectral theory for integrable automorphism groupsof C∗-algebras, J. Funct. Anal. 172 (2000), 404–465.

[5] P. Green, C∗-algebras of transformation groups with smooth orbit space, Pacific J. Math. 72

(1977), 71–97.[6] A. an Huef, S. Kaliszewski, I. Raeburn and D. P. Williams, Extension problems for represen-

tations of crossed-product C∗-algebras, J. Operator Theory 62 (2009), 171–198.[7] , Naturality of Rieffel’s Morita equivalence for proper actions, Algebr. Represent.

Theory, to appear.[8] , Fixed-point algebras for proper actions and crossed products by homogeneous spaces,

preprint; arXiv:math/0907.0681.[9] A. an Huef and I. Raeburn, Mansfield’s imprimitivity theorem for arbitrary closed subgroups,

Proc. Amer. Math. Soc. 132 (2004), 1153–1162.[10] A. an Huef, I. Raeburn and D. P. Williams, Proper actions on imprimitivity bimodules and

decompositions of Morita equivalences, J. Funct. Anal. 200 (2003), 401–428.[11] , A symmetric imprimitivity theorem for commuting proper actions, Canad. J. Math.

57 (2005), 983–1011.[12] S. Kaliszewski and J. Quigg, Imprimitivity for C∗-coactions of non-amenable groups, Math.

Proc. Camb. Phil. Soc. 123 (1998), 101–118.

23

16 AN HUEF, RAEBURN, AND WILLIAMS

[13] , Categorical Landstad duality for actions, Indiana Univ. Math. J. 58 (2009), 415–441.[14] S. Kaliszewski, J. Quigg and I. Raeburn, Duality of restriction and induction for C∗-

coactions, Trans. Amer. Math. Soc. 349 (1997), 2085–2113.[15] , Proper actions, fixed-point algebras and naturality in nonabelian duality, J. Funct.

Anal. 254 (2008), 2949–2968.[16] A. Kishimoto and H. Takai, Some remarks on C∗-dynamical systems with a compact abelian

group, Publ. Res. Inst. Math. Sci. 14 (1978), 383–397.

[17] N. P. Landsman, Bicategories of operator algebras and Poisson manifolds, MathematicalPhysics in Mathematics and Physics, Fields Inst. Commun., vol. 30, Amer. Math. Soc.,Providence, 2001, pages 271–286.

[18] , Quantized reduction as a tensor product, Quantization of Singular Symplectic Quo-tients, Progress in Math., vol. 198, Birkhuser, Basel, 2001, pages 137–180.

[19] M. B. Landstad, Duality theory for covariant systems, Trans. Amer. Math. Soc. 248 (1979),223–267.

[20] G. W. Mackey, Imprimitivity for representations of locally compact groups, Proc. Nat. Acad.Sci. USA 35 (1949), 537–545.

[21] S. Mac Lane, Categories for the Working Mathematician, Graduate Texts in Math, vol. 5,Springer, Berlin, 1971.

[22] K. Mansfield, Induced representations of crossed products by coactions, J. Funct. Anal. 97(1991), 112–161.

[23] D. Marelli and I. Raeburn, Proper actions which are not saturated, Proc. Amer. Math. Soc.137 (2009), 2273–2283.

[24] R. Meyer, Generalised fixed-point algebras and square-integrable group actions, J. Funct.Anal. 186 (2001), 167–195.

[25] J. W. Negrepontis, Duality in analysis from the point of view of triples, J. Algebra 15 (1971),228–253.

[26] D. Olesen and G. K. Pedersen, Applications of the Connes spectrum to C∗-dynamical systems,J. Funct. Anal. 30 (1978), 179–197.

[27] , Applications of the Connes spectrum to C∗-dynamical systems II, J. Funct. Anal.36 (1980), 18–32.

[28] G. K. Pedersen, Pullback and pushout constructions in C∗-algebra theory, J. Funct. Anal.167 (1999), 243–344.

[29] N. C. Phillips, Equivariant K-Theory and Freeness of Group Actions on C∗-Algebras, LectureNotes in Math., vol. 1274, Springer, Berlin, 1987.

[30] J. C. Quigg, Landstad duality for C∗-coactions, Math. Scand. 71 (1992), 277–294.[31] , Full and reduced C∗-coactions, Math. Proc. Camb. Phil. Soc. 116 (1994), 435–450.[32] J. C. Quigg and I. Raeburn, Induced C∗-algebras and Landstad duality for twisted coactions,

Trans. Amer. Math. Soc. 347 (1995), 2885–2915.[33] I. Raeburn, On crossed products and Takai duality, Proc. Edinburgh Math. Soc. 31 (1988),

321–330.[34] I. Raeburn and D. P. Williams, Pull-backs of C∗-algebras and crossed products by certain

diagonal actions, Trans. Amer. Math. Soc. 287 (1985), 755–777.[35] , Morita Equivalence and Continuous-Trace C∗-Algebras, Math. Surveys and Mono-

graphs, vol. 60, Amer. Math. Soc., Providence, 1998.[36] M. A. Rieffel, Induced representations of C∗-algebras, Adv. in Math. 13 (1974), 176–257.[37] , Unitary representations of group extensions; an algebraic approach to the theory

of Mackey and Blattner, Studies in Analysis, Adv. in Math. Suppl. Stud., vol. 4, AcademicPress, New York-London, 1979, pages 43–82.

[38] , Proper actions of groups on C∗-algebras, Mappings of Operator Algebras, Progressin Math., vol. 84, Birkhauser, Boston, 1990, pages 141–182.

[39] , Integrable and proper actions on C∗-algebras, and square-integrable representationsof groups, Expositiones Math. 22 (2004), 1–53.

[40] J. Schweizer, Crossed products by C∗-correspondences and Cuntz-Pimsner algebras, C∗-Algebras (Munster, 1999), Springer, Berlin, 2000, pages 203–226.

[41] J.-L. Vallin, C∗-algebres de Hopf et C∗-algebres de Kac, Proc. London Math. Soc. 50 (1985),131–174.

[42] D. P. Williams, Crossed Products of C∗-Algebras, Math. Surveys and Monographs, vol. 134,Amer. Math. Soc., Providence, 2007.

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[43] S. L. Woronowicz, Pseudospaces, pseudogroups and Pontriagin duality, Mathematical Prob-lems in Theoretical Physics, Lecture Notes in Phys., vol. 116, Springer, Berlin, 1980,pages 407–412.

[44] , Unbounded elements affiliated with C∗-algebras and noncompact quantum groups,Comm. Math. Phys. 136 (1991), 399–432.

School of Mathematics and Statistics, The University of New South Wales, Sydney,

NSW 2052, Australia

Current address: Department of Mathematics and Statistics, University of Otago, PO Box56, Dunedin 9054, New Zealand

E-mail address: astrid@maths.otago.ac.nz

School of Mathematics and Applied Statistics, University of Wollongong, NSW

2522, Australia

E-mail address: raeburn@uow.edu.au

Department of Mathematics, Dartmouth College, Hanover, NH 03755, USA

E-mail address: dana.williams@dartmouth.edu

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Proceedings of Symposia in Pure Mathematics

Twists of K-theory and TMF

Matthew Ando, Andrew J. Blumberg, and David Gepner

Abstract. We explore an approach to twisted generalized cohomology from

the point of view of stable homotopy theory and ∞-category theory providedby [ABGHR]. We explain the relationship to the twisted K-theory providedby Fredholm bundles. We show how this approach allows us to twist ellip-tic cohomology by degree four classes, and more generally by maps to the

four-stage Postnikov system BO〈0 . . . 4〉. We also discuss Poincare duality and

umkehr maps in this setting.

Contents

1. Introduction2. Classical Examples of Twisted Generalized Cohomology3. Bundles of Module Spectra4. The Generalized Thom Spectrum5. Twisted Generalized Cohomology6. Multiplicative Orientations and Comparison of Thom Spectra7. Application: K(Z, 3), Twisted K-theory, and the Spinc Orientation8. Application: Degree-four Cohomology and Twisted Elliptic Cohomology9. Application: Poincare Duality and Twisted Umkehr Maps10. Motivation: D-brane Charges in K-theory11. An Elliptic Cohomology Analogue12. Twists of Equivariant Elliptic CohomologyReferences

1. Introduction

In [ABGHR], we and our co-authors generalize the classical notion of Thomspectrum. Let R be an A∞ ring spectrum: it has a space of units GL1R whichdeloops to give a classifying space BGL1R. To a space X and a map

ξ : X → BGL1R

M. Ando was supported in part by NSF grant DMS-0705233.A. J. Blumberg was supported in part by NSF grant DMS-0906105.

c©0000 (copyright holder)

1

Proceedings of Symposia in Pure MathematicsVolume 81, 2010

c©2010 American Mathematical Society

27

27303240424552545658595961

2 ANDO, BLUMBERG, AND GEPNER

we associate an R-module Thom spectrum Xξ. Letting S denote the sphere spec-trum, one finds that BGL1S is the classifying space for stable spherical fibrations ofvirtual rank 0, and Xξ is equivalent to the classical Thom spectrum of the sphericalfibration classified by ξ (as in, for example, [LMSM86]).

We remark in the introduction to [ABGHR] that BGL1R classifies the twistsof R-theory. More precisely, we define the ξ-twisted R-homology of X to be

Rk(X)ξdef= π0R-mod(ΣkR,Xξ) ∼= πkX

ξ

and the ξ-twisted cohomology to be

Rk(X)ξdef= π0R-mod(Xξ,ΣkR),

where here Σk denotes the k-fold suspension, or equivalently smashing with Sk.In this paper, we expand on that remark, explaining how this definition gener-

alizes both singular cohomology with local coefficients and the twists of K-theorystudied by [DK70, Ros89, AS04]. The key maneuver is to focus on the ∞-categorical approach to Thom spectra developed in [ABGHR] (where by ∞-categories we mean the quasicategories of [Joy02, HTT]). We show that the∞-category LineR of R-modules L which admit a weak equivalence R L is amodel for BGL1R: we have a weak equivalence of spaces

|LineR| BGL1R.

One appeal of LineR is that, by construction, it classifies what one might call“homotopy local systems” of free rank-one R-modules. This flexible notion gener-alizes both classical local coefficient systems and bundles of spaces (such as bundlesof Fredholm operators). As one might expect, our work is closely related to theparametrized spectra of May and Sigurdsson [MS06]; we discuss the relationshipfurther in Section 3.4.

As applications of our approach to twisted generalized cohomology, we explainhow the twisting of K-theory by degree three cohomology is related to the Spinc

orientation of Atiyah-Bott-Shapiro. Similarly, recall that there is a map (uniqueup to homotopy)

BSpinλ−→ K(Z, 4)

whose restriction toBSU is the second Chern class. The fiber of λ is calledBString,and if V is a Spin vector bundle on X, then a String structure on V is a trivial-ization of λ(V ); that is, a map g in the diagram

BString

π

XV

g

BSpin,

together with a homotopy πg ⇒ V.The work of Ando, Hopkins, and Rezk [AHR] constructs an E∞ String ori-

entation of tmf , the spectrum of topological modular forms. (The discussion inthis paper applies equally to the connective spectrum tmf and to the periodicspectrum TMF .) Associating to a vector bundle its underlying spherical fibrationgives a map BSpin → BGL1S, and associated to the unit S → tmf is a mapBGL1S → BGL1tmf. Composing these, we have a map

k : BSpin → BGL1tmf.

28

TWISTS OF K-THEORY AND TMF 3

We show that the E∞ String orientation of tmf of [AHR] implies the following.

Theorem 1.1.

(1) tmf admits twists by degree-four integral cohomology. More precisely,there is a map h : K(Z, 4) → BGL1tmf making the diagram

BSpink−−−−→ BGL1tmf

λ

=

K(Z, 4)h−−−−→ BGL1tmf

commute up to homotopy. Thus given a map z : X → K(Z, 4) (represent-ing a class in H4(X,Z)) we can define

tmf∗(X)zdef= tmf∗(X)hz.

(2) If V is a Spin-bundle over X, classified by

XV−→ BSpin,

then a homotopy hλ(V ) ⇒ k(V ) determines an isomorphism

tmf∗(XV ) ∼= tmf∗(X)λ(V ).

of modules over tmf∗(X).

Theorem 1.1 is well-known to the experts (e.g. Hopkins, Lurie, Rezk, andStrickland). As the reader will see, with the approach to twisted cohomology pre-sented here, it is an immediate consequence of the String orientation.

We also explain how twisted generalized cohomology is related to Poincareduality. We briefly describe some work in preparation, concerning twisted umkehrmaps in generalized cohomology. As special cases, we recover the twisted K-theoryumkehr map constructed by Carey and Wang, and we construct an umkehr mapin twisted elliptic cohomology. As we explain, our interest in the twisted ellipticcohomology umkehr map arose from conversations with Hisham Sati.

Finally, we report two applications of twisted equivariant elliptic cohomology:we recall a result (due independently to the first author [And00] and Jacob Lurie),relating twisted equivariant elliptic cohomology to representations of loop groups,and we explain work of the first author and John Greenlees, relating twisted equi-variant elliptic cohomology to the equivariant sigma orientation.

Remark 1.2. If E is a cohomology theory and X is a space, then E∗(X) willrefer to the unreduced cohomology. If Z is a spectrum, then we write E∗(Z) forthe spectrum cohomology. We write Σ∞

+ for the functor

Σ∞+ : (spaces)

disjoint basepoint−−−−−−−−−−−→ (pointed spaces)Σ∞−−→ (spectra),

so we have by definition

E∗(X) ∼= E∗(Σ∞+ X),

while the reduced cohomology is

E∗(X) ∼= E∗(Σ∞X).

If V is a vector bundle over X of rank r, then we write XV for its Thom spectrum:this is equivalent to the suspension spectrum of the Thom space, so E∗(XV ) is the

29

4 ANDO, BLUMBERG, AND GEPNER

reduced cohomology of the Thom space. Thus the Thom isomorphism, if it exists,takes the form

E∗(X) ∼= E∗+r(XV ).

Remark 1.3. The ∞-category LineR is not the largest category we could use toconstruct twists of R-theory. If R is an E∞ ring spectrum, i.e. a commutative ringspectrum, then we could consider the ∞-category Pic(R), consisting of invertible R-modules: R-modules L for which there exists an R-module M such that L∧RM R.

AcknowledgmentsThis paper is the basis for a talk given by the first author at the CBMS confer-

ence on C∗-algebras, topology, and physics at Texas Christian University in May2009. We thank the organizers, Bob Doran and Greg Friedman, for the opportu-nity. It is a pleasure to acknowledge stimulating conversations with Alan Carey,Dan Freed, and Hisham Sati which directly influenced this write-up.

2. Classical Examples of Twisted Generalized Cohomology

2.1. Geometric models for twisted K-theory. LetH be a complex Hilbertspace, and let F be its space of Fredholm operators. Then F is a representing spacefor K-theory: Atiyah showed [Ati69] that

K(X) ∼= π0 map(X,F) = π0Γ(X ×F → X).

Atiyah and Segal [AS04] develop the following approach to twisted K-theory. Theunitary group U = U(H) of H acts on the space F of Fredholm operators byconjugation. Associated to a principal PU -bundle P → X, then, we can form thebundle

ξ = P ×PU F → X

with fiber F . They define the P -twisted K-theory of X to be

K(X)P = π0Γ(ξ → X).

Thus one twists K(X) by PU -bundles over X; isomorphism classes of these areclassified by π0 map(X,BPU); as BPU is a model for K(Z, 3), we have haveπ0 map(X,BPU) ∼= H3(X;Z).

We warn the reader that this summary neglects important and delicate issueswhich Atiyah and Segal address with care, for example concerning the choice oftopology on U and PU .

Another approach to twisted K-theory passes through algebraic K-theory;again we neglect important operator-theoretic matters, referring the reader to[Ros89, BM00, BCMMS] for details. Conjugation induces an action of PUon the algebra K of compact operators on H. Thus from the PU -bundle P → X wecan form the bundle P ×PU K. Let A = Γ(P ×PU K → X). This is a (non-unital)C∗-algebra, and

K(X)P ∼= K(A).

If P is the trivial bundle, then A ∼= map(X,K), and we have isomorphisms

K(A) ∼= K(map(X,C)) ∼= K(X).

Both of these approaches to twisted K-theory are based on the idea that from aPU -bundle we can build a bundle of copies of the representing space for K-theory,

30

TWISTS OF K-THEORY AND TMF 5

and both have had a number of successes. They demand a good deal of informationabout K-theory, and they exploit features of models of K-theory which may notbe available in other cohomology theories.

May and Sigurdsson show how to implement the construction of Atiyah andSegal in the setting of their theory of parametrized stable homotopy theory [MS06,§22]. Specifically, they give a construction of certain twisted cohomology theoriesassociated to parametrized spectra, and explain how the Atiyah-Segal definition fitsinto their framework. However, the approach of May and Sigurdsson also takes ad-vantage of good features of known models for K-theory which may not be availablein other cohomology theories.

In this paper we explain another way to locate twisted K-theory in stable ho-motopy theory. Our constructions continue to demand a good deal of K-theory, forexample that it be an A∞ or E∞ ring spectrum, but many generalized cohomologytheories E satisfy our demands, and so our approach works in those cases as well.

Our approach incorporates and generalizes the construction of Thom spectraof vector bundles, and so clarifies standard results concerning twisted E-theory,such as the relationship to the Thom isomorphism and Poincare duality. It alsogeneralizes the classical notion of (co)homology with local coefficients, as we nowexplain.

2.2. Cohomology with local coefficients. Let X be a space, and letΠ≤1(X) be its fundamental groupoid. We recall that a local coefficient systemon X is a functor1

A : Π≤1(X) → (Abelian groups).

Given a local system A on X, we can form the twisted singular homologyH∗(X; A) and cohomology H∗(X; A).

Example 2.1. For example, if π : E → X is a Serre fibration, then associatingto a point p ∈ X the fiber Fp = π−1(p) gives rise to a representation

F• : Π≤1(X) −→ Ho(spaces).

(Here Ho denotes the homotopy category obtained by inverting the weak equiva-lences.) Applying singular cohomology in degree r produces the local coefficientsystem Hr(F•).

Example 2.2. If V is a vector bundle overX of rank r, then taking the fiberwiseone-point compactification V + provides a Serre fibration, and so we have the localcoefficient system

(2.3) Hr(V +• );Z

whose value at p ∈ X is the cohomology group Hr(V +p ;Z).

From the Serre spectral sequence it follows that there is an isomorphism

H∗(X; Hr(V +• )) ∼= H∗+r(XV ;Z)

between the twisted cohomology of X with coefficients in the system Hr(V +• )

and the cohomology of the Thom spectrum of V .

1Since the fundamental groupoid is a groupoid, it is equivalent to consider covariant orcontravariant functors.

31

6 ANDO, BLUMBERG, AND GEPNER

An orientation of V is a trivialization of the local system (2.3), that is, anisomorphism of functors

Hr(V +• ) ∼= Z

(where Z denotes the evident constant functor). It follows immediately that anorientation of V determines a Thom isomorphism

H∗(X;Z) ∼= H∗+r(XV ;Z).

3. Bundles of Module Spectra

3.1. The problem. Let X be a space. We seek a notion of local system ofspectra ξ on X, generalizing the bundles of Fredholm operators in §2.1 and thelocal systems of §2.2. In particular, if E → X is a Serre fibration as in Example2.1, then the classical local system Hr(F•) should arise from a bundle of spectraF• ∧HZ by passing to homotopy groups.

From this example, we quickly see that while it is reasonable to ask ξ to as-sociate to each point p ∈ X a spectrum ξp, it is too much to expect to associateto a path γ : I → X from p to q an isomorphism of fibers; instead, we expect ahomotopy equivalence

ξγ : ξp → ξq.

Moreover, an (endpoint-preserving) homotopy of paths H : γ → γ′ should give riseto a path

ξH : ξγ → ξγ′

in the space of homotopy equivalences from ξp to ξq.These homotopy coherence issues quickly lead one to consider representations

of not merely the fundamental groupoid Π≤1(X), but the whole fundamental ∞-groupoid Π≤∞(X), that is, the singular complex SingX. Quasicategories make itboth natural and inevitable to consider such representations.

3.2. ∞-categories from spaces and from simplicial model categories.Recall that a quasicategory is a simplicial set which has fillings for all inner horns.Thus one source of quasicategories is spaces. If X is a space, then its singularcomplex SingX is a Kan complex: it has fillings for all horns. From the pointof view of quasicategories, where 1-simplices correspond to morphisms, this meansthat all morphisms are invertible up to (coherent higher) homotopy. Thus Kancomplexes may be identified with “∞-groupoids”.

We also recall (from [HTT, Appendix A and 1.1.5.9]) how simplicial modelcategories give rise to quasicategories. This procedure is an important source ofquasicategories, and it provides intuition about how quasicategories encode homo-topy theory.

If M is a simplicial model category, then we can define M to be the fullsubcategory consisting of cofibrant and fibrant objects. The simplicial nerve ofM ,

C = NM ,

is the quasicategory associated to M .By construction, C is the simplicial set in which

(1) the vertices C0 are cofibrant-fibrant objects of M ;

32

TWISTS OF K-THEORY AND TMF 7

(2) C1 consists of maps

L → M

between cofibrant-fibrant objects;(3) C2 consists of diagrams (not necessarily commutative)

Lf

h

M

g

N,

together with a homotopy from gf to h in the mapping space (simplicialset) M (L,M);

and so forth.In particular, in C the equivalences correspond to weak equivalences in M ,

that is, homotopy equivalences. Thus we may sometimes refer to the equivalencesin a quasicategory as weak equivalences or homotopy equivalences.

A simplicial model category M has an associated homotopy category hoM ,and an ∞-category C has a homotopy category hoC . As one would expect, thereis an equivalence of categories (enriched over the homotopy category of spaces)

hoM hoNM .

By analogy to the model category situation, if C is a quasicategory and hoC D ,then we shall say that C is a “model for D”.

3.3. The ∞-category of A-modules. Let S be a symmetric monoidal ∞-category of spectra. Lurie constructs such an ∞-category from scratch ([DAGI]introduces an ∞-category of spectra, which is shown to be monoidal in [DAGII],and symmetric monoidal in [DAGIII]). Lurie shows that his ∞-category is equiv-alent to the symmetric monoidal ∞-category arising from the symmetric spectraof [HSS00], and so by [MMSS01] it is equivalent to the symmetric monoidal ∞-categories of spectra arising from various classical symmetric monoidal simplicialmodel categories of spectra. Let S be the sphere spectrum.

Definition 3.1. An S-algebra is a monoid (strictly speaking, an algebra,since the relevant monoidal structure is not given by the cartesian product) inS . We write Alg(S) for the ∞-category of S-algebras, and CommAlg(S) for the∞-category of commutative S-algebras.

Using [DAGII, DAGIII] and [MMSS01] as above, one learns that the sym-metric monoidal structure on S is such that Alg(S) is a model for A∞ ring spec-tra, and CommAlg(S) is a model for E∞ ring spectra, so the reader is free touse his or her favorite method to produce ∞-categories equivalent to Alg(S) andCommAlg(S).

Definition 3.2. If A is an S-algebra, we let ModA be the ∞-category of A-modules.

Example 3.3. An S-module is just a spectrum, and so ModS is the∞-categoryof spectra.

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8 ANDO, BLUMBERG, AND GEPNER

3.4. Bundles of spaces and spectra. The purpose of this section is to intro-duce the ∞-categorical model of parametrized spectra we work with in this paperand compare it to the May-Sigurdsson notion of parametrized spectra [MS06].

We begin by reviewing some models for the ∞-category of spaces over a cofi-brant topological space X. On the one hand, we have the topological model cat-egory T /X of spaces over X, obtained from the topological model category ofspaces by forming the slice category; i.e., the weak equivalences and fibrations aredetermined by the forgetful functor to spaces. We will refer to this model struc-ture as the “standard” model structure on T /X. This is Quillen equivalent to thecorresponding simplicial model category structure on simplicial sets over SingX,which in turn is Quillen equivalent to the simplicial model category of simplicialpresheaves on the simplicial category C[SingX] (with, say, the projective modelstructure) [HTT, §2.2.1.2]. Here C denotes the left adjoint to the simplicial nerve;it associates a simplicial category to a simplicial set [HTT, §1.1.5].

Remark 3.4. The Quillen equivalence between simplicial presheaves and para-metrized spaces depends on the fact that the base is an ∞-groupoid (Kan complex)as opposed to an ∞-category; there is a more general theory of “right fibrations”(and, dually, “left fibrations”), but over a Kan complex a right fibration is a leftfibration (and conversely) and therefore a Kan fibration.

On the level of ∞-categories, this yields an equivalence

St : NSetΔ/ SingX −→ Fun(SingXop,NSetΔ);

the map, called the straightening functor, rigidifies a fibration over SingX into apresheaf of ∞-groupoids on SingX whose value at the point x is equivalent to thefiber over x [HTT, §3.2.1].

A distinct benefit of the presheaf approach is a particularly straightforwardtreatment of the base-change adjunctions. Given a map of spaces f : Y → X, wemay restrict a presheaf of ∞-groupoids F on SingX to a presheaf of ∞-groupoidsf∗F on Sing Y . This gives a functor, on the level of ∞-categories, from spacesover X to spaces over Y , such that the fiber of f∗F over the point y of Y isequivalent to the fiber of F over f(y). Moreover, f∗ admits both a left adjointf! and a right adjoint f∗, given by left and right Kan extension along the mapSing Y op → SingXop, respectively. Note that this is left and right Kan extension inthe ∞-categorical sense, which amounts to homotopy left and right Kan extensionon the level of simplicial categories or model categories. On the level of modelcategories of presheaves, there is an additional subtlety:

f∗ : Fun(C[SingXop], SetΔ) −→ Fun(C[Sing Y op], SetΔ)

is a right Quillen functor for the projective model structure, with (derived) leftadjoint f!, and a left Quillen functor for the injectivemodel structure, with (derived)right adjoint f∗, on the above categories of (simplicial) presheaves. Of course theidentity adjunction gives a Quillen equivalence between these two model structures,but nevertheless one is forced to switch back and forth between projective andinjective model structures if one wishes to simultaneously consider both base-changeadjunctions.

Now we may stabilize either of the equivalent ∞-categories

NSetΔ/ SingX Fun(SingXop,NSetΔ)

34

TWISTS OF K-THEORY AND TMF 9

by forming the ∞-category of spectrum objects in C∗; here C∗ denotes the ∞-category of pointed objects in C . If C is an ∞-category with finite limits, then sois C∗, and Stab(C ) is defined as the inverse limit of the tower

Stab(C ) = lim· · · Ω−→ C∗Ω−→ C∗

associated to the loops endomorphism Ω: C∗ → C∗ of C∗. In other words, a spec-trum object in an ∞-category C (with finite limits) is a sequence of pointed objectsA = A0, A1, . . . together with equivalences An ΩAn+1 for each natural numbern.

Thus, our category of parametrized spectra is the stabilizationStab(N(SetΔ/ SingX)). For our purposes, it turns out to be much moreconvenient to use the presheaf model; there is an equivalence of ∞-categories

Stab(N(SetΔ/ SingX)) Fun(SingXop,S ).

Note that a functor F : SingX → S associates to each point x of X a spectrum Fx,to each path x0 → x1 in X a map of spectra (necessarily a homotopy equivalence)Fx0

→ Fx1, and so on for higher-dimensional simplices of X.

Given a presentable ∞-category C , the stabilization Stab(C ) is itself pre-sentable, and the functor Ω∞ : Stab(C ) → C admits a left adjoint Σ∞ : C →Stab(C ) [DAGI, Proposition 15.4]. Just as Ω∞ is natural in presentable ∞-categories and right adjoint functors, dually, Σ∞ is natural in presentable ∞-categories and left adjoint functors [DAGI, Corollary 15.5]. In particular, givena map of spaces f : Y → X, the adjoint pairs (f!, f

∗) and (f∗, f∗) defined aboveextend to the stabilizations, yielding a restriction functor

f∗ : Fun(SingXop,S ) −→ Fun(Sing Y op,S )

which admits a left adjoint f! and a right adjoint f∗, again given by left and rightKan extension, respectively.

We can also formally stabilize suitable model categories, using Hovey’s workon spectra in general model categories [Hov01]. Specifically, given a left propercellular model category C and an endofunctor of C , Hovey constructs a cellularmodel category SpNC of spectra. When C is additionally a simplicial symmetricmonoidal model category, the endofunctor given by the tensor with S1 yields asimplicial symmetric monoidal model category of symmetric spectra SpΣC (as well

as a simplicial model category SpNC of prespectra). These models of the stabiliza-tion are functorial in left Quillen functors which are suitably compatible with therespective endofunctors (see [Hov01, 5.2]).

In order to compare our model of parametrized spectra over X to the May-Sigurdsson model, we use the following consistency result.

Proposition 3.5. Let C be a left proper cellular simplicial model category andwrite SpNC for the cellular simplicial model category of spectra generated by thetensor with S1. Then there is an equivalence of ∞-categories

N(SpNC ) Stab(NC ).

Proof. The functors Evn : SpNC → C , which associate to a spectrum its

nth-space An, induce a functor (of ∞-categories)

f : N(SpNC ) → lim· · · Ω−→ NC ∗

Ω−→ NC ∗ Stab(NC )

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10 ANDO, BLUMBERG, AND GEPNER

which is evidently essentially surjective. To see that it is fully faithful, it sufficesto check that for cofibrant-fibrant spectrum objects A and B in SpNC , there is anequivalence of mapping spaces

map(A,B) holim· · · Ω−→ map(A1, B1)Ω−→ map(A0, B0),

where Ω: map(An+1, Bn+1) → map(An, Bn) sends An+1 → Bn+1 to

An ΩAn+1 → ΩBn+1 Bn.

Since any cofibrant A is a retract of a cellular object, inductively we can reduce tothe case in which A = FmX, i.e., the shifted suspension spectrum on a cofibrantobject X of C∗. Then map(A,B) map(X,Bm) by adjunction. The latter is inturn equivalent to map(Σn−mX,Bn), where we interpret Σ

n−mX = ∗ for m > n, inwhich case the homotopy limit is equivalent to that of the homotopically constant(above degree n) tower whose nth term is map(Σn−mX,Bn).

We now recall the May-Sigurdsson setup. Given a space X, let (T /X)∗ denotethe category of spaces over and under X (ex-spaces). Although this category has amodel structure induced by the standard model structure on T /X, one of the keyinsights of May and Sigurdsson is that for the purposes of parametrized homotopytheory it is essential to work with a variant they call the qf -model structure [MS06,6.2.6]. This model structure is Quillen equivalent to the standard model structureon ex-spaces [MS06, 6.2.7]; however, its cofiber and fiber sequences are compatiblewith classical notions of cofibration and fibration (described in terms of extensionand lifting properties).

May and Sigurdsson then construct a stable model structure on the categoriesSX of orthogonal spectra in (T /X)∗ [MS06, 12.3.10]. This model structure isbased on the qf -model structure on ex-spaces, leveraging the diagrammatic view-point of [MMSS01, MM02]. Similarly, they construct a stable model structureon the category PX of prespectra in (T /X)∗; the forgetful functor SX → PX isa Quillen equivalence [MS06, 12.3.10].

Using [MS06, 12.3.14], we see that after passing to ∞-categories the category

PX is in turn equivalent to the category SpN(T /X)∗; the formal stabilization ofthe qf -model structure on (T /X)∗ with respect to the fiberwise smash with S1.Using Proposition 3.5 and the fact that the qf -model structure is Quillen equivalentto the standard model structure, we obtain equivalences of ∞-categories

N(SX) → N(PX) → N(SpN(T /X)∗)

→ Stab(N(T /X)) → Stab(N(SetΔ/ SingX)) → Fun(SingXop,S ).

Thus we obtain the following comparison theorem.

Theorem 3.6. There is an equivalence of ∞-categories between the simplicialnerve of the May-Sigurdsson category of parametrized orthogonal spectra N(SX)

and the ∞-category Fun(SingXop,S ) of presheaves on X with values in spectra.

Furthermore, the derived base-change functors we construct via the stabiliza-tion of the presheaves agree with the derived base-change functors constructed byMay and Sigurdsson. To see this, observe that it suffices to check this for f∗; com-patibility then follows formally for the adjoints f∗ and f!. Moreover, since f∗ on thecategories of spectra is obtained as the suspension of f∗ on spaces, we can reduce to

36

TWISTS OF K-THEORY AND TMF 11

checking that the right derived functor of f∗ : (T /X)∗ → (T /Y )∗ in the qf -modelstructure is compatible with the right derived functor of

f∗ : Fun(C(SingXop), SetΔ) → Fun(C(Sing Y op), SetΔ)

in the projective model structure. By the work of [MS06, §9.3], it suffices tocheck the compatibility for f∗ in the q-model structure. Since both versions of f∗

that arise here are Quillen right adjoints, this amounts to the verification that thediagram

Fun(C(SingXop), SetΔ)

Un

f∗ Fun(C(Sing Y op), SetΔ)

Un

SetΔ/Xf∗

SetΔ/Y

commutes when applied to fibrant objects, where here Un denotes the unstraight-ening functor (which is the right adjoint of the Quillen equivalence). Finally, thisfollows from [HTT, 2.2.1.1].

3.5. Bundles of A-modules and A-lines. If X is a space, let SingX be itssingular complex. The work of the previous section justifies the following definition.

Definition 3.7. A bundle or homotopy local system of A-modules over X is amap of simplicial sets

f : SingX → ModA.

Similarly if Y is any ∞-groupoid, then a bundle of A-modules over Y is just a mapof simplicial sets

f : Y → ModA.

Thus f assigns

(0) to each point p ∈ X an A-module f(p);(1) to each path γ from p to q a map of A-modules

(3.8) f(γ) : f(p) → f(q);

(2) to each 2-simplex σ : Δ2 → X, say

p

σ01

σ02

qσ12

r,

a path f(σ) in ModA(f(p), f(r)) from f(σ12)f(σ01) to f(σ02);

and so forth.Recall [HTT, 1.2.7.3] that if Y is a simplicial set and C is an ∞-category, then

the simplicial mapping space CY is the ∞-category Fun(Y, C) of functors from Y toC.

Definition 3.9. The ∞-category of bundles of A-modules over X is the sim-plicial mapping space

ModXAdef= Fun(SingX,ModA).

37

12 ANDO, BLUMBERG, AND GEPNER

Remark 3.10. We have not set up the framework necessary to work directlywith the bundle of A-modules associated to f : SingX → ModA (although seeTheorem 3.6). Nonetheless, the notation of bundle and pullback is compelling, andso we write M for the identity map

ModA → ModA,

and if f : SingX → ModA is a map of ∞-categories, then we may write f∗M as asynonym for f , when we want to emphasize its bundle aspect.

Recall that SingX is a Kan complex or ∞-groupoid: it satisfies the extensioncondition for all horns. Viewing an ∞-category as a model for a homotopy theory,an∞-groupoid models a homotopy theory in which all the morphisms are homotopyequivalences.

In particular, the map f(γ) in (3.8) is necessarily an equivalence: the A-modulesf(p) will vary through weak equivalences as p varies over a path component of X.We shall be particularly interested in the case that these fibers are free rank-oneA-modules.

Definition 3.11. An A-line is an A-module L which admits a weak equivalence

L−→ A.

The ∞-category LineA is the maximal ∞-groupoid in ModA generated by the A-lines. We write j for the inclusion

j : LineA → ModA

and Ldef= j∗M for the tautological bundle of A-lines over LineA.

By construction LineA is a Kan complex, and we regard it as the classifyingspace for bundles of A-lines. If X is a space, then a map

f : SingX → LineA.

assigns

(0) to each point p ∈ X an A-line f(p);(1) to each path γ from p to q an equivalence map of A-lines

f(γ) : f(p) f(q);

(2) to each 2-simplex σ : Δ2 → X, say

p

σ01

σ02

qσ12

r,

a path f(σ) in LineA(f(p), f(r)) from f(σ12)f(σ01) to f(σ02);

and so forth.

Definition 3.12. The simplicial mapping space LineXA = Fun(SingX,LineA)is an ∞-category (in fact, a Kan complex); we call it the the ∞-category or spaceof A-lines over X.

We develop twisted A-theory starting from LineA in §5. Before doing so, webriefly discuss other aspects of the ∞-category LineA.

38

TWISTS OF K-THEORY AND TMF 13

3.6. LineA and GL1A. By construction, LineA is connected, and so equivalentto the maximal ∞-groupoid BAut(A) on the single A-module A. As we discussin [ABGHR, §6], it is an important point that the space of morphisms Aut(A) =LineA(A,A) is not a group, or even a monoid, but instead merely a group-like A∞space.

Nevertheless, LineA is not only the classifying space for bundles of A-lines, butit is a delooping of Aut(A). To see this, let Triv(A) be the ∞-category of A-lines

L, equipped with an equivalence L−→ A. Then [ABGHR, Prop. 7.38] Triv(A) is

contractible, and the map

Triv(A) → LineA

is a Kan fibration, with fiber Aut(A).Classical infinite loop space theory provides another model for homotopy type

Aut(A). Namely, let A be an A∞ ring spectrum in the sense of [LMSM86]: soπ0Ω

∞A is a ring. Let GL1A be the pull-back in the diagram

GL1A −−−−→ Ω∞A

(π0Ω∞A)× −−−−→ π0Ω

∞A.

Then GL1A is a group-like A∞ space: π0GL1A is a group. We show that

GL1A |Aut(A)|.Since the geometric realization of a Kan fibration is a Serre fibration [Qui68], thefibration

Aut(A) → Triv(A) → LineA

gives rise to a fibration

(3.13) GL1A |Aut(A)| → |Triv(A)| ∗ → |LineA|.Thus |LineA| provides a model for the delooping BGL1A. It has the virtue that wehave already given a precise description of the vertices of the simplicial mappingspace

LineXA = map(SingX,LineA) map(X, |LineA|).

Example 3.14. If S is the sphere spectrum, then Ω∞S is the space QS0 =Ω∞Σ∞S0, and

GL1S = Q±1S0,

i.e., the unit components. The space BGL1S |LineS | is the classifying space forstable spherical fibrations of virtual rank 0. It follows the space of S-lines over Xis homotopy equivalent to the space of spherical fibration of virtual rank 0.

Example 3.15. The classical J-homomorphism is a map

J : O → GL1S,

which deloops to give a map

BJ : BO → BGL1S.

One sees that this is the map which takes a virtual vector bundle of rank 0 to itsassociated stable spherical fibration; we may regard this as associating to a vectorbundle its bundle of S-lines.

39

14 ANDO, BLUMBERG, AND GEPNER

Example 3.16. If A is an S-algebra, then the unit of A induces a map

BGL1S → BGL1A.

In our setting, this map arises from the map of ∞-categories

ModS → ModA

given by M → M ⊗S A = M ∧S A, which restricts to give a map of ∞-categories

LineS → LineA.

Example 3.17 ([MQRT77]). LetHZ be the integral Eilenberg-MacLane spec-trum. Then Ω∞HZ K(Z, 0) Z, and so GL1HZ ±1 Z/2, andBGL1HZ BZ/2 K(Z/2, 1).

Remark 3.18. If A = K, the spectrum representing complex K-theory, thenAut(K) has the homotopy type of the space of K-module equivalences K → K.Atiyah and Segal [AS04] build twists of K-theory from PU -bundles. They remarkthat one can more generally build twists of K-theory from G-bundles, where Gis the group of strict K-module automorphisms of K-theory. Our space Aut(K)generalizes this idea.

Remark 3.19. As we explain in [ABGHR, §6], for many algebras A (includingthe sphere S), the group Autstrict(A) of strict A-module automorphisms ofA cannotprovide a sufficiently rich theory of bundles of A-modules. For example, Lewis’sTheorem [Lew91] implies that there is no model for the sphere spectrum S suchthat the classifying space BAutstrict(S) classifies stable spherical fibrations. (Seealso [MS06, §22.2] for discussion of this issue.)

4. The Generalized Thom Spectrum

Let A be an S-algebra, let X be a space, and let f be a bundle of A-lines overX, that is, a map of simplicial sets

f : SingX → LineA.

Although the ∞-category LineA is not cocomplete (it doesn’t even have sums),the ∞-category ModA of A-modules is complete and cocomplete. This allows us in[ABGHR] to make the following definition.

Definition 4.1. The Thom spectrum of f is the colimit

Xf def= colim

(SingX

f−→ LineAj−→ ModA

).

Equivalently, Xf is the left Kan extension Lπjf in the diagram

Xf

π

LineAj

ModA.

∗Lp(fj)

The colimit and left Kan extension here are ∞-categorical colimits: they aregeneralizations of the notion of homotopy colimit and homotopy left Kan extension.It is an important achievement of ∞-category theory to give a sensible definitionof these colimits.

40

TWISTS OF K-THEORY AND TMF 15

Let AX : X → ∗ → LineA be the map which picks out A, considered as theconstant A-line over X. The colimit means that we have an equivalence of mappingspaces

ModA(Xf , A) ModXA (f∗j∗M , AX).

Notice also that we have a natural inclusion

(4.2) LineXA (f∗L , AX) → ModXA (f∗j∗M , AX) :

a map of bundles of A-modules f∗L → AX is a map of bundles of A-lines if it isan equivalence over every point of X, and one checks that the inclusion (4.2) is theinclusion of a set of path components.

Definition 4.3. The space of orientations of Xf is the pull-back in the dia-gram

(4.4)

orient(Xf , A) −−−−→ ModA(Xf , A)

LineXA (f∗L , AX) −−−−→ ModXA (f∗j∗M , AX).

That is, the space of orientations orient(Xf , A) is the subspace of A-module mapsXf → A which correspond, under the equivalence

ModA(Xf , A) ModXA (f∗j∗M , AX),

to fiberwise equivalences f∗L → AX .

This appealing notion of orientation expresses orientations as fiberwise equiv-alences of bundles of spectra. The following results from [ABGHR] explain howour Thom spectra and orientations generalize the classical notions.

Theorem 4.5. Suppose that

ξ : SingX → LineS

corresponds to a mapg : X → BGL1S.

Then Xξ is equivalent to the classical Thom spectrum Xg of the spherical fibrationclassified by g. It follows (see Example 3.16) that if f is the composition

f : SingXξ−→ LineS → LineA,

then Xf Xξ∧SA Xg∧SA is equivalent to the classical Thom spectrum tensoredwith A.

We can then study the space of orientations of Xf via the equivalences

(4.6) ModA(Xf , A) ModA(X

g ∧S A,A) ModS(Xg, A),

and we find that

Proposition 4.7. A map α : Xf → A ∈ ModA(Xf , A) is in orient(Xf , A)

if and only if it corresponds to an orientation β : Xg → A of the classical Thomspectrum, that is, if and only if

(X+z−→ A) → (Xg Δ−→ X+ ∧Xg z∧β−−→ A ∧A → A)

induces an isomorphismA∗(X+) ∼= A∗(Xg).

41

16 ANDO, BLUMBERG, AND GEPNER

Our theory leads to an obstruction theory for orientations. Let

mapf (SingX,Triv(A))

be the simplicial set which is the pull-back in the diagram

mapf (SingX,Triv(A)) −−−−→ map(SingX,Triv(A))

f −−−−→ map(SingX,LineA).

That is, mapf (SingX,Triv(A)) is the mapping simplicial set of lifts in the diagram

(4.8) Triv(A)

SingX

f LineA.

The obstruction theory for orientations of the bundle of A-modules is given bythe following.

Theorem 4.9. Let f : SingX → LineA be a bundle of A-lines over X, and letXf be the associated A-module Thom spectrum. Then there is an equivalence

mapf (SingX,Triv(A)) LineXA (f, ι) orient(Xf , A).

In particular, the bundle f∗L admits an orientation if and only if f is null-homotopic.

Example 4.10. This theorem recovers and slightly generalizes the obstructiontheory of [MQRT77] (which treats the case that A is a E∞ ring spectrum, thatis, a commutative S-algebra). Let g : X → BGL1S be a stable spherical fibration.Then g admits a Thom isomorphism in A-theory if and only if the composition

Xg−→ BGL1S |LineS | → |LineA| BGL1A

is null.

Example 4.11. This example appears in [MQRT77]. Let HZ be the integralEilenberg-MacLane spectrum. From Example 3.17 we have BGL1HZ K(Z/2, 1).The obstruction to orienting a vector bundle V/X in singular cohomology is themap

XV−→ BO

BJ−−→ BGL1S −→ BGL1HZ K(Z/2, 1);

this is just the first Stiefel-Whitney class.

5. Twisted Generalized Cohomology

Now we consider twisted generalized cohomology in the language of sections 3and 4. Let A be an S-algebra, and let

f : SingX → LineA

or, equivalently, f : X → BGL1A (see §3.6) classify a bundle of A-lines over X. Asin Definition 4.1, let

Xf = colim(SingX

f−→ LineAj−→ ModA

)

42

TWISTS OF K-THEORY AND TMF 17

be the indicated A-module. We think of Xf as the f -twisted cohomology objectassociated to the bundle f , and we make the following

Definition 5.1. The f-twisted A homology and cohomology groups of X are

An(X)fdef= π0ModA(X

f ,ΣnA)

An(X)fdef= π0ModA(Σ

nA,Xf ).

Equivalently, we have

An(X)f = π−nFA(Xf , A)

An(X)f = πnFA(A,Xf ) ∼= πnXf .

Here if V and W are A-modules, then FA(V,W ) is the function spectrum ofA-module maps from V to W : it is a spectrum such that

Ω∞FA(V,W ) ModA(V,W ).

Thus for n ≥ 0,

πnFA(V,W ) ∼= πnModA(V,W ) ∼= ModA(ΣnV,W ) ∼= ModA(V,Σ

−nW ).

Example 5.2. Suppose that V is a vector bundle over X. Then we can formthe map

j(V ) : XV−→ BO

BJ−−→ BGL1S −→ BGL1A.

and also the twisted cohomology

A∗(X)j(V ) = π0ModA(Xj(V ),Σ∗A).

Since by Theorem 4.5

Xj(V ) XV ∧A,

we have

(5.3) A∗(X)j(V )∼= π0ModS(X

V ,Σ∗A) = A∗(XV ),

so in this case the twisted cohomology is just the cohomology of the Thom spectrum.

The definition is not quite a direct generalization of that Atiyah and Segal.Let SX : SingX → ModS be the constant functor which attaches to each point ofX the sphere spectrum S. Theorem 3.6 and the work of [MS06, §22] allow us todescribe their construction as attaching to the bundle of A-lines f∗L over X thegroup

A(X)f,AS = π0Γ(f/X) ∼= π0ModXS (SX , f∗L ).

To compare this definition to ours, we must assume (as is the case for K-theory,for example) that A is a commutative S-algebra. In that case, if L is an A-module,then the dual spectrum

L∨ def= FA(L,A)

is again an A-module. As in the case of classical commutative rings, the operationL → L∨ defines an involution

LineA(− )∨−−−−→ LineA

on the ∞-category of A-lines, such that

L∨ ∧A M FA(L,M);

43

18 ANDO, BLUMBERG, AND GEPNER

in particular

ModS(S,L∨) ModA(A,L∨) ModA(L,A).

Remark 5.4. As we have written it, we have an evident functor LineA →(LineA)

op. As LineA is an ∞-groupoid, it admits an equivalence LineA LineopA .

This is an ∞-categorical approach to the following: if A is a commutativeS-algebra then GL1A is a sort of commutative group. More precisely, it is a com-mutative group-like monoid in the ∞-category of spaces, or equivalently it is agroup-like E∞-space. As such it has an involution

−1: GL1A → GL1A

which deloops to a map

B(−1) : BGL1A → BGL1A.

In any case, given a map

f : SingX → LineA,

classifying the bundle f∗L , we may form the map

−f = f∨ : SingXf−→ LineA

(− )∨−−−−→ LineA

so that (−f)∗L is the fiberwise dual of f∗L , and then one has

Γ((−f)∗L ) = ModXS (SX , (−f)∗L )

ModXA (AX , (−f)∗L )

ModXA (f∗L , A)

ModA(Xf , A).

That is, the cohomology object we associate to f : X → BGL1A is the one whichAtiyah and Segal associate to −f : X → BGL1A,

A∗(X)f ∼= A∗(X)−f,AS.

Of course Atiyah and Segal also explain how to construct a K-line from aPU -bundle over X: in our language, they construct a map

BPU K(Z, 3) → BGL1K.

From our point of view the existence of this map can be phrased as a questionabout the Spinc orientation of complex K-theory.

To see this, recall that Atiyah, Bott, and Shapiro [ABS64] produce a Thomisomorphism in complex K-theory for Spinc-bundles. According to Theorem 4.9,

44

TWISTS OF K-THEORY AND TMF 19

this corresponds to the arrow labeled ABS in the diagram

K(Z, 2)

GL1K

BSpinc ABS

Triv(K) ∗

BSOBJ

βw2

LineS(− )∧K

LineK BGL1K

K(Z, 3)BABS BGL1K

The map ABS induces a map

(5.5) ABS : K(Z, 2) → GL1K,

and the diagram suggests that we ask whether this map deloops to give a map

B(ABS) : K(Z, 3) → BGL1K,

as indicated. Not surprisingly, this question is related to the multiplicative proper-ties of the orientation.

The Thom spectrum associatedMSpinc associated to BSpinc is a commutativeS-algebra, as isK-theory. The construction of Atiyah-Bott-Shapiro produces a mapof spectra

t : MSpinc → K,

and Michael Joachim [Joa04] shows that t can be refined to a map of commutativeS-algebras. As we shall see in §6, it follows that K(Z, 2) → GL1K is a map ofinfinite loop spaces.

We use these ideas to twist K-theory by maps X → K(Z, 3) in §7.

6. Multiplicative Orientations and Comparison of Thom Spectra

The most familiar orientations are exponential: the Thom class of a Whitneysum is the product of the Thom classes. For example, consider the case of Spinbundles and real K-theory, KO. Atiyah, Bott, and Shapiro show that the Diracoperator associates to a spin vector bundle V → X a Thom class t(V ) ∈ KO(XV ).If MSpin is the Thom spectrum of the universal Spin bundle over BSpin, then wecan view their construction as corresponding to a map of spectra

t : MSpin → KO.

If W → Y is another spin vector bundle, then

(X × Y )V⊕W XV ∧ Y W ,

and it turns out that with respect to the resulting isomorphism

KO((X × Y )V⊕W ) ∼= KO(XV ∧ Y W ),

one has

(6.1) t(V ⊕W ) = t(V ) ∧ t(W ).

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20 ANDO, BLUMBERG, AND GEPNER

Now the sum of vector bundles gives MSpin the structure of a ring spectrum,and so the multiplicative property (6.1) (together with a unit condition, which saysthat t(0) = 1) corresponds to the fact that

t : MSpin → KO

is a map of monoids in the homotopy category of spectra.It is important that t is in fact a map of commutative monoids in the∞-category

of spectra. More precisely, MSpin and KO are both commutative S-algebras, andit turns out [Joa04, AHR] that t is a map of commutative S-algebras.

The construction of classical Thom spectra such as MSpin, MSO, MU ascommutative S-algebras (equivalently, E∞ ring spectra) is due to [MQRT77,LMSM86]. In this section, we discuss the theory from the ∞-categorical pointof view. We’ll see (Remark 6.23) that this gives a way to think about the compar-ison of our Thom spectrum to classical constructions. It also provides some toolswe use to build twists of K-theory from PU -bundles.

We begin with a question. Suppose that A is a commutative S-algebra. Underwhat conditions on a map f should we expect that the Thom

Xf = colim(SingXf−→ LineA

j−→ ModA)

is a commutative A-algebra? And in that situation, how do we understand A-algebra maps out of Xf?

In the context of ∞-categories, spaces play the role which sets play in thecontext of classical categories, and so we begin by studying the situation of adiscrete commutative ring R and a set X. In that case, an R-line is just a freerank-one R-module, and a bundle of R-lines ξ over X is just a collection of R-lines,indexed by the points x ∈ X. We can think of this as a functor

ξ : X → LineR

from X, considered as a discrete category, to the category of R-lines: free rank-oneR-modules and isomorphisms. The “Thom spectrum”

Xξ = colim(X

ξ−→ LineR −→ ModR

)

is easily seen to be the sum

Xξ ∼=⊕x∈X

ξx.

Now suppose that R is a commutative ring, so that ModR is a symmetricmonoidal category, and LineR is the maximal sub-groupoid of ModR generatedby R. If X is a discrete abelian group, then we may consider X as a symmetricmonoidal category with objects the elements of X. It is then not difficult to checkthe following.

Proposition 6.2. If ξ : X → LineR is a map of symmetric monoidal categories,then Xξ has structure of a commutative R-algebra.

The analogue of this result holds in the ∞-categorical setting; see Theorem6.21 below. It is possible to give a direct proof; instead we sketch the circuitousproof given in [ABGHR], as some of the results which arise along the way will beuseful in sections 7 and 8.

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TWISTS OF K-THEORY AND TMF 21

We begin with another construction of the R-module Xξ in the discrete asso-ciative case. Let GL1R be the group of units of R. Note that the free abelian groupfunctor

(6.3) Z : (sets) → (abelian goups)

induces a functor

(6.4) Z : (groups)(rings) : GL1

whose right adjoint is GL1. In particular, we have a natural map of rings

Z[GL1R] → R

and so the colimit-preserving functor

(GL1R-sets)Z[− ]⊗Z[GL1R]R−−−−−−−−−−→ ModR.

This functor restricts to an equivalence of categories

(6.5) Z[− ]⊗Z[GL1R] R : Tors(GL1R)LineR : T .

Here Tors(GL1R) is the category of GL1R-torsors, and the inverse equivalence isthe functor T which associates to an R-line L the GL1R-torsor

T (L)def= LineR(R,L) ∼= u ∈ L|Ru ∼= L.

That is, we have the following diagram of categories which commutes up tonatural isomorphism

LineR

T∼=

ModR

Tors(GL1R)

Z[− ]⊗Z[GL1R]R

(GL1R-sets).

Z[− ]⊗Z[GL1R]R

Moreover, the vertical arrows preserve colimits, and the left vertical arrows comprisean equivalence.

If ξ is a bundle of R-lines over X, we write P (ξ) for the GL1R-set

P (ξ) = colim(X

ξ−→ LineRT−→ Tors(GL1R) −→ (GL1R-sets)

).

That is, P (ξ) is the GL1R-torsor over X whose fiber at x ∈ X is P (ξ)x = T (ξx).

Proposition 6.6. There is a natural isomorphism of R-modules

Xξ ∼= Z[P (ξ)]⊗Z[GL1R] R.

Proof. We have

Xξ = colim(X

ξ−→ LineR → ModR

)

∼= colim

(X

Tξ−−→ Tors(GL1R) −→ (GL1R-sets)Z[− ]⊗Z[GL1R]R−−−−−−−−−−→ ModR

)

∼= Z[P (ξ)]⊗Z[GL1R] R.

47

22 ANDO, BLUMBERG, AND GEPNER

Figure 1. Selected instances of the analogySets:Spaces::Categories:∞-categories

Categorical notion ∞-categorical notion Alternate name/description

Category ∞-categorySet ∞-groupoid Space/Kan complexMonoid monoidal ∞-groupoid A∞ space; ∗-algebraGroup group-like monoidal group-like A∞ space

∞-groupoidAbelian group group-like symmetric group-like E∞ space;

monoidal ∞-groupoid (−1)-connected spectrumAbelian group spectrumThe ring Z The sphere spectrum SRing Monoid in spectra S-algebra or A∞ ring spectrumCommutative ring Commutative monoid in spectra Commutative S-algebra or

E∞ ring spectrumThe functor Z The functor Σ∞

+

ModR ModALineR LineA BGL1AGL1R LineA(A,A) = AutA(A) GL1AGL1R-set GL1A-space A∞ GL1A-spaceGL1R-torsors Tors(GL1A)⊗ ∧

Now suppose that R is a commutative ring, and X is an abelian group. Ifξ : X → LineR is a symmetric monoidal functor, then

GL1R → P (ξ) → X

is an extension of abelian groups. The adjunction (6.4) restricts further to anadjunction

(6.7) Z : (abelian groups)(commutative rings) : GL1,

and so we have maps of commutative rings

Z[P (ξ)] ← Z[GL1R] → R.

The isomorphism of Proposition 6.6 has the following consequence.

Proposition 6.8. If R is a commutative ring and ξ : X → LineR is a symmet-ric monoidal functor, then Xξ is a commutative R-algebra; indeed, it is the pushoutin the category of commutative rings

Z[GL1R] −−−−→ R

Z[P (ξ)] −−−−→ Xξ.

The preceding discussion generalizes elegantly and directly to spaces and spec-tra. The generalization illustrates how spaces play the role in ∞-categories that setsplay in categories. The reader may find it useful to consult the table in Figure 6.

48

TWISTS OF K-THEORY AND TMF 23

Let T be an ∞-category of spaces. The analogue of the adjunction (6.3) is

(6.9) Σ∞+ : T

S : Ω∞.

Definition 6.10. Let Alg(∗) be the ∞-category of monoids in the ∞-categoryof T .

We introduce the awkward name ∗-algebra to emphasize that these are moregeneral than monoids with respect to the classical product of topological spaces.The symmetric monoidal structure on T is such that Alg(∗) is a model for the∞-category of A∞ spaces. A ∗-algebra X is group-like if π0X is a group.

Definition 6.11. We write Alg(∗)× for the ∞-category of group-like monoids.

The adjunction (6.9) restricts to an adjunction

(6.12) Σ∞+ : Alg(∗)×

Alg(S) : GL1.

Remark 6.13. If X is a group-like monoid in T , then SingX is a group-likemonoidal∞-groupoid. One way to see that a group-like A∞ space is the appropriategeneralization of a group is to observe that if Z is an object in a category, thenEnd(Z) is a monoid, and Aut(Z) is a group. If Z is an object in an ∞-category,then End(Z) is a monoidal ∞-groupoid, and Aut(Z) is group-like.

In particular, as we have already discussed in §3.6, one construction of theright adjoint GL1 is the following. If A is an S-algebra, then it has an ∞-categoryof modules ModA. In ModA we have the maximal sub-∞-groupoid LineA whoseobjects are weakly equivalent to A. We can define

GL1A = Aut(A) = LineA(A,A)

to be the subspace of (the geometric realization of) ModA(A,A) consisting of ho-motopy equivalences: it is a group-like ∗-algebra. We also write BGL1A for thefull ∞-subcategory of LineA on the single object A: this is the ∞-groupoid withAutA as its simplicial set of of morphisms.

Since GL1A is a (group-like) monoid in the symmetric monoidal ∞-category ofspaces, we can form the ∞-category of GL1A-spaces, and then define Tors(GL1A)to be the maximal subgroupoid whose objects are GL1A-spaces P which admit anequivalence of GL1A-spaces

GL1A P.

The adjunction (6.12) provides a map of S-algebras

Σ∞+ GL1A → A,

and so a (∞-category) colimit-preserving functor

Σ∞+ (− ) ∧Σ∞

+ GL1A A : (GL1A-spaces) −→ ModA,

which restricts to an equivalence of ∞-categories

Σ∞+ (− ) ∧Σ∞

+ GL1A A : Tors(GL1A) → LineA.

The inverse equivalence is the functor

T : LineA → Tors(GL1A)

which to an A-line L associates the GL1A-torsor

T (L)def= LineA(A,L).

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24 ANDO, BLUMBERG, AND GEPNER

Putting all these together, we have the homotopy commutative diagram of∞-categories

LineA

T

ModA

Tors(GL1A)

Σ∞+ (− )∧Σ∞

+GL1AA

(GL1A-spaces),

Σ∞+ (− )∧Σ∞

+GL1AA

in which the vertical arrows preserve ∞-categorical colimits, and the left verticalarrows comprise an equivalence.

Now let X be a space, and let

ξ : X → LineA

be a bundle of A-lines over X. Recall that

Xξ = colim(SingX

ξ−→ LineA −→ ModA

).

On the other hand, let

P (ξ) = colim(SingX

ξ−→ LineAT−→ Tors(GL1A) −→ (GL1A-spaces)

).

We have the following analogue of Proposition 6.6.

Proposition 6.14. There is a natural equivalence of A-modules

(6.15) Xξ Σ∞+ P (ξ) ∧Σ∞

+ GL1A A.

Now we turn to the commutative case.

Definition 6.16. We write CommAlg(∗) for the ∞-category of commutativemonoids in T . It is equivalent to the nerve of the simplicial category of (cofibrantand fibrant) E∞ spaces. We write CommAlg(∗)× for the ∞-category of group-likecommutative monoids, which models group-like E∞ spaces.

The adjunction (6.12) restricts to the analogue of the adjunction (6.7), namely

(6.17) Σ∞+ : CommAlg(∗)×

CommAlg(S) : GL1

The reader will notice that in the table in Figure 6, we mention two modelsfor “abelian groups”, namely, group-like E∞ spaces and spectra. It is a classicaltheorem of May [May72, May74], reviewed for example in [ABGHR, §3], thatthe functor Ω∞ induces an equivalence of ∞-categories

Ω∞ : ((−1)-connected spectra) CommAlg(∗)×,and so we may rewrite the adjunction (6.17) as(6.18)

Σ∞+ Ω∞ : ((−1)-connected spectra)

Ω∞ CommAlg(∗)×

Σ∞+

CommAlg(S) : gl1

GL1

(The left adjoints are written on top, but the pair of adjoints on the left is anequivalence of ∞-categories). Note that we have introduced the functor gl1, withthe property that if A is a commutative S-algebra, then

GL1A Ω∞gl1A.

We also define bgl1A = Σgl1A: then Ω∞bgl1A BGL1A.

50

TWISTS OF K-THEORY AND TMF 25

Thus a map of spectra

f : b → bgl1A

may be viewed equivalently as a map of group-like commutative ∗-algebras

Ω∞f : B = Ω∞b → BGL1A

or as a map of symmetric monoidal ∞-groupoids

(6.19) ξ : SingB −→ LineA.

Form the pull-back diagram

(6.20)

gl1A gl1A

p −−−−→ egl1A ∗

bf−−−−→ bgl1A.

One checks [ABGHR, Lemma 8.23] that

P (ξ) Ω∞p,

and so we have the following.

Proposition 6.21. The Thom spectrum of the map ξ of symmetric monoidal∞-groupoids (6.19) is a commutative A-algebra; indeed, we have

Xξ Σ∞+ P (ξ) ∧Σ∞

+ GL1A A Σ∞+ Ω∞p ∧Σ∞

+ Ω∞gl1A A.

Example 6.22. Taking A to be the sphere spectrum in Proposition 6.21, werecover the result of [LMSM86] that the Thom spectrum of an ∞-loop map

B → BGL1S

is a commutative S-algebra.

Remark 6.23. The formula (6.15) provides one way to see that our Thomspectrum coincides with the classical Thom spectrum of [MQRT77, LMSM86].One way to compute the smash product in (6.15) is to realize it as a two-sidedbar construction [EKMM96, Proposition 7.5]. In particular if ξ : X → BGL1Sclassifies a spherical fibration, then one expects

(6.24) Xξ Σ∞+ P (ξ) ∧Σ∞

+ GL1S S B(Σ∞+ P (ξ),Σ∞

+ GL1S, S).

It is not difficult to see that the constructions of [MQRT77, LMSM86] providecareful models for the two-sided bar construction on the right-hand side of (6.24).Note that some care is required to make this proposal precise; see §8.6 of [ABGHR]for details.

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26 ANDO, BLUMBERG, AND GEPNER

7. Application: K(Z, 3), Twisted K-theory, and the Spinc Orientation

7.1. Recall that BSpinc participates in a fibration of infinite loop spaces

(7.1) K(Z, 2) → BSpinc → BSObw2−−→ K(Z, 3),

where bw2 is the composite of the usual w2 with the Z-Bockstein

BSOw2−−→ K(Z/2, 2)

b−→ K(Z, 3).

Passing to Thom spectra in (7.1) we have a map of commutative S-algebras

Σ∞+ K(Z, 2) → MSpinc.

It’s a theorem of Joachim [Joa04] that the orientation

MSpinc → K

of Atiyah-Bott-Shapiro is map of commutative S-algebras, and so we have a se-quence of maps of commutative S-algebras

(7.2) Σ∞+ K(Z, 2) → MSpinc → K.

The (Σ∞+ Ω∞, gl1) adjunction (6.18) produces from (7.2) a map of infinite loop

spacesK(Z, 2) → GL1K

which we deloop once to view as a map

T : K(Z, 3) → BGL1K.

That is, the fact that the Atiyah-Bott-Shapiro orientation is a map of (commuta-tive) S-algebras implies that we have a homotopy-commutative diagram

(7.3)

K(Z, 2) −−−−→ GL1K

BSpinc −−−−→ ∗

BSOj−−−−→ BGL1K

βw2

=

K(Z, 3)T−−−−→ BGL1K.

Now suppose given a map α : X → K(Z, 3). Then we may form the Thomspectrum XTα, and define

Kn(X)αdef= π0ModK(XTα,ΣnK).

We then have the following.

Proposition 7.4. A map α : X → K(Z, 3) gives rise to a twist K∗(X)α of theK-theory of X. A choice of homotopy

Tβw2 ⇒ j

in the diagram (7.3) above determines, for every oriented vector bundle V over X,an isomorphism

(7.5) K∗(X)βw2(V )∼= K∗(XV ).

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TWISTS OF K-THEORY AND TMF 27

In this way, the characteristic class βw2(V ) determines the K-theory of the Thomspectrum of V .

Proof. We prove the isomorphism (7.5) to show how simple it is from thispoint of view. Consider the homotopy commutative diagram

XV−−−−→ BSO

j−−−−→ BGL1K

βw2

=

K(Z, 3)T−−−−→ BGL1K.

Omitting the gradings, we have

K0(X)βw2(V ) = π0ModK(XTβw2(V ),K)

∼= π0ModK(Xj(V ),K)

∼= π0ModK(XV ∧K,K)

∼= π0 S (XV ,K)

= K0(XV ).

The first isomorphism uses the construction of Xξ together with the fact thatTβw2 and j are homotopic as maps BSO → BGL1K. The second isomorphism isTheorem 4.5.

Remark 7.6. We needn’t have started with an oriented bundle. For example,let F be the fiber in the sequence

F → BSpinc → BO.

This is a fibration of infinite loop spaces, and so it deloops to give

F → BSpinc → BOγ−→ BF.

The same argument produces an E∞ map

Σ∞+ F → MSpinc → K

whose adjoint

F → GL1K

deloops to

ζ : BF → BGL1K,

and if V is any vector bundle then

K(XV ) ∼= K(X)ζγ(V ).

7.2. Khorami’s theorem. At this point we are in a position to state a re-markable result of M. Khorami [Kho10]. K(Z, 2) is a group-like commutative∗-algebra, and so we have a bundle

K(Z, 2) → EK(Z, 2) ∗ → BK(Z, 2) K(Z, 3).

(One can build this bundle a number of ways: by modeling K(Z, 2) PU asmentioned in §2.1, or using infinite loop space theory, or using the ∞-categorytechnology described above.)

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28 ANDO, BLUMBERG, AND GEPNER

Given a map ζ : X → K(Z, 3), we can form the pull-back K(Z, 2)-bundle

Q −−−−→ EK(Z, 2)

Xζ−−−−→ K(Z, 3).

On the other hand, let P be the pull-back

P −−−−→ EGL1K

XTζ−−−−→ BGL1K

as in Proposition 6.21. One can check that

Σ∞+ P Σ∞

+ Q ∧Σ∞+ K(Z,2) Σ

∞+ GL1K,

and so the Thom spectrum whose homotopy calculates the ζ-twisted K-theory is

XTζ Σ∞+ P ∧Σ∞

+ GL1K K

Σ∞+ Q ∧Σ∞

+ K(Z,2) Σ∞+ GL1K ∧Σ∞

+ GL1K K

Σ∞+ Q ∧Σ∞

+ K(Z,2) K,

(7.7)

where the map of commutative S-algebras Σ∞+ K(Z, 2) → K is (7.2).

The formula (7.7) implies that there is a spectral sequence (see for example[EKMM96, Theorem 4.1])

(7.8) TorK∗K(Z,2)∗ (K∗Q,K∗) ⇒ π∗X

Tζ ∼= K∗(X)ζ .

Note that K∗K(Z, 2) ∼= K∗β1, β2, . . ., so K∗ is not a flat K∗K(Z, 2)-module.Nevertheless Khorami proves the following.

Theorem 7.9. In (7.8) one has Torq = 0 for q > 0, and so

K∗(X)ζ ∼= K∗Q⊗K∗K(Z,2) K∗.

8. Application: Degree-four Cohomology and Twisted EllipticCohomology

The arguments of §7.1 apply equally well to String structures and the spectrumof topological modular forms.

Recall that spin vector bundles admit a degree four characteristic class λ, whichwe may view as a map of infinite loop spaces

BSpinλ−→ K(Z, 4).

Indeed this map detects the generator of H4BSpin ∼= Z. The fiber of λ is calledBString, and so we have maps of infinite loop spaces

K(Z, 3) −→ BString → BSpinλ−→ K(Z, 4).

Passing to Thom spectra, we get maps of commutative S-algebras

Σ∞+ K(Z, 3) → MString → MSpin.

Let tmf be the spectrum of topological modular forms [Hop02]. Ando, Hopkins,and Rezk [AHR] have produced a map of commutative S-algebras

MStringσ−→ tmf,

54

TWISTS OF K-THEORY AND TMF 29

and so we have a map of commutative S-algebras

Σ∞+ K(Z, 3) → tmf,

whose adjoint (see (6.12,6.17,6.18))

K(Z, 3) → GL1tmf

deloops to

T : K(Z, 4) → BGL1tmf.

By construction, the map T makes the diagram

(8.1)

K(Z, 3) −−−−→ GL1tmf

BString −−−−→ ∗

BSpinj−−−−→ BGL1tmf

λ

=

K(Z, 4)T−−−−→ BGL1tmf

commute up to homotopy. If ζ : X → K(Z, 4) is a map, then we may define

tmf(X)kζdef= π0Modtmf (X

Tζ ,Σktmf),

and so we have the following.

Proposition 8.2. A map ζ : X → K(Z, 4) gives rise to a twist tmf∗(X)ζ ofthe tmf -theory of X. A choice of homotopy

Tλ ⇒ j

in the diagram (8.1) above determines, for every map

V : X −→ BSpin,

an isomorphism of tmf∗(X)-modules

(8.3) tmf∗(X)λ(V )∼= tmf∗(XV ).

In this way, the characteristic class λ(V ) determines the tmf -cohomology of theThom spectrum of V .

Remark 8.4. As in Remark 7.6, we could have started with the fiber F in thefibration sequence of infinite loop spaces

F → BString → BOγ−→ BF.

We have a map of commutative S-algebras

Σ∞+ F → MString → tmf

whose adjoint

F → GL1tmf

deloops to

ζ : BF → BGL1tmf,

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30 ANDO, BLUMBERG, AND GEPNER

and if V : X → BO classifies a virtual vector bundle, then

tmf∗(XV ) ∼= tmf∗(X)ζγ(V )

9. Application: Poincare Duality and Twisted Umkehr Maps

Let M be a compact smooth manifold with tangent bundle T of rank d. EmbedM in RN , and then perform the Pontrjagin-Thom construction: collapse to a pointthe complement of a tubular neighborhood of M . If ν is the normal bundle of theembedding, this gives a map

SN → Mν .

Desuspending N times then yields a map

(9.1) μ : S0 → M−T .

As usual, the Thom spectrum admits a relative diagonal map

M−T Δ−→ Σ∞+ M ∧M−T

(this is the map which gives the cohomology of the Thom spectrum the structureof a module over the cohomology of the base).

If E is a spectrum, then to a map

f : M−T → E

we can associate the composition

S0 μ−→ M−T Δ−→ Σ∞+ M ∧M−T 1∧f−−→−→ Σ∞

+ M ∧E.

Milnor-Spanier-Atiyah duality says that this procedure yields an isomorphism

E∗(M) ∼= E−∗(M−T ).

In the presence of a Thom isomorphism

E−∗(M−T ) ∼= Ed−∗(M)

we have Poincare duality

E∗(M) ∼= Ed−∗(M).

Without a Thom isomorphism, we choose a map α

Mα−→ BO

classifying d− T , and then we define τ (−T ) to be the composition

τ (−T ) : Mα−→ BO

j−→ BGL1S −→ BGL1E.

Then, following Example 5.2, we have

Ed−∗(M)τ(−T )∼= Ed−∗(Md−T ) ∼= E−∗(M−T ) ∼= E∗(M).

Combining this with the results of sections 7.1 and 8, we have the following.

Proposition 9.2. Suppose that M is an oriented compact manifold of dimen-sion d. Then

K∗(M) ∼= Kd−∗(M)−βw2(M).

If M is a spin manifold, then

tmf∗(M) ∼= tmfd−∗(M)−λ(M).

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TWISTS OF K-THEORY AND TMF 31

9.1. Twisted umkehr maps. In this section we sketch the construction ofsome umkehr maps in twisted generalized cohomology. Note that similar construc-tions are studied in [CW, MS06, Wal06]. Also, since this paper was written, wehave learned that Bunke, Schneider, and Spitzweck have independently developeda similar approach to twisted umkehr maps.

Suppose that we have a family of compact spaces over X, that is, a map of∞-categories

ζ : SingX → (compact spaces).

We also have the trivial map

∗X : SingX∗−→ (compact spaces).

If M is a compact space, let

DM = F (Σ∞+ M,S0)

be the Spanier-Whitehead dual of M+: this is a functor of ∞-categories

D : (compact spaces)op → S .

The projectionM → ∗

gives rise to a map of spectra

(9.3) S0 ∼= D∗ → DM.

Indeed if M is a compact manifold with tangent bundle T , then Milnor-Spanier-Atiyah duality says that DM M−T , in such a way that the Pontrjagin-Thommap (9.1) identifies with (9.3).

In any case, let S0X = D∗X . We have a natural map

u : S0X → Dζ

of bundles of spectra over X. Essentially, we are applying the map (9.3) fiberwise.It follows from Proposition 7.7 of [ABGHR] that

XS0X Σ∞

+ X.

As for XDζ , in a forthcoming paper we prove the following.

Proposition 9.4. Suppose that ζ arises from a bundle

Yf−→ X

of compact manifolds, and let Tf be its bundle of tangents along the fiber. Then

XDζ Y −Tf .

In particular, passing to Thom spectra on u gives a map of spectra

(9.5) t : Σ∞+ X → Y −Tf .

This map is equivalent to the classical stable transfer map associated to f .

The map t, and indeed the idea that it arises from applying the map (9.1)fiberwise, is classical; see for example [BG75]. Casting it in our setting enables usto construct twisted versions.

More precisely, suppose that R is a commutative S-algebra, and suppose givena bundle of R-lines over X

ξ : SingX −→ LineR.

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32 ANDO, BLUMBERG, AND GEPNER

We then have a map of bundles of R-lines over X

u ∧ ξ : ξ S0X ∧ ξ → Dζ ∧ ξ.

Thus we have constructed a twisted umkehr map

R∗(XDζ)ξ → R∗(X)ξ.

In the situation of Proposition 9.4, we have a twisted transfer map

(9.6) R∗(Y −Tf )ξ → R∗(X)ξ.

About this we show the following.

Proposition 9.7. Suppose that

YTf−−→ BO −→ BGL1S −→ BGL1R,

regarded as a map Sing Y → LineR, is homotopic to ξf : Sing Y −→ SingX → LineR.A choice of homotopy determines an isomorphism

R∗(Y ) ∼= R∗(Y −Tf )ξ,

and composing with the twisted transfer (9.6) we have a twisted umkehr map R∗(Y ) →R∗(X)ξ.

10. Motivation: D-brane Charges in K-theory

10.1. The Freed-Witten anomaly. Let j : D → X be an embedded sub-manifold, let ν be the normal bundle of j, and suppose that D carries a complexvector bundle ξ.

Suppose moreover that ν carries a Spinc-structure. Then we can form theK-theory push-forward

j! : K(D) → K(X).

In that situation Minasian and Moore and Witten discovered that it is sensible tothink of the K-theory class

j!(ξ) ∈ K(X)

as the “charge” of the D-brane D with Chan-Paton bundle ξ.If ν does not carry a Spinc-structure, then we still have the Pontrjagin-Thom

constructionX → Dν .

Suppose we have a map H : X → K(Z, 3) making the diagram

Dν−−−−→ BSO

j

bw2

XH−−−−→ K(Z, 3)

commute up to homotopy. According to Proposition 9.7, a homotopy

c : bw2 Hj

determines an isomorphism

K∗(D) ∼= K∗(Dν)−H

(since ν = −Tj), and then we have a twisted umkehr map

(10.1) j! : K∗(D) −→ K∗(X)−H .

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TWISTS OF K-THEORY AND TMF 33

The class j!(ξ) ∈ K∗−H(X) is evidently an analogue of the charge in this situ-

ation. The discovery of the condition that there exists a class H on X such thatH|D = W3(ν) is due to Freed and Witten [FW99].

Although we discovered this push-forward in an attempt to understand Freedand Witten’s condition, we were not the first: it appeared, formulated this way, ina paper of Carey and Wang [CW]. An important contribution of their work is theconstruction, using the twisted K-theory of [AS04], of the umkehr map (10.1).

11. An Elliptic Cohomology Analogue

Now suppose that we are given an embedding of manifolds

j : M → Y,

and that ν = ν(j) is equipped with a Spin structure. Suppose we have a map Hmaking the diagram

(11.1)

Mν−−−−→ BSpin

j

λ

YH−−−−→ K(Z, 4)

commute up to homotopy. By Proposition 9.7, a homotopy c : λν Hj determinesan isomorphism

tmf∗(M) tmf∗(Dν)−H ,

and then we have a homomorphism of tmf∗(Y )-modules

(11.2) tmf∗(M) tmf∗(Dν)−H → tmf∗(Y )−H .

Remark 11.3. The data of a configuration like (12.3) together with the homo-topy c was studied by Wang [Wan08], who calls it a twisted String structure. Infact, it was predicted by Kriz and Sati [KS04, Sat] that tmf should be the naturalreceptacle for M -brane charges.

Remark 11.4. The authors are grateful to Hisham Sati for suggesting thatwe think about diagrams like (11.1). The on-going investigation of the resultingtwisted umkehr maps is joint work with him.

12. Twists of Equivariant Elliptic Cohomology

At present we do not know how to twist equivariant cohomology theories ingeneral; for that matter, equivariant Thom spectra are poorly understood. How-ever, twists by degree four Borel cohomology play an important role in equivariantelliptic cohomology. We review two instances to give the reader a taste of thesubject.

Let G be a connected and compact Lie group. In 1994, Grojnowski sketchedthe construction of a G-equivariant elliptic cohomology EG, based on a complexelliptic curve of the form Cq = C/Λ ∼= C×/qZ; more generally, the constructioncan be used to give a theory for the universal curve over the complex upper half-plane (Grojnowski’s paper is now available [Gro07]). In the case of the circle,Greenlees [Gre05] has given a complete construction of a rational S1-equivariantelliptic spectrum.

Note also that Jacob Lurie has obtained analogous and sharper results aboutequivariant elliptic cohomology, in the context of his derived elliptic curves.

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34 ANDO, BLUMBERG, AND GEPNER

The functor EG takes its values in sheaves of OMG-modules, where MG is the

complex abelian variety

MG = (T ⊗Z Cq)/W.

Note that the completion of MG at the origin is spf E(BG); in general one hasEG(X)∧0

∼= E(EG×GX), where E is the non-equivariant elliptic cohomology asso-ciated to Cq.

Grojnowski points out that a construction of Looijenga [Loo76] (see also[And03, §5]) associates to a class c ∈ H4(BG) a line bundle A(c) over MG. Thusif X is any G-space, then we can form the OMG

-module

EG(X)cdef= EG(X)⊗A(c)

This EG(X)-module is a twisted form of EG(X).

12.1. Representations of loop groups. Already the case of a point is inter-esting: one learns that twisted equivariant elliptic cohomology carries the charactersof representations of loop groups. Suppose that G is a simple and simply connectedLie group, such as SU(d) or Spin(2d). Then

H4(BG) ∼= Z.

We then have the following result, due independently to the first author [And00](who learned it from Grojnowski) and, in a much more precise form involvingderived equivariant elliptic cohomology, Jacob Lurie.

Proposition 12.1. Let G be a simple and simply connected compact Lie group,and let φ ∈ H4(BG;Z) ∼= Z. The character of a representation of the loop groupLG of level φ is a section of A(φ), and the Kac character formula shows that wehave an isomorphism

Rφ(LG) ∼= Γ(EG(∗)φ)after tensoring with Z((q)).

It is fun to compare this result to the work of Freed, Hopkins, and Teleman (forexample [FHT]), who show that Rφ(LG) is the twisted G-equivariant K-theory ofG. Thus we have a map

ΓEG(∗)φ → KG(G)φ.

This map is an instance of the relationship between elliptic cohomology and theorbifold K-theory of the free loop space. We hope to provide a more extensivediscussion in the future.

12.2. The equivariant sigma orientation. Let T be the circle group, andsuppose that V/X is a T-equivariant vector bundle with structure group G (in thissection we suppose that G = Spin(2d) or G = SU(d)). Let P/X be the associatedprincipal bundle. Then

EG×T(P ) ∼= ET(X)

is a sheaf of EG(∗) = OMG-algebras, and so we can twist ET(X) by A(c) for

c ∈ H4(BG) ∼= Z. Let c be the generator corresponding to c2 if G = SU(d) orthe “half Pontrjagin class” λ if G = Spin(2d). Note that c determines a Borelequivariant class cT.

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TWISTS OF K-THEORY AND TMF 35

In [And03, AG], the authors show first of all that the twist ET(X) ⊗ A(c)depends only on the equivariant degree-four class cT(V ) ∈ H4

T(X;Z), and so we

may define

ET(X)cT(V ) = ET(X)⊗A(c).

Second, they show that the Weierstrass sigma function leads to an isomorphism

(12.2) ET(X)cT(V )∼= ET(X

V );

this is an analytic and equivariant form of the isomorphism (8.3).In the case that cT(V ) = 0 we conclude that

(12.3) ET(X) ∼= ET(XV );

this is the T-equivariant sigma orientation in this context. More precisely, we havethe following.

Proposition 12.4. Let V/X be an S1-equivariant SU vector bundle. LetcT2 (V ) ∈ H4

T(X) be the equivariant second Chern class of V . Let ET denote Gro-

jnowski’s or Greenlees’s T-equivariant elliptic cohomology, associated to the thecomplex analytic elliptic curve C. Then there is a canonical isomorphism

ET(X)cT2(V )∼= ET(X

V ),

natural in V/X. In particular if V0 and V1 are two such bundles with cT2 (V0) =cT2 (V1), and

W = V0 − V1,

then there is a canonical isomorphism

ET(X) ∼= ET(XW ).

Remark 12.5. In [And03] the author constructs the T-equivariant sigma ori-entation in Grojnowski’s equivariant elliptic cohomology, for Spin and SU bundles.The construction was motivated by the Proposition stated above, which howeverwas given as Conjecture 1.14. In [AG] the authors construct the T-equivariantsigma orientation for Greenlees’s equivariant elliptic cohomology, for T-equivariantSU -bundles. Proposition 12.4 appears there as Theorem 11.17. It should not bedifficult to adapt the methods of these two papers to the case of Spin bundles.

Remark 12.6. The careful reader will note that in [AG] we show how to twistE∗

T(X) by cT2(V ) ∈ H4

T(X;Z): we do not there discuss twisting by general elements

of H4T(X;Z). The construction of such general twists of ET(X) is the subject of

on-going work of the first author and Bert Guillou.

Remark 12.7. Lurie has obtained similar and sharper results for the ellipticcohomology associated to a derived elliptic curve. In particular he can constructthe sigma orientation and twists by H4

T(X;Z).

References

[ABGHR] Matthew Ando, Andrew J. Blumberg, David Gepner, Michael Hopkins, and CharlesRezk. Units of ring spectra and Thom spectra, arxiv:0810.4535v3.

[ABS64] Michael F. Atiyah, Raoul Bott, and Arnold Shapiro. Clifford modules. Topology, 3suppl. 1:3–38, 1964.

[AG] Matthew Ando and J. P. C. Greenlees. Circle-equivariant classifying spaces and therational equivariant sigma genus, http://arxiv.org/abs/0705.2687v2. Submitted.

61

36 ANDO, BLUMBERG, AND GEPNER

[AHR] Matthew Ando, Michael J. Hopkins, and Charles Rezk. Multiplicativeorientations of KO and of the spectrum of topological modular forms,http://www.math.uiuc.edu/˜mando/papers/koandtmf.pdf. Preprint.

[And00] Matthew Ando. Power operations in elliptic cohomology and representations of loopgroups. Trans. Amer. Math. Soc., 352(12):5619–5666, 2000.

[And03] Matthew Ando. The sigma orientation for analytic circle-equivariant elliptic cohomol-ogy. Geometry and Topology, 7:91–153, 2003, arXiv:math.AT/0201092.

[AS04] Michael Atiyah and Graeme Segal. Twisted K-theory. Ukr. Mat. Visn., 1(3):287–330,2004, arXiv:math/0407054.

[Ati69] M. F. Atiyah. Algebraic topology and operators in Hilbert space. In Lectures in Mod-ern Analysis and Applications. I, pages 101–121. Springer, Berlin, 1969.

[BG75] J. C. Becker and D. H. Gottlieb. The transfer map and fiber bundles. Topology, 14:1–12, 1975.

[BCMMS] P. Bouwknegt and A. L. Carey and V. Mathai and M. K. Murray and D. Stevenson.Twisted K-theory and K-theory of bundle gerbes. Commun. Math. Phys., 228:17–49,2002, http://arxiv.org/abs/hep-th/0106194.

[BM00] Peter Bouwknegt and Varghese Mathai. D-branes, B-fields and twisted K-theory. J.High Energy Phys., (3):Paper 7, 11, 2000, http://arxiv.org/abs/hep-th/0002023v3.

[CW] Alan L. Carey and Bai-Ling Wang. Thom isomorphism and push-forward map intwisted K-theory, arxiv:math/0507414.

[DK70] P. Donovan and M. Karoubi. Graded Brauer groups and K-theory with local coeffi-

cients. Inst. Hautes Etudes Sci. Publ. Math., (38):5–25, 1970.[EKMM96] A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May. Rings, modules, and alge-

bras in stable homotopy theory, volume 47 of Mathematical surveys and monographs.American Math. Society, 1996.

[FHT] Daniel S. Freed, Michael J. Hopkins, and Constantin Teleman. Twisted K-theory andLoop Group Representations. arXiv:math.AT/0312155.

[FW99] Daniel S. Freed and Edward Witten. Anomalies in string theory with D-branes. AsianJ. Math., 3(4):819–851, 1999.

[Gre05] J. P. C. Greenlees. Rational S1-equivariant elliptic cohomology. Topology, 44(6):1213–1279, 2005.

[Gro07] Ian Grojnowski. Delocalized equivariant elliptic cohomology. In Elliptic cohomology:geometry, applications, and higher chromatic analogues, volume 342 of London Math-ematical Society Lecture Notes. Cambridge University Press, 2007.

[Hop02] M. J. Hopkins. Algebraic topology and modular forms. In Proceedings of the Interna-tional Congress of Mathematicians, Vol. I (Beijing, 2002), pages 291–317, Beijing,2002. Higher Ed. Press, arXiv:math.AT/0212397.

[Hov01] M. Hovey. Spectra and symmetric spectra in general model categories. J. Pure Appl.Algebra, 165(1):63–127, 2001.

[HSS00] Mark Hovey, Brooke Shipley, and Jeff Smith. Symmetric spectra. J. Amer. Math.Soc., 13(1):149–208, 2000.

[Joa04] Michael Joachim. Higher coherences for equivariant K-theory. In Structured ring spec-tra, volume 315 of London Math. Soc. Lecture Note Ser., pages 87–114. CambridgeUniv. Press, Cambridge, 2004.

[Joy02] A. Joyal. Quasi-categories and Kan complexes. J. Pure Appl. Algebra, 175(1-3):207–222, 2002. Special volume celebrating the 70th birthday of Professor Max Kelly.

[Kho10] Mehdi Khorami. A universal coefficient theorem for twisted K-theory.arXiv:math.AT/10014790

[KS04] Igor Kriz and Hisham Sati. M-theory, type IIA superstrings, and elliptic coho-mology. Adv. Theor. Math. Phys., 8(2):345–394, 2004, http://arxiv.org/abs/hep-th/0404013v3.

[Lew91] L. G. Lewis, Jr. Is there a convenient category of spectra? J. Pure Appl. Algebra,73:233–246, 1991.

[LMSM86] L. G. Lewis, Jr., J. P. May, M. Steinberger, and J. E. McClure. Equivariant stablehomotopy theory, volume 1213 of Lecture Notes in Mathematics. Springer-Verlag,Berlin, 1986. With contributions by J. E. McClure.

[Loo76] Eduard Looijenga. Root systems and elliptic curves. Inventiones Math., 38, 1976.

62

TWISTS OF K-THEORY AND TMF 37

[DAGI] Jacob Lurie. Derived algebraic geometry I: stable ∞-categories,arXiv:math.CT/0608040.

[DAGII] Jacob Lurie. Derived algebraic geometry II: noncommutative algebra,arXiv:math.CT/0702229.

[DAGIII] Jacob Lurie. Derived algebraic geometry III: commutative algebra,arXiv:math.CT/0703204.

[HTT] Jacob Lurie. Higher Topos Theory. AIM 2006 -20, arXiv:math.CT/0608040.

[May72] J. P. May. The geometry of iterated loop spaces. Springer-Verlag, Berlin, 1972. Lec-tures Notes in Mathematics, Vol. 271.

[May74] J. P. May. E∞ spaces, group completions, and permutative categories. In New devel-opments in topology (Proc. Sympos. Algebraic Topology, Oxford, 1972), pages 61–93.London Math. Soc. Lecture Note Ser., No. 11. Cambridge Univ. Press, London, 1974.

[MQRT77] J. P. May. E∞ ring spaces and E∞ ring spectra. Springer-Verlag, Berlin, 1977. Withcontributions by Frank Quinn, Nigel Ray, and Jørgen Tornehave, Lecture Notes inMathematics, Vol. 577.

[MM02] M. A. Mandell and J. P. May. Equivariant orthogonal spectra and S-modules. Mem.Amer. Math. Soc., 159(755), 2002.

[MMSS01] M. A. Mandell, J. P. May, S. Schwede, and B. Shipley. Model categories of diagramspectra. Proc. London Math. Soc. (3), 82(2):441–512, 2001.

[MS06] J. P. May and J. Sigurdsson. Parametrized homotopy theory, volume 132 of Mathe-matical Surveys and Monographs. American Mathematical Society, Providence, RI,2006.

[Qui68] Daniel G. Quillen. The geometric realization of a Kan fibration is a Serre fibration.Proc. Amer. Math. Soc., 19:1499–1500, 1968.

[Ros89] Jonathan Rosenberg. Continuous-trace algebras from the bundle theoretic point ofview. J. Austral. Math. Soc. Ser. A, 47(3):368–381, 1989.

[Sat] Hisham Sati. Geometric and topological structures related to M-branes. These pro-ceedings.

[Wal06] Robert Waldmuller. Products and push-forwards in parametrised cohomology theories.PhD thesis.

[Wan08] Bai-Ling Wang. Geometric cycles, index theory and twisted K-homology. J. Noncom-mut. Geom., 2(4):497–552, 2008.

Department of Mathematics, The University of Illinois at Urbana-Champaign, Ur-

bana IL 61801, USA

E-mail address: mando@math.uiuc.edu

Department of Mathematics, University of Texas, Austin, TX 78703

E-mail address: blumberg@math.utexas.edu

Department of Mathematics, The University of Illinois at Chicago, Chicago IL

60607, USA

E-mail address: gepner@uic.edu

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Proceedings of Symposia in Pure Mathematics

Division Algebras and Supersymmetry I

John C. Baez and John Huerta

Abstract. Supersymmetry is deeply related to division algebras. For exam-

ple, nonabelian Yang–Mills fields minimally coupled to massless spinors aresupersymmetric if and only if the dimension of spacetime is 3, 4, 6, or 10. Thesame is true for the Green–Schwarz superstring. In both cases, supersymmetryrelies on the vanishing of a certain trilinear expression involving a spinor field.

The reason for this, in turn, is the existence of normed division algebras in

dimensions two less, namely 1, 2, 4 and 8: the real numbers, complex numbers,quaternions and octonions. Here we provide a self-contained account of howthis works.

1. Introduction

There is a deep relation between supersymmetry and the four normed divisionalgebras: the real numbers R, the complex numbers C, the quaternions H, and theoctonions O. This is visible in the study of superstrings, supermembranes, andsupergravity, but perhaps most simply in supersymmetric Yang–Mills theory. Inany dimension, we may consider a Yang–Mills field coupled to a massless spinortransforming in the adjoint representation of the gauge group. These fields aredescribed by the Lagrangian:

L = −1

4〈F, F 〉+ 1

2〈ψ, /DAψ〉.

Here A is a connection on a bundle with semisimple gauge group G, F is thecurvature of A, ψ is a g-valued spinor field, and /DA is the covariant Dirac operatorassociated with A. It is well-known that this theory is supersymmetric if and onlyif the dimension of spacetime is 3, 4, 6, or 10. Our goal here is to present a self-contained proof of the ‘if’ part of this result, based on the theory of normed divisionalgebras.

This result goes back to the work of Brink, Schwarz, and Sherk [3] and others.The book by Green, Schwarz and Witten [10] contains a standard proof basedon the properties of Clifford algebras in various dimensions. But Evans [7] hasshown that the supersymmetry of L in dimension n + 2 implies the existence ofa normed division algebra of dimension n. Conversely, Kugo and Townsend [12]showed how spinors in dimension 3, 4, 6, and 10 derive special properties from thenormed division algebras R, C, H and O. They formulated a supersymmetric model

c©2010 John C. Baez and John Huerta

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c©2010 American Mathematical Society

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2 JOHN C. BAEZ AND JOHN HUERTA

in 6 dimensions using the quaternions, H. They also speculated about a similarformalism in 10 dimensions using the octonions, O.

Shortly after Kugo and Townsend’s work, Sudbery [17] used division algebrasto construct vectors, spinors and Lorentz groups in Minkowski spacetimes of di-mensions 3, 4, 6, and 10. He then refined his construction with Chung [4], andwith Manogue [13] he used these ideas to give an octonionic proof of the supersym-metry of the above Lagrangian in dimension 10. This proof was later simplified byManogue, Dray and Janesky [5]. In the meantime, Schray [14] applied the sametools to the superparticle.

All this work has made it quite clear that normed division algebras explain whythe above theory is supersymmetric in dimensions 3, 4, 6, and 10. Technically, whatwe need to check for supersymmetry is that δL is a total divergence with respectto the supersymmetry transformation

δA = ε · ψδψ = 1

2Fε

for any constant spinor field ε. (We explain the notation here later; we assume noprior understanding of supersymmetry or normed division algebras.) A calculationthat works in any dimension shows that

δL = triψ + divergence

where triψ is a certain expression depending in a trilinear way on ψ and linearlyon ε.

So, the marvelous fact that needs to be understood is that triψ = 0 in dimen-sions 3, 4, 6, and 10, thanks to special properties of the normed division algebras R,C, H and O. Indeed, this fact is responsible for supersymmetry, not only for Yang–Mills fields in these dimensions, but also for superstrings! The same term triψshows up as the obstruction to supersymmetry in the Green–Schwarz Lagrangianfor classical superstrings [9, 10]. So, the vanishing of this term deserves to beunderstood: clearly, simply, and in as many ways as possible.

Unfortunately, many important pieces of the story are scattered throughoutthe literature. The treatment of Deligne and Freed [6] is self-contained, and ituses normed division algebras, but it does not use ‘purely equational reasoning’: itproves triψ = 0 by first showing that the double cover of the Lorentz group actstransitively on the set of nonzero spinors in dimensions 3, 4, 6, and 10. Whilethis geometrical argument is beautiful and insightful, a purely equational approachhas its own charm. The line of work carried out by Fairlie, Manogue, Sudbery,Dray, and collaborators [5, 8, 13, 14] has shown that the equation triψ = 0 canbe derived from the complete antisymmetry of another trilinear expression, the‘associator’

[a, b, c] = (ab)c− a(bc)

in the normed division algebra. Our desire here is to merely present this argumentas clearly as we can.

So, here we present an equational proof that triψ = 0 in dimensions 3, 4,6, and 10, based on the complete antisymmetry of the associator for the normeddivision algebras K = R, C, H and O. In Section 2 we review the properties ofnormed division algebras that we will need. In Section 3 we start by recalling howto interpret vectors as 2 × 2 hermitian matrices with entries in K, and spinors aselements of K2. We then use this language to describe the basic operations involving

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DIVISION ALGEBRAS AND SUPERSYMMETRY I 3

vectors, spinors and scalars. These include an operation that takes two spinors ψand φ and forms a vector ψ ·φ, and an operation that takes a vector A and a spinorψ and forms a spinor Aψ. In Section 4 we prove the fundamental identity thatholds only in Minkowski spaces of dimensions 3, 4, 6 and 10:

(ψ · ψ)ψ = 0.

Following Schray [14], we call this the ‘3-ψ’s rule’. In Section 5 we introduce alittle superalgebra, and explain why we should treat K as an ‘odd’, or ‘fermionic’,super vector space. In Section 6 we formulate pure super-Yang–Mills theory interms of normed division algebras, completely avoiding the use of gamma matrices.We explain how the term triψ arises as the obstruction to supersymmetry in thistheory. Finally, we use the 3-ψ’s rule to prove that triψ = 0 in dimensions 3, 4, 6and 10.

2. Normed Division Algebras

By a classic theorem of Hurwitz [11], there are only four normed divisionalgebras: the real numbers, R, the complex numbers, C, the quaternions, H, andthe octonions, O. These algebras have dimension 1, 2, 4, and 8. For an overview ofthis subject, including a Clifford algebra proof of Hurwitz’s theorem, see [1]. Herewe introduce the bare minimum of material needed to reach our goal.

A normed division algebra K is a (finite-dimensional, possibly nonassocia-tive) real algebra equipped with a multiplicative unit 1 and a norm | · | satisfying:

|ab| = |a||b|for all a, b ∈ K. Note this implies that K has no zero divisors. We will freely identifyR1 ⊆ K with R.

In all cases, this norm can be defined using conjugation. Every normed divisionalgebra has a conjugation operator—a linear operator ∗ : K → K satisfying

a∗∗ = a, (ab)∗ = b∗a∗

for all a, b ∈ K. Conjugation lets us decompose each element of K into real andimaginary parts, as follows:

Re(a) =a+ a∗

2, Im(a) =

a− a∗

2.

Conjugating changes the sign of the imaginary part and leaves the real part fixed.We can write the norm as

|a| =√aa∗ =

√a∗a.

This norm can be polarized to give an inner product on K:

(a, b) = Re(ab∗) = Re(a∗b).

The algebras R, C and H are associative. The octonions O are not. Yet theycome close: the subalgebra generated by any two octonions is associative. Anotherway to express this fact uses the associator:

[a, b, c] = (ab)c− a(bc),

a trilinear map K ⊗ K ⊗ K → K. A theorem due to Artin [15] states that for anyalgebra, the subalgebra generated by any two elements is associative if and only ifthe associator is alternating (that is, completely antisymmetric in its three argu-ments). An algebra with this property is thus called alternative. The octonions

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4 JOHN C. BAEZ AND JOHN HUERTA

O are alternative, and so of course are R, C and H: for these three the associatorsimply vanishes!

In what follows, our calculations make heavy use of the fact that all four normeddivision algebras are alternative. Besides this, the properties we require are:

Proposition 1. The associator changes sign when one of its entries is conju-gated.

Proof. Since the subalgebra generated by any two elements is associative, andreal elements of K lie in every subalgebra, [a, b, c] = 0 if any one of a, b, c is real. Itfollows that [a, b, c] = [Im(a), Im(b), Im(c)], which yields the desired result.

Proposition 2. The associator is purely imaginary.

Proof. Since (ab)∗ = b∗a∗, a calculation shows [a, b, c]∗ = −[c∗, b∗, a∗]. Byalternativity this equals [a∗, b∗, c∗], which in turn equals −[a, b, c] by the aboveproposition. So, [a, b, c] is purely imaginary.

For any square matrix A with entries in K, we define its trace tr(A) to be thesum of its diagonal entries. This trace lacks the usual cyclic property, because K

is noncommutative, so in general tr(AB) = tr(BA). Luckily, taking the real partrestores this property:

Proposition 3. Let a, b, and c be elements of K. Then

Re((ab)c) = Re(a(bc))

and this quantity is invariant under cyclic permutations of a, b, and c.

Proof. Proposition 2 implies that Re((ab)c) = Re(a(bc)). For the cyclic prop-erty, it then suffices to prove Re(ab) = Re(ba). Since (a, b) = (b, a) and the innerproduct is defined by (a, b) = Re(ab∗) = Re(a∗b), we see:

Re(ab∗) = Re(b∗a).

The desired result follows upon substituting b∗ for b.

Proposition 4. Let A, B, and C be k × , × m and m × k matrices withentries in K. Then

Re tr((AB)C) = Re tr(A(BC))

and this quantity is invariant under cyclic permutations of A, B, and C. We callthis quantity the real trace Re tr(ABC).

Proof. This follows from the previous proposition and the definition of thetrace.

The reader will have noticed three trilinears in this section: the associator[a, b, c], the real part Re((ab)c), and the real trace Re tr(ABC). This is no coinci-dence, as they all relate to the star of the show, triψ. In fact:

triψ = Re tr(ψ†(ε · ψ)ψ).

for some suitable matrices ψ†, ε · ψ and ψ. Of course, we have not yet said how toconstruct these. We turn to this now.

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DIVISION ALGEBRAS AND SUPERSYMMETRY I 5

3. Vectors, Spinors and Intertwiners

It is well-known [1, 12, 17] that given a normed division algebra K of dimensionn, one can construct (n+2)-dimensional Minkowski spacetime as the space of 2×2hermitian matrices with entries in K, with the determinant giving the Minkowskimetric. Spinors can then be described as elements of K2. Our goal here is toprovide self-contained proofs of these facts, and then develop all the basic operationsinvolving vectors, spinors and scalars using this language.

To begin, let K[m] denote the space of m×m matrices with entries in K. GivenA ∈ K[m], define its hermitian adjoint A† to be its conjugate transpose:

A† = (A∗)T .

We say such a matrix is hermitian if A = A†. Now take the 2 × 2 hermitianmatrices:

h2(K) =

(t+ x yy∗ t− x

): t, x ∈ R, y ∈ K

.

This is an (n + 2)-dimensional real vector space. Moreover, the usual formula forthe determinant of a matrix gives the Minkowski norm on this vector space:

− det

(t+ x yy∗ t− x

)= −t2 + x2 + |y|2.

We insert a minus sign to obtain the signature (n + 1, 1). Note this formula isunambiguous even if K is noncommutative or nonassociative.

It follows that the double cover of the Lorentz group, Spin(n + 1, 1), acts onh2(K) via determinant-preserving linear transformations. Since this is the ‘vector’representation, we will often call h2(K) simply V . The Minkowski metric

g : V ⊗ V → R

is given by

g(A,A) = − det(A).

There is also a nice formula for the inner product of two different vectors. Thisinvolves the trace reversal of A ∈ h2(K), introduced by Schray [14] and definedas follows:

A = A− (trA)1.

Note we indeed have tr(A) = −tr(A). Also note that

A =

(t+ x yy∗ t− x

)=⇒ A =

(−t+ x yy∗ −t− x

)

so trace reversal is really time reversal. Moreover:

Proposition 5. For any vectors A,B ∈ V = h2(K), we have

AA = AA = − det(A)1

and1

2Re tr(AB) =

1

2Re tr(AB) = g(A,B)

Proof. We check the first equation by a quick calculation. Taking the realtrace and dividing by 2 gives

1

2Re tr(AA) =

1

2Re tr(AA) = − det(A) = g(A,A).

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6 JOHN C. BAEZ AND JOHN HUERTA

Then we use the polarization identity, which says that two symmetric bilinear formsthat give the same quadratic form must be equal.

Next we consider spinors. As real vector spaces, the spinor representations S+

and S− are both just K2. However, they differ as representations of Spin(n+ 1, 1).To construct these representations, we begin by defining ways for vectors to act onspinors:

γ : V ⊗ S+ → S−A⊗ ψ → Aψ.

andγ : V ⊗ S− → S+

A⊗ ψ → Aψ.

We can also think of these as maps that send elements of V to linear operators:

γ : V → Hom(S+, S−),γ : V → Hom(S−, S+).

Here a word of caution is needed: since K may be nonassociative, 2 × 2 matriceswith entries in K cannot be identified with linear operators on K2 in the usual way.They certainly induce linear operators via left multiplication:

LA(ψ) = Aψ.

Indeed, this is how γ and γ turn elements of V into linear operators:

γ(A) = LA,γ(A) = LA.

However, because of nonassociativity, composing such linear operators is differentfrom multiplying the matrices:

LALB(ψ) = A(Bψ) = (AB)ψ = LAB(ψ).

Since vectors act on elements of S+ to give elements of S− and vice versa, theymap the space S+⊕S− to itself. This gives rise to an action of the Clifford algebraCliff(V ) on S+ ⊕ S−:

Proposition 6. The vectors V = h2(K) act on the spinors S+⊕S− = K2⊕K2

via the map

Γ: V → End(S+ ⊕ S−)

given by

Γ(A)(ψ, φ) = (Aφ, Aψ).

Furthermore, Γ(A) satisfies the Clifford algebra relation:

Γ(A)2 = g(A,A)1

and so extends to a homomorphism Γ: Cliff(V ) → End(S+ ⊕ S−), i.e. a represen-tation of the Clifford algebra Cliff(V ) on S+ ⊕ S−.

Proof. Suppose A ∈ V and Ψ = (ψ, φ) ∈ S+ ⊕ S−. We need to check that

Γ(A)2(Ψ) = − det(A)Ψ.

Here we must be mindful of nonassociativity: we have

Γ(A)2(Ψ) = (A(Aψ), A(Aφ)).

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DIVISION ALGEBRAS AND SUPERSYMMETRY I 7

Yet it is easy to check that the expressions A(Aψ) and A(Aφ) involve multiplyingat most two different nonreal elements of K. These associate, since K is alternative,so in fact

Γ(A)2(Ψ) = ((AA)ψ, (AA)φ).

To conclude, we use Proposition 5.

The action of a vector swaps S+ and S−, so acting by vectors twice sends S+

to itself and S− to itself. This means that while S+ and S− are not modules for theClifford algebra Cliff(V ), they are both modules for the even part of the Cliffordalgebra, generated by products of pairs of vectors. The group Spin(n + 1, 1) livesin this even part: as is well-known, it is generated by products of pairs of unitvectors in V : that is, vectors A with g(A,A) = ±1. As a result, S+ and S− areboth representations of Spin(n+ 1, 1).

While we will not need this in what follows, one can check that:

• WhenK = R, S+∼= S− is the Majorana spinor representation of Spin(2, 1).

• WhenK = C, S+∼= S− is the Majorana spinor representation of Spin(3, 1).

• WhenK = H, S+ and S− are the Weyl spinor representations of Spin(5, 1).• When K = O, S+ and S− are the Majorana–Weyl spinor representationsof Spin(9, 1).

This counts as a consistency check, because these are precisely the kinds of spinorrepresentations that go into pure super-Yang–Mills theory. But it is important tonote that the differences between these spinor representations are irrelevant to ourargument. What matters is how they are the same—they can all be defined on K2.

Now that we have representations of Spin(n+ 1, 1) on V , S+ and S−, we needto develop the Spin(n + 1, 1)-equivariant maps that relate them. Ultimately, todefine the Lagrangian for pure super-Yang–Mills theory, we need:

• An invariant pairing:

〈−,−〉 : S+ ⊗ S− → R.

• An equivariant map that turns pairs of spinors into vectors:

· : S± ⊗ S± → V.

Another name for an equivariant map between group representations is an ‘inter-twining operator’. As a first step, we show that the action of vectors on spinors isitself an intertwining operator:

Proposition 7. The maps

γ : V ⊗ S+ → S−A⊗ ψ → Aψ

andγ : V ⊗ S− → S+

A⊗ ψ → Aψ

are equivariant with respect to the action of Spin(n+ 1, 1).

Proof. Both γ and γ are restrictions of the map

Γ: V ⊗ (S+ ⊕ S−) → S+ ⊕ S−,

so it suffices to check that Γ is equivariant. In fact, we will do this not just forthe action of Spin(n+ 1, 1), but for the larger group Pin(n+ 1, 1), the subgroup of

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8 JOHN C. BAEZ AND JOHN HUERTA

Cliff(V ) generated by unit vectors in V . This group acts on V with unit vectorsacting by conjugation on V ⊆ Cliff(V ). It acts on S+ ⊕ S− with a unit vector Bacting by Γ(B). Both of these representations of Pin(n+ 1, 1) restrict to the usualrepresentations of Spin(n+ 1, 1).

Thus, we compute:

Γ(BAB−1)Γ(B)Ψ = Γ(B)(Γ(A)Ψ).

Here it is important to note that the conjugation BAB−1 is taking place in theassociative algebra Cliff(V ), not in the algebra of matrices. This equation saysthat Γ is indeed Pin(n+ 1, 1)-equivariant, as claimed.

Now we exhibit the key tool: the pairing between S+ and S−:

Proposition 8. The pairing

〈−,−〉 : S+ ⊗ S− → R

ψ ⊗ φ → Re(ψ†φ)

is invariant under the action of Spin(n+ 1, 1).

Proof. Given A ∈ V , we use the fact that the associator is purely imaginaryto show that

Re((Aφ)†(Aψ)

)= Re

((φ†A)(Aψ)

)= Re

(φ†(A(Aψ))

).

As in the proof of the Clifford relation, it is easy to check that the column vectorA(Aψ) involves at most two nonreal elements of K and equals g(A,A)ψ. So:

〈γ(A)φ, γ(A)ψ〉 = g(A,A)〈ψ, φ〉.In particular when A is a unit vector, acting by A swaps the order of ψ and φand changes the sign at most. 〈−,−〉 is thus invariant under the group in Cliff(V )generated by products of pairs of unit vectors, which is Spin(n+ 1, 1).

With this pairing in hand, there is a manifestly equivariant way to turn a pairof spinors into a vector. Given ψ, φ ∈ S+, there is a unique vector ψ ·φ whose innerproduct with any vector A is given by

g(ψ · φ,A) = 〈ψ, γ(A)φ〉.Similarly, given ψ, φ ∈ S−, we define ψ · φ ∈ V by demanding

g(ψ · φ,A) = 〈γ(A)ψ, φ〉for all A ∈ V . This gives us maps

S± ⊗ S± → V

which are manifestly equivariant.On the other hand, because S± = K2 and V = h2(K), there is also a naive way

to turn a pair of spinors into a vector using matrix operations: just multiply thecolumn vector ψ by the row vector φ† and then take the hermitian part:

ψφ† + φψ† ∈ h2(K),

or perhaps its trace reversal:

˜ψφ† + φψ† ∈ h2(K).

In fact, these naive guesses match the manifestly equivariant approach describedabove:

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DIVISION ALGEBRAS AND SUPERSYMMETRY I 9

Proposition 9. The maps · : S± ⊗ S± → V are given by:

· : S+ ⊗ S+ → V

ψ ⊗ φ → ˜ψφ† + φψ†

· : S− ⊗ S− → Vψ ⊗ φ → ψφ† + φψ†.

These maps are equivariant with respect to the action of Spin(n+ 1, 1).

Proof. First suppose ψ, φ ∈ S+. We have already seen that the map· : S+ ⊗ S+ → V is equivariant. We only need to show that this map has thedesired form. We start by using some definitions:

g(ψ · φ,A) = 〈ψ, γ(A)φ〉 = Re(ψ†(Aφ)) = Re tr(ψ†Aφ).

We thus haveg(ψ · φ,A) = Re tr(ψ†Aφ) = Re tr(φ†Aψ),

where in the last step we took the adjoint of the inside. Applying the cyclic propertyof the real trace, we obtain

g(ψ · φ,A) = Re tr(φψ†A) = Re tr(ψφ†A).

Averaging gives

g(ψ · φ,A) =1

2Re tr((ψφ† + φψ†)A).

On the other hand, Proposition 5 implies that

g(ψ · φ,A) =1

2Re tr((ψ · φ)A).

Since both these equations hold for all A, we must have

ψ · φ = ψφ† + φψ†.

Doing trace reversal twice gets us back where we started, so

ψ · φ = ˜ψφ† + φψ†

as desired. A similar calculation shows that if ψ, φ ∈ S−, then ψ·φ = ψφ†+φψ†.

Map Division algebra notation Index notation

g : V ⊗ V → R 12Re tr(AB) AμBμ

γ : V ⊗S+→S− Aψ γμAμψ

γ : V ⊗S−→S+ Aψ γμAμψ

· : S+⊗S+→ V ˜ψφ† + φψ† ψγμφ

· : S−⊗S−→ V ψφ† + φψ† ψγμφ

〈−,−〉 : S+⊗S−→ R Re(ψ†φ) ψφ

Table 1. Division algebra notation vs. index notation

We can summarize our work so far with a table of the basic bilinear maps involvingvectors, spinors and scalars. Table 1 shows how to translate between divisionalgebra notation and something more closely resembling standard physics notation.

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10 JOHN C. BAEZ AND JOHN HUERTA

In this table the adjoint spinor ψ denotes the spinor dual to ψ under the pairing〈−,−〉. The gamma matrix γμ denotes a Clifford algebra generator acting on S+,while γμ denotes the same element acting on S−. Of course γ is not standardphysics notation; the standard notation for this depends on which of the four caseswe are considering: R, C, H or O.

4. The 3-ψ’s Rule

Now we prove the fundamental identity that makes supersymmetry tick indimensions 3, 4, 6, and 10. This identity was dubbed the ‘3-ψ’s rule’ by Schray[14]. The following proof is based on an argument in the appendix of the paper byDray, Janesky and Manogue [5]. Note that it is really the alternative law, ratherthan the normed division algebra axioms, that does the job:

Theorem 10. Suppose ψ ∈ S+. Then (ψ ·ψ)ψ = 0. Similarly, if φ ∈ S−, then

(φ · φ)φ = 0.

Proof. Suppose ψ ∈ S+. By definition,

(ψ · ψ)ψ = 2(ψψ†)ψ = 2(ψψ† − tr(ψψ†)1)ψ.

It is easy to check that tr(ψψ†) = ψ†ψ, so

(ψ · ψ)ψ = 2((ψψ†)ψ − (ψ†ψ)ψ).

Since ψ†ψ is a real number, it commutes with ψ:

(ψ · ψ)ψ = 2((ψψ†)ψ − ψ(ψ†ψ)).

Since K is alternative, every subalgebra of K generated by two elements is asso-ciative. Since ψ ∈ K2 is built from just two elements of K, the right-hand sidevanishes. The proof of the identity for φ ∈ S− is similar.

It will be useful to state this result in a somewhat more elaborate form. Tosave space we only give this version for spinors in S+, though an analogous resultholds for spinors in S−:

Theorem 11. Define a map

T : S+ ⊗ S+ ⊗ S+ → S−ψ ⊗ φ⊗ χ → (ψ · φ)χ+ (φ · χ)ψ + (χ · ψ)φ.

Then T = 0.

Proof. It is easy to check that ψ · φ = φ · ψ for all ψ, φ ∈ S+, so the mapT is completely symmetric in its three arguments. Just as any symmetric bilin-ear form B(x, y) can be recovered from the corresponding quadratic form B(x, x)by polarization, so too can any symmetric trilinear form be recovered from thecorresponding cubic form. Since T (ψ, ψ, ψ) = 0 by Theorem 10, it follows thatT = 0.

To see how this theorem is the key to supersymmetry for super-Yang–Millstheory, we need a little superalgebra.

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DIVISION ALGEBRAS AND SUPERSYMMETRY I 11

5. Superalgebra

So far we have used normed division algebras to construct a number of alge-braic structures: vectors as elements of h2(K), spinors as elements of K2, and thevarious bilinear maps involving vectors, spinors, and scalars. However, to describesupersymmetry, we also need superalgebra. Specifically, we need anticommutingspinors. Physically, this is because spinors are fermions, so we need them to satisfyanticommutation relations. Mathematically, this means that we will do our algebrain the category of ‘super vector spaces’, SuperVect, rather than the category ofvector spaces, Vect.

A super vector space is a Z2-graded vector space V = V0 ⊕ V1 where V0 iscalled the even or bosonic part, and V1 is called the odd or fermionic part. LikeVect, SuperVect is a symmetric monoidal category [2]. It has:

• Z2-graded vector spaces as objects;• Grade-preserving linear maps as morphisms;• A tensor product ⊗ that has the following grading: if V = V0 ⊕ V1 and

W = W0⊕W1, then (V ⊗W )0 = (V0⊗W0)⊕ (V1 ⊗W1) and (V ⊗W )1 =(V0 ⊗W1)⊕ (V1 ⊗W0);

• A braiding

BV,W : V ⊗W → W ⊗ V

defined as follows: v ∈ V and w ∈ W are of grade p and q, then

BV,W (v ⊗ w) = (−1)pqw ⊗ v.

The braiding encodes the ‘the rule of signs’: in any calculation, when two oddelements are interchanged, we introduce a minus sign.

In what follows we treat the normed division algebra K as an odd super vectorspace. This turns out to force the spinor representations S± to be odd and thevector representation V to be even, as follows.

There is an obvious notion of direct sums for super vector spaces, with

(V ⊕W )0 = V0 ⊕W0, (V ⊕W )1 = V1 ⊕W1

and also an obvious notion of duals, with

(V ∗)0 = (V0)∗, (V ∗)1 = (V1)

∗.

We say a super vector space V is even if it equals its even part (V = V0), and oddif it equals its odd part (V = V1). Any subspace U ⊆ V of an even (resp. odd)super vector space becomes a super vector space which is again even (resp. odd).

We treat the spinor representations S± as super vector spaces using the factthat they are the direct sum of two copies of K. Since K is odd, so are S+ and S−.Since K2 is odd, so is its dual. This in turn forces the space of linear maps fromK2 to itself, End(K2) = K2 ⊗ (K2)∗, to be even. This even space contains the 2× 2matrices K[2] as the subspace of maps realized by left multiplication:

K[2] → End(K2)A → LA.

K[2] is thus even. Finally, this forces the subspace of hermitian 2 × 2 matrices,h2(K), to be even. So, the vector representation V is even. All this matches theusual rules in physics, where spinors are fermionic and vectors are bosonic.

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12 JOHN C. BAEZ AND JOHN HUERTA

6. Super-Yang–Mills Theory

We are now ready to give a division algebra interpretation of the pure super-Yang–Mills Lagrangian

L = −1

4〈F, F 〉+ 1

2〈ψ, /DAψ〉

and use this to prove its supersymmetry. For simplicity, we shall work over Minkowskispacetime, M . This allows us to treat all bundles as trivial, sections as functions,and connections as g-valued 1-forms.

At the outset, we fix an invariant inner product on g, the Lie algebra of asemisimple Lie group G. We shall use the following standard tools from differentialgeometry to construct L, none of which need involve spinors or division algebratechnology:

• A connection A on a principal G-bundle over M . Since the bundle istrivial we think of this connection as a g-valued 1-form.

• The exterior covariant derivative dA = d+ [A,−] on g-valued p-forms.• The curvature F = dA+ 1

2 [A,A], which is a g-valued 2-form.• The usual pointwise inner product 〈F, F 〉 on g-valued 2-forms, definedusing the Minkowski metric on M and the invariant inner product on g.

We also need the following spinorial tools. Recall from the preceding section thatS+ and S− are odd objects in SuperVect. So, whenever we switch two spinors, weintroduce a minus sign.

• A g-valued section ψ of a spin bundle over M . Note that this is, in fact,just a function:

ψ : M → S± ⊗ g.

We call the collection of all such functions Γ(S± ⊗ g).• The covariant Dirac operator /DA derived from the connection A. Ofcourse,

/DA : Γ(S± ⊗ g) → Γ(S∓ ⊗ g)

and in fact,

/DA = /∂ +A.

• A bilinear pairing

〈−,−〉 : Γ(S+ ⊗ g)⊗ Γ(S− ⊗ g) → C∞(M)

built pointwise using our pairing

〈−,−〉 : S+ ⊗ S− → R

and the invariant inner product on g.

The basic fields in our theory are a connection on a principal G-bundle, whichwe think of as a g-valued 1-form:

A : M → V ∗ ⊗ g.

and a g-valued spinor field, which we think of as a S+ ⊗ g-valued function on M :

ψ : M → S+ ⊗ g.

All our arguments would work just as well with S− replacing S+.

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DIVISION ALGEBRAS AND SUPERSYMMETRY I 13

To show that L is supersymmetric, we need to show δL is a total divergencewhen δ is the following supersymmetry transformation:

δA = ε · ψ

δψ =1

2Fε

where ε is an arbitrary constant spinor field, treated as odd, but not g-valued. Bya supersymmetry transformation we mean that computationally we treat δ asa derivation. So, it is linear:

δ(αf + βg) = αδf + βδg

where α, β ∈ R, and it satisfies the product rule:

δ(fg) = δ(f)g + fδg.

For a more formal definition of ‘supersymmetry transformation’ see [6].The above equations require further explanation. The dot in ε · ψ denotes

an operation that combines the spinor ε with the g-valued spinor ψ to produce ag-valued 1-form. We build this from our basic intertwiner

· : S+ ⊗ S+ → V.

We identify V with V ∗ using the Minkowski inner product g, obtaining

· : S+ ⊗ S+ → V ∗.

Then we tensor both sides with g. This gives us a way to act by a spinor fieldon a g-valued spinor field to obtain a g-valued 1-form. We take the liberty of alsodenoting this with a dot:

· : Γ(S+)⊗ Γ(S+ ⊗ g) → Ω1(M, g).

We also need to explain how the 2-form F acts on the constant spinor field ε.Using the Minkowski metric, we can identify differential forms on M with sectionsof the Clifford algebra bundle over M :

Ω∗(M) ∼= Cliff(M).

Using this, differential forms act on spinor fields. Tensoring with g, we obtain a wayfor g-valued differential forms like F to act on spinor fields like ε to give g-valuedspinor fields like Fε.

Let us now apply the supersymmetry transformation to each term in the La-grangian. First, the bosonic term:

Proposition 12. The bosonic term has:

δ〈F, F 〉 = 2(−1)n+1 〈ψ, ( dA F )ε〉+ divergence.

Proof. By the symmetry of the inner product, we get:

δ〈F, F 〉 = 2〈F, δF 〉.Using the handy formula δF = dAδA, we have:

〈F, δF 〉 = 〈F, dAδA〉.Now the adjoint of the operator dA is dA , up to a pesky sign: if ν is a g-valued(p− 1)-form and μ is a g-valued p-form, we have

〈μ, dAν〉 = (−1)dp+d+1+s〈 dA μ, ν〉+ divergence

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14 JOHN C. BAEZ AND JOHN HUERTA

where d is the dimension of spacetime and s is the signature, i.e., the number ofminus signs in the diagonalized metric. It follows that

〈F, δF 〉 = 〈F, dAδA〉 = (−1)n 〈 dA F, δA〉+ divergence

where n is the dimension of K. By the definition of δA, we get

〈 dA F, δA〉 = 〈 dA F, ε · ψ〉.Now we can use division algebra technology to show:

〈 dA F, ε · ψ〉 =1

2Re tr

(( dA F )(εψ† + ψε†)

)= −〈ψ, ( dA F )ε〉,

using the cyclic property of the real trace in the last step, and introducing a minussign in accordance with the sign rule. Putting everything together, we obtain thedesired result.

Even though this proposition involved the bosonic term only, division algebratechnology was still a useful tool in its proof. This is even more true in the nextproposition, which deals with the the fermionic term:

Proposition 13. The fermionic term has:

δ〈ψ, /DAψ〉 = 〈ψ, /DA(Fε)〉+ triψ + divergence

where

triψ = 〈ψ, (ε · ψ)ψ〉.

Proof. It is easy to compute:

δ〈ψ, /DAψ〉 = 〈δψ, /DAψ〉+ 〈ψ, δ /DAψ〉+ 〈ψ, /DAδψ〉.Now we insert δ /DA = δA = ε · ψ, and thus see that the penultimate term is thetrilinear one:

triψ = 〈ψ, (ε · ψ)ψ〉.So, let us concern ourselves with the remaining terms:

〈δψ, /DAψ〉+ 〈ψ, /DAδψ〉.A computation using the product rule shows that the divergence of the 1-form ψ ·φis given by −〈φ, /DAψ〉 + 〈ψ, /DAφ〉, where the minus sign on the first term arisesfrom using the sign rule with these odd spinors. In the terms under consideration,we can use this identity to move /DA onto δψ:

〈δψ, /DAψ〉+ 〈ψ, /DAδψ〉 = 2〈ψ, /DAδψ〉+ divergence.

Substituting δψ = 12Fε, we obtain the desired result.

Using these two propositions, it is immediate that

δL = −1

4δ〈F, F 〉+ 1

2δ〈ψ, /DAψ〉

=1

2(−1)n〈ψ, ( dA F )ε〉+ 1

2〈ψ, /DA(Fε)〉+ 1

2triψ + divergence.

All that remains to show is that /DA(Fε) = (−1)n+1( dA F ) ε. Indeed, Snyggshows (Eq. 7.6 in [16]) that for an ordinary, non-g-valued p-form F

/∂(Fε) = (dF )ε+ (−1)d+dp+s( d F )ε

7878

DIVISION ALGEBRAS AND SUPERSYMMETRY I 15

where d is the dimension of spacetime and s is the signature. This is easily gener-alized to covariant derivatives and g-valued p-forms:

/DA(Fε) = (dAF )ε+ (−1)d+dp+s( dA F )ε.

In particular, when F is the curvature 2-form, the first term vanishes by the Bianchiidentity dAF = 0, and we are left with:

/DA(Fε) = (−1)n+1( dA F )ε

where n is the dimension of K. We have thus shown:

Proposition 14. Under supersymmetry transformations, the Lagrangian Lhas:

δL =1

2triψ + divergence.

The above result actually holds in every dimension, though our proof useddivision algebras and was thus adapted to the dimensions of interest: 3, 4, 6, and10. The next result is where division algebra technology becomes really crucial:

Proposition 15. For Minkowski spacetimes of dimensions 3, 4, 6, and 10,triψ = 0.

Proof. At each point, we can write

ψ =∑

ψa ⊗ ga,

where ψa ∈ S+ and ga ∈ g. When we insert this into triψ, we see that

triψ =∑

〈ψa, (ε · ψb)ψc〉 〈ga, [gb, gc]〉.

Since 〈ga, [gb, gc]〉 is totally antisymmetric, this implies triψ = 0 for all ε if and onlyif the part of 〈ψa, (ε · ψb)ψc〉 that is antisymmetric in a, b and c vanishes for all ε.Yet these spinors are odd; for even spinors, we require the part of 〈ψa, (ε · ψb)ψc〉that is symmetric in a, b and c to vanish for all ε.

Now let us bring in some division algebra technology to remove our dependenceon ε. While we do this, let us replace ψa with ψ, ψb with φ, and ψc with χ to lessenthe clutter of indices. Substituting in the formulas from Table 1, we have

〈ψ, (ε · φ)χ〉 = Re(ψ†( ˜εφ† + φε†)χ)

= Re tr(ψ†(εφ† + φε† − ε†φ− φ†ε)χ)

= 〈ε, (ψ · χ)φ〉,where again we have employed the cyclic symmetry of the real trace, along withthe identity:

tr(εφ† + φε†) = Re tr(εφ† + φε†) = φ†ε+ ε†φ.

This real quantity commutes and associates in any expression. So, if we seek toshow that the part of 〈ψ, (ε · φ)χ〉 that is totally symmetric in ψ, φ and χ vanishesfor all ε, it is equivalent to show the totally symmetric part of (φ · χ)ψ vanishes.And since the dot operation in φ · χ is symmetric, this follows immediately fromour main result, Theorem 11.

Acknowledgements. We thank Geoffrey Dixon, Tevian Dray, and CorinneManogue for helpful conversations and correspondence. We also thank An Huang,Theo Johnson-Freyd and David Speyer for catching some errors. This work waspartially supported by an FQXi grant.

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16 JOHN C. BAEZ AND JOHN HUERTA

References

1. J. C. Baez, The octonions, Bull. Amer. Math. Soc. 39 (2002), 145–205. Also available asarXiv:math/0105155.

2. J. C. Baez and M. Stay, Physics, topology, logic and computation: a Rosetta Stone, to appearin New Structures For Physics, ed. Bob Coecke. Also available as arXiv:0903.0340.

3. L. Brink, J. Schwarz and J. Scherk, Supersymmetric Yang–Mills theory, Nucl. Phys. B121(1977), 77–92.

4. K.-W. Chung and A. Sudbery, Octonions and the Lorentz and conformal groups of ten-dimensional space-time, Phys. Lett. B 198 (1987), 161–164.

5. T. Dray, J. Janesky and C. A. Manogue, Octonionic hermitian matrices with non-real eigenval-ues, Adv. Appl. Clifford Algebras 10 (2000), 193–216. Also available as arXiv:math/0006069.

6. P. Deligne et al, eds., Quantum Fields and Strings: A Course for Mathematicians, Volume1, Amer. Math. Soc., Providence, Rhode Island, 1999.

7. J. M. Evans, Supersymmetric Yang–Mills theories and division algebras, Nucl. Phys. B298(1988), 92–108. Also available as 〈http://ccdb4fs.kek.jp/cgi-bin/img index?8801412〉.

8. D. B. Fairlie and C. A. Manogue, A parameterization of the covariant superstring, Phys. Rev.

D36 (1987), 475–479.9. M. Green and J. Schwarz, Covariant description of superstrings, Phys. Lett. B136 (1984),

367–370.10. M. Green, J. Schwarz and E. Witten, Superstring Theory, Volume 1, Cambridge U. Press,

Cambridge, 1987. Appendix 4.A: Super Yang–Mills theories, pp. 244–247. Section 5.1.2: Thesupersymmetric string action, pp. 253–255.

11. A. Hurwitz, Uber die Composition der quadratischen Formen von beliebig vielen Variabeln,Nachr. Ges. Wiss. Gottingen (1898), 309–316.

12. T. Kugo and P. Townsend, Supersymmetry and the division algebras, Nucl. Phys. B221(1983), 357–380. Also available at 〈http://ccdb4fs.kek.jp/cgi-bin/img index?198301032〉.

13. C. A. Manogue and A. Sudbery, General solutions of covariant superstring equations of motion,Phys. Rev. D 12 (1989), 4073–4077.

14. J. Schray, The general classical solution of the superparticle, Class. Quant. Grav. 13 (1996),27–38. Also available as arXiv:hep-th/9407045.

15. R. D. Schafer, Introduction to Non-Associative Algebras, Dover, New York, 1995.16. J. Snygg, Clifford Algebra: a Computational Tool for Physicists, Oxford U. Press, Oxford,

1997.17. A. Sudbery, Division algebras, (pseudo)orthogonal groups and spinors, Jour. Phys. A17

(1984), 939–955.

Department of Mathematics, University of California, Riverside, CA 92521 USA

E-mail address: baez@math.ucr.edu

Department of Mathematics, University of California, Riverside, CA 92521 USA

E-mail address: huerta@math.ucr.edu

8080

Proceedings of Symposia in Pure Mathematics

K-homology and D-branes

Paul Baum

1. Introduction

K-homology is the dual theory to K-theory. In algebraic geometry [14] [7],the K-homology of a (possibly singular) projective variety X is the Grothendieckgroup of coherent algebraic sheaves on X. In topology, there are three ways todefine K-homology. First, K-homology is the homology theory determined by theBott spectrum. Second, K-homology is the group of geometric K-cycles introducedby Baum-Douglas [6]. Third, using functional analysis, K-homology is the groupof abstract elliptic operators as in the work of M. F. Atiyah [1], Brown-Douglas-Fillmore [15], and G. Kasparov [21].

The D-branes of string theory [31] are twisted geometric K-cycles which areendowed with some additional structure. The charge of a D-brane is the elementin the twisted K-homology of spacetime determined by the underlying twisted K-cycle of the D-brane. Essentially, the Baum-Douglas theory [6] was rediscovered interms of constraints on open strings. The aim of this expository note is to brieflydescribe this development.

Thanks go to the referee and to Serge Ballif for helpful comments and technicalassistance.

2. K-cycles and K-homology

Let X be a locally compact Hausdorff topological space. For example, X canbe a locally finite CW-complex.

A K-cycle on X is a triple (M,E,ϕ) such that:

• M is a compact Spinc manifold (without boundary).• E is a C vector bundle on M .• ϕ : M → X is a continuous map from M to X.

On the collection of allK-cycles onX impose the equivalence relation generatedby the three elementary moves:

• bordism• direct sum-disjoint union• vector bundle modification

PB was partially supported by an NSF grant.

c©0000 (copyright holder)

1

Proceedings of Symposia in Pure MathematicsVolume 81, 2010

c©2010 American Mathematical Society

81

2 PAUL BAUM

Precise definitions of these three elementary moves are as follows.Isomorphism. Two K-cycles (M,E,ϕ), (M ′, E′, ϕ′) on X are isomorphic if

there is a diffeomorphism f : M → M ′ such that f preserves the Spinc structureson M and M ′, the C vector bundles E, f∗(E′) are isomorphic, and the diagram

Mf

ϕ

M ′

ϕ′

X

commutes.Bordism. Two K-cycles (M0, E0, ϕ0) and (M1, E1, ϕ1) are bordant if there

exists a triple (W,E,Ψ) such that W is a compact Spinc manifold with boundary,E is a C vector bundle on W , Ψ: W → X is a continuous map from W to X and

(∂W,E|∂W,Ψ|∂W ) ∼= (M0, E0, ϕ0) ∪ (−M1, E1, ϕ1).

Note that here ∂W is given the Spinc structure that it receives from W and −M1

denotes M1 with its Spinc structure reversed. ∪ is disjoint union.Direct sum-disjoint union. Let (M,E,ϕ) be a K-cycle on X and let E′ be

a C vector bundle on M , then

(M,E,ϕ) ∪ (M,E′, ϕ) ∼ (M,E ⊕ E′, ϕ).

Vector bundle modification. Let (M,E,ϕ) be a K-cycle on X and let Fbe a Spinc vector bundle on M . Assume that F has even dimensional fibers. Then

(M,E,ϕ) ∼ (S(F ⊕ θ1), β ⊗ ρ∗E,ϕ ρ).Here θ1 is the trivial R line bundle on M (θ1 = M × R), S(F ⊕ θ1) is the unitsphere bundle of F ⊕θ1, and ρ : S(F ⊕θ1) → M is the projection of S(F ⊕θ1) ontoM . Since F has even dimensional fibers, S(F ⊕ θ1) is a sphere bundle over M witheven dimensional spheres as fibers. S(F ⊕ θ1) is a Spinc manifold as it is given theSpinc structure resulting from the Spinc structure of M and the Spinc structure ofF . β is the Thom isomorphism C vector bundle on S(F ⊕ θ1) determined by theSpinc structure of F . When restricted to any fiber of ρ : S(F ⊕ θ1) → M , β is theBott generator vector bundle of that even dimensional sphere.

Denote the collection of all K-cycles on X by (M,E,ϕ). The K-homologyof X, denoted K∗(X), is

K∗(X) := (M,E,ϕ)/ ∼ .

K∗(X) is an abelian group. Addition is disjoint union.

(M,E,ϕ) + (M ′, E′, ϕ′) = (M ∪M ′, E ∪ E′, ϕ ∪ ϕ′).

The negative of (M,E,ϕ) is (−M,E,ϕ). As above, −M is M with its Spinc

structure reversed. The zero element of K∗(X) is given by any (M,E,ϕ) whichbounds.

With j = 0, 1 let Kj(X) be the subgroup of K∗(X) given by all (M,E,ϕ) suchthat every connected component of M has its dimension congruent to j modulo 2.Then:

K∗(X) = K0(X)⊕K1(X)

82

K-HOMOLOGY AND D-BRANES 3

Let (M,E,ϕ) be a K-cycle on X such that every connected component of Mhas its dimension congruent to j modulo 2. DE denotes the Dirac operator of Mtensored with E. DE yields an element of the Kasparov [21] K-homology groupKKj(C(M),C).

[DE ] ∈ KKj(C(M),C).

The map of C∗ algebras C0(X) → C(M) given by ϕ : M → X induces a homomor-phism of abelian groups

KKj(C(M),C) → KKj(C0(X),C)

As usual [29] C(M) is the C∗ algebra of all continuous complex-valued functionson M , and C0(X) is the C∗ algebra of all continuous complex-valued vanishing-at-infinity functions on X. Denote by ϕ∗[DE ] the element of KKj(C0(X),C) obtainedby applying this homomophism to [DE ] ∈ KKj(C(M),C)

Then (M,E,ϕ) → ϕ∗[DE ] is a homomophism of abelian groups mappingKj(X) to KKj(C0(X),C). Denote this by

η : Kj(X) → KKj(C0(X),C).

The Kasparov K-homology of X , KKj(C0(X),C), is not a compactly sup-ported theory. By a slight abuse of notation, KKj

c (C0(X),C) will denote the directlimit over all compact subsets Δ of X of KKj(C(Δ),C),

KKjc (C0(X),C) := lim−→

Δ⊂XΔ compact

KKj(C(Δ),C).

Thus KK∗c (C0(X),C) is Kasparov K-homology with compact supports. For any

compact subset Δ of X, the inclusion Δ ⊂ X gives a homomorphism of abeliangroups

KKj(C(Δ),C) → KKj(C0(X),C).

These fit together to yield a homomorphism of abelian groups :

KKjc (C0(X),C) → KKj(C0(X),C).

It is immediate that

η : Kj(X) → KKj(C0(X),C)).

factors through KKjc (C0(X),C), so we obtain :

η : Kj(X) → KKjc (C0(X),C).

Theorem (P. Baum + R. Douglas [6], P. Baum + N. Higson + T. Schick [9],P. Baum + N. Higson + T. Schick [10], P. Baum + H. Oyono-Oyono + T. Schick[11]). Let X be a locally finite CW complex. Then

η : Kj(X) → KKjc (C0(X),C)

is an isomorphism of abelian groups. j = 0, 1.

Remark. For X any CW complex, there is a natural isomorphism of K∗(X)to the K-homology of X defined, as in homotopy theory, via the Bott spectrum.Denote homotopy theory K-homology by Kh

∗ (X). To explicitly describe the isomor-phism

Kh∗ (X) ∼= K∗(X)

it will be convenient to recall the

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Lemma 1. Let B be a Banach algebra with unit 1B. Then the open unit ballin B centered at 1B consists of invertible elements.

For a proof see any introductory text on Banach Algebras, e.g. [22]. To definethe isomorphism of abelian groups

Kh∗ (X) −→ K∗(X)

proceed as follows. Let H be a separable (but not finite dimensional) Hilbert space.L(H) denotes the C∗ algebra of all bounded operators T : H → H. Within L(H)there is the open set Fred(H) of all Fredholm operators. According to [2] and [19],Fred(H) can be taken to be Z×BU. So starting with a continuous map

f : Sn −→ (X × Fred(H))/(X × I)

a K-cycle on X must be constructed. Here Sn is the n-sphere, I is the identityoperator of H and (X×Fred(H))/(X×I) is X×Fred(H) with X×I collapsed to apoint. This collapsing creates the basepoint, denoted p0, in (X×Fred(H))/(X×I).The map f is assumed to send the basepoint of Sn to p0.

Within Sn set

Σ = f−1(p0)

Choose an open set U in Sn such that U contains Σ and U has a smooth boundary.

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K-HOMOLOGY AND D-BRANES 5

Thus Sn − U is a (codimension zero) smooth submanifold with boundary in Sn.Set Ω = Sn − U and consider the restriction of f to Ω.

f : Ω −→ (X × Fred(H))/(X × I).

Composing with the projections of X × Fred(H) onto X and Fred(H) then giveswell-defined continuous maps

fX : Ω −→ X

fFred(H) : Ω −→ Fred(H).

Let M be the manifold obtained by ”doubling” Ω along ∂Ω. Thus M is formedby taking the disjoint union of two copies of Ω and then identifying the two copiesof ∂Ω,

M = Ω ∪∂Ω Ω.

M is a closed manifold and is a π manifold , i.e. the tangent bundle TM isstably trivial. Hence a fortiori M is a compact Spinc manifold without boundary.Let

ϕ : M −→ X

be the composition

M = Ω ∪∂Ω Ω −→ Ω −→ X

where the first map is the standard map of a ”doubled” space back to the originalspace and the second map is fX .

Now use fFred(H) and the above lemma to define a continuous map

Ψ: M = Ω ∪∂Ω Ω −→ Fred(H).

On the first copy of Ω, Ψ is fFred(H). The normal bundle ν of ∂Ω in the secondcopy of Ω is trivial. So there is the emdedding

∂Ω× [0, 1] → second copy of Ω.

The open set U above can be chosen small enough so that fFred(H) maps ∂U = ∂Ωto the open unit ball in L(H) centered at I. The above lemma then applies to givethe evident linear homotopy from the restrriction of fFred(H) to ∂Ω to the constantmap sending all of ∂Ω to I. Ψ on ∂Ω × [0, 1] is this homotopy. On the remainderof the second copy of Ω, Ψ is the constant map sending every point to I.

Since Fred(H) is Z × BU , Ψ determines an element in the K-theory of M .Denote this K-theory element by E − F where E and F are C vector bundles onM . Then the required K-cycle on X is

(M,E,ϕ) (−M,F, ϕ)

where is disjoint union and −M is M with its Spinc structure reversed.The inverse homomorphism of abelian groups

Kh∗ (X) ←− K∗(X)

is defined as follows. Given a K-cycle (M,E,ϕ) on X, embed M in Rn with evencodimension. On the normal bundle of the embedding, the pull-backs of the two12 -Spin vector bundles associated to the normal bundle become isomorphic (viaClifford multiplication) off the zero section of the normal bundle. Similarly whenthe two 1

2 -Spin vector bundles are tensored wiuth the pull-back of E. Since Fred(H)is Z × BU this produces a continuous map from the normal bundle to Fred(H)

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which maps the exterior of the unit ball bundle to I. Combining this with theprojection of the unit ball bundle onto M yields a continuous map

f : Sn −→ (M × Fred(H))/(M × I)

ϕ : M → X and the identity map of Fred(H) to itself give the evident map

(M × Fred(H))/(M × I) −→ (X × Fred(H))/(X × I)

The required map

f : Sn −→ (X × Fred(H))/(X × I)

is then the composition of these two.

3. String Theory and D-branes

In string theory [16], [27], [26] [13] [24] a spacetime X is fixed. Often Xis a Spin manifold of dimension 10, although there are other possibilities for X.Spacetime X comes equipped with its B-field and H-flux. The B-field, denotedB, is a locally defined 2-form and the H-flux, denoted H, is a globally definedclosed 3-form on X with H = dB. Note that since B is only locally defined,the de Rham cohomology class determined by H can be non-zero. This de Rhamcohomology class is required to satisfy a condition which essentially (i.e. up to anormalizing constant) asserts that the de Rham cohomology class of H is in theintegral cohomology group H3(X,Z). Hence spacetime X comes equipped with anelement in H3(X,Z).

An elementary particle in spacetime X is either a 1-sphere

S1 = (t1, t2) ∈ R2 | t21 + t22 = 1or a unit interval [0, 1] = t ∈ R | 0 ≤ t ≤ 1 which moves in time within Xsweeping out a two dimensional manifold or manifold with boundary known as theworldsheet of the particle. The S1 case is the case of “closed strings” and the [0, 1]case is the case of “open strings”. In the open string case, the two endpoints of theinterval [0, 1] are required to remain on a sub-manifold of X. This sub-manifold ofX is known as the worldvolume of the D-brane giving the constraint condition onthe open string.

A D-brane, however, consists of more than just its worldvolume. If M denotesthe worldvolume of a D-brane, then on M (due to the mixing of states in quantummechanics [26]) a C vector bundle E appears known as the Chan-Paton bundle.Also, a D-brane will have additional structure. M will be something like a Spinc

manifold, a connection will be given for the Chan-Paton bundle E etc.Consider Type II superstring theory on a spacetime X with all background

supergravity form fields turned off. In this special case, spacetime X is a Spinmanifold of dimension 10 and B and H are both zero. In this setting, [25] a firstapproximation to a definition of D-brane is that a D-brane is a triple (M,E,ϕ)such that:

• M is a compact Spinc manifold (without boundary).• E is a C vector bundle on M .• ϕ : M → X is an embedding of M into X.

In other words, a D-brane is a K-cycle (M,E,ϕ) such that ϕ : M → X is anembedding of M into X. The element in K∗(X) determined by a D-brane (M,E,ϕ)is the charge of the D-brane. Each of the three elementary moves:

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K-HOMOLOGY AND D-BRANES 7

• bordism• direct sum-disjoint union• vector bundle modification

used to define K∗(X) now takes on a physical meaning. For example, “direct sum-disjoint union” is closely related to “gauge symmetry enhancement for coincidentbranes” [25], [30] and “vector bundle modification” connects to the ”dielectriceffect” [25], [23]. Two D-branes have the same charge if and only if it is possible topass from one D-brane to the other by a finite sequence of these three elementarymoves.

Now consider Type II superstring theory where the background supergravityform fields are not turned off. The de Rham cohomology class of the H-flux mightbe non-zero. If so, this creates an anomaly and the D-branes must satisfy [18] ananomaly cancellation condition. The appropriate anomaly cancellation is achievedby using twisted K-homology, which will now be introduced.

4. Compact Operator Vector Bundles

Let H be a separable (but not finite dimensional) Hilbert space. U(H) isthe group of unitary operators U : H → H. U(H) is topologized by the weakoperator topology. U(H) is a topological group, and as a topological space U(H)is contractible. S1 K(Z, 1) where K(Z, n) denotes an n-th Eilenberg-MacLanespace for Z i.e. K(Z, n) is a topological space, having the homotopy type of a CWcomplex, whose n-th homotopy group is Z and all other homotopy groups are zero.

πjK(Z, n) =

Z, j = n,

0, j = n.

The homotopy sequence of the fibration

S1 −→ U(H) −→ PU(H)

implies thatPU(H) K(Z, 2).

From this we obtain a fibration

PU(H) −→ EPU(H) −→ K(Z, 3)

This is the universal principal fibration with PU(H) as structure group.

K(H) denotes the set of all compact operators T : H → H. U(H) acts on K(H)by conjugation :

U(H)×K(H) −→ K(H)

(U, T ) −→ UTU∗ = UTU−1.

This action factors through PU(H), yielding an action of PU(H) on K(H)

PU(H)×K(H) −→ K(H).

When combined with the above universal principal PU(H) bundle, this actionof PU(H) on K(H) gives a fibration

K(H) −→ EPU(H)×PU(H) K(H) −→ K(Z, 3).

In this fibration each fiber is a C∗ algebra which is (non-canonically) isomorphic toK(H).

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A twisted topological space X is a topological space X with a given continuousmap

α : X −→ K(Z, 3).

If X has the homotopy type of a CW complex, then the set of homotopy classesof continuous maps from X to K(Z, 3) is in bijection with H3(X,Z) — so one mightbe tempted to define a twisted topological space as a topological space X togetherwith a given element of H3(X,Z). It might, in fact, be possible to proceed thisway — but a much greater degree of mathematical precision is achieved by takingthe twisting structure (as above) to be a continuous map α : X −→ K(Z, 3). Thispoint will be explained below.

If X is a twisted topological space, then the given twisting map

α : X −→ K(Z, 3)

can be used to pull back the fibration

K(H) −→ EPU(H)×PU(H) K(H) −→ K(Z, 3).

Thus a fibration

K(H) −→ T −→ X

is obtained with base space X. Each fiber is a C∗ algebra which is (non-canonically)isomorphic to K(H).

If X is Hausdorff and locally compact, consider the C∗ algebra, Cα0 (X) con-

sisting of all continuous vanishing-at-infinity sections of the fibration

K(H) −→ T −→ X.

The twisted K-theory of X [3] [4] [20] is, by definition, the K-theory of the C∗

algebra Cα0 (X). As usual [29], this K-theory is denoted K∗C

α0 (X). The twisted

K-homology of X [28] (with compact supports), denoted KKjc (C

α0 (X),C), is

KKjc (C

α0 (X),C) := lim−→

Δ⊂XΔ compact

KKj(Cα(Δ),C).

Here the direct limit is taken over all compact subsets Δ of X, and Cα(Δ) is theC∗ algebra consisting of all continuous sections defined on the compact set Δ ofthe fibration

K(H) −→ T −→ X.

Remark. Let α : X −→ K(Z, 3) and γ : X −→ K(Z, 3) be two continuousmaps from X to K(Z, 3). If α and γ are homotopic, then K∗C

α0 (X) and K∗C

γ0 (X)

are isomorphic. The isomorphism depends on the choice of homotopy from α toγ and thus is not canonical. So if a twisted topological space were defined as atopological space X together with a given element of H3(X,Z), then the twisted K-theory [3] [4] would not be a well-defined (i.e. functorial) invariant of X. Similarlyfor the twisted K-homology. Hence the abelian group of which the charge of a D-brane is an element (and therefore also the charge itself) would not be well-defined.

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K-HOMOLOGY AND D-BRANES 9

5. Twisted K-cycles

Let X be a twisted topological space with given twisting map

α : X −→ K(Z, 3).

M denotes a compact oriented manifold (without boundary) for which a con-tinuous map ν : M −→ BSO has been fixed such that ν is a classifying map for the(oriented) stable normal bundle of M . BSO = lim

n→∞BSO(n).

Recall that the third Stiefel-Whitney class determines (up to homotopy) acontinuous map

π : BSO −→ K(Z, 3)

Fix once and for all such a continuous map π.With M, ν as above, a twisted K-cycle for X is a triple (M,E,ϕ) such that

• E is a C vector bundle on M .• ϕ : M → X is a continuous map from M to X.• The diagram

Mν

ϕ

BSO

π

X α K(Z, 3)

commutes.

LetW3(M) be the third (integral) Stiefel-Whitney class ofM and let [α] ∈ H3(X,Z)be the element in the third (integral) cohomology group of M determined by thetwisting map α : X −→ K(Z, 3). Commutativity of the diagram implies

ϕ∗([α]) +W3(M) = 0.

This is the Freed-Witten anomaly cancellation condition for D-branes in Type IIsuperstring theory [18].

These D-branes [31] are twisted K-cycles for spacetime X which are endowedwith some extra structure. In particular, the charge of a D-brane is the elementin the twisted K-homology of spacetime X determined by the underlying twistedK-cycle of the D-brane.

Consider the special case when the twisting map

α : X −→ K(Z, 3)

is the constant map which sends every point of X to one point of K(Z, 3). Recallthat for n ≥ 3 π1(SO(n)) = Z/2Z. Spin(n) is the unique non-trivial two-foldcover of SO(n), i.e. Spin(n) is the universal covering group of SO(n). Spinc(n) :=S1 ×Z/2Z Spin(n). Thus, by definition, there is a short exact sequence of compactconnected Lie groups

1 −→ S1 −→ Spinc(n) −→ SO(n) −→ 1.

Passing to classifying spaces this gives a fibration

BS1 −→ BSpinc(n) −→ BSO(n).

Since BS1 = K(Z, 2), a standard topological construction yields a fibration

BSpinc(n) −→ BSO(n) −→ K(Z, 3)

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in which the map BSO(n) −→ K(Z, 3) is the map determined by the third Stiefel-Whitney class.

Letting n go to infinity, this gives a fibration

BSpinc −→ BSO −→ K(Z, 3).

We may assume that the map

π : BSO −→ K(Z, 3)

fixed above is the projection of BSO onto K(Z, 3). If the twisting map

α : X −→ K(Z, 3)

is the constant map, then commutativity of the diagram

Mν

ϕ

BSO

π

X α K(Z, 3)

implies thatν : M −→ BSO

maps M to BSpinc. This is tantamount to giving a Spinc structure for M . So whenthe twisting map

α : X −→ K(Z, 3)

is the constant map, twisted K-cycles coincide with ordinary (i.e. untwisted) K-cycles as defined above.

Remark. If the twisting map

α : X −→ K(Z, 3)

is only homotopic to a constant map and the diagram

Mν

ϕ

BSO

π

X α K(Z, 3)

is only commutative up to homotopy, then a Spinc structure for M is not unam-biguously determined. Homotopy-commutativity of the diagram implies that theFreed-Witten anomaly cancellation condition

ϕ∗([α]) +W3(M) = 0

is valid, and α null-homotopic implies

[α] = 0

Therefore, W3(M) = 0, so M is Spinc-able (i.e. M admits a Spinc structure),but there is insufficient information for a Spinc structure on M to be definitelydetermined. So again it is mathematically more precise to use maps and truecommutativity of diagrams rather than homotopy classes of maps and homotopy-commutative diagrams. In [28] this issue is dealt with by requiring that a homotopyof α ϕ to π ν be part of the given data. In the present note, however, the point

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K-HOMOLOGY AND D-BRANES 11

of view is that the diagrams are required to truly commute at the level of spacesand maps.

Suppose now that the twisting map α : X −→ K(Z, 3) is not the constant map.With α fixed, consider the collection of all the twisted K-cycles (M,E,ϕ). Pro-ceeding as in the untwisted case, impose on this collection the equivalence relationgenerated by the three elementary moves:

• bordism• direct sum-disjoint union• vector bundle modification

With addition given by disjoint union, there are then two abelian groups:

Kαj (X) j = 0, 1

If X is a Spin or Spinc manifold, then the given Spin or Spinc structure for Xdetermines Poincare duality isomorphisms

Kαj (X) ∼= Kj+εC

α0 (X) j = 0, 1.

and

KKjc (C

α0 (X),C) ∼= Kj+εC

α0 (X) j = 0, 1.

Here ε is the dimension of X modulo 2. Combining these two isomorphisms givesan isomorphism

Kαj (X) ∼= KKj

c (Cα0 (X),C) j = 0, 1

This last isomorphism (unlike the two preceding isomorphisms) does not dependon the choice of Spin or Spinc structure for X — which gives plausibility to

Conjecture. Let X be a locally finite CW complex with given twisting map

α : X −→ K(Z, 3).

Then, (as in the untwisted case) there is a natural isomorphism of abelian groups

Kαj (X) ∼= KKj

c (Cα0 (X),C) j = 0, 1.

Outline of Proof. Let (M,E,ϕ) be a twistedK-cycle forX. With notationas above, consider the commutative diagram

Mν

ϕ

BSO

π

X α K(Z, 3)

Since ν is a classifying map for the (oriented) stable normal bundle of M ,

[π ν] = −W3(M) ∈ H3(M,Z).

This implies that M has a fundamental class, denoted M ∈ KK∗(Cπν(M),C).Commutativity of the diagram is precisely what is needed to define a homomor-phism of abelian groups

ϕ∗ : KKj(Cπν(M),C) −→ KKjc (C

α0 (X),C).

The conjectured isomorphism

Kαj (X) −→ KKj

c (Cα0 (X),C) j = 0, 1

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12 PAUL BAUM

will then be

(M,E,ϕ) → ϕ∗(E ∩ M)where E ∩M is the cap product of E ∈ K0(M) with M ∈ KK∗(Cπν(M),C).

This conjecture will be taken up in [12]. In future versions of string theory, itis possible that spacetime X might not be a Spin or Spinc manifold (e.g. spacetimemight be an orbifold [17] [24] [13] , rather than a manifold). If so, then thisconjecture will be relevant to developing the theory.

Appendix: K-theory versus K-homology

If spacetime X is a Spin or Spinc manifold , then (as noted above) there is thePoincare duality isomorphism

Kαj (X) ∼= Kj+εC

α0 (X) j = 0, 1 ε = dim(X) mod 2.

Thus the twisted K-theory, K∗Cα0 (X) of X could be used as the abelian group for

which the charge of a D-brane is an element. It seems more natural (and closer tothe basic geometry-physics) to use twisted K-homology.

Here’s a low-dimensional classical example. Let X be the torus S1 × S1.H1(X,Z) is a free abelian group on two generators a, b indicated below :

Consider an embedded S1 in X, which winds 8 times in the a direction and 1time in the −b direction.

The “charge” of this “D-brane” is 8a − b and it seems more natural to viewthis as an element of the homology group H1(X,Z) rather than to invoke Poincareduality and take this to be an element of the cohomology group H1(X,Z). In manylow-dimensional examples (such as this one) the Chern character maps K-homology–K-theory to ordinary integral homology – cohomology and hence gives an isomor-phism of K-homology –K-theory to ordinary integral homology – cohomology.

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K-HOMOLOGY AND D-BRANES 13

Some interesting examples of K-homology are calculated in [25]. In general, asobserved in [27] [5], K-homology – K-theory is conceptually simpler than ordinaryhomology – cohomology.

It is possible that in future versions of string theory spacetime X might havesingularities [17] [24] [13] and thus Poincare duality might not be valid. If so, thena decision will have to be made on K-theory versus K-homology. In extendingRiemann-Roch [7] [8] [6] from non-singular projective algebraic varieties to pro-jective algebraic varieties which may have singularities, such a decision had to bemade — and it was K-homology which, at the end of the day, played the morefundamental role.

References

[1] Michael Atiyah, Global theory of elliptic operators. 1970 Proc. Internat. Conf. on FunctionalAnalysis and Related Topics (Tokyo, 1969) pp. 21–30 Univ. of Tokyo Press, Tokyo.

[2] Micahl Atiyah, K-theory, W. A. Benjamin Inc. New York, 1967.[3] Michael Atiyah and Graeme Segal, Twisted K-theory. Ukr. Mat. Visn. 1 (2004), no. 3, 287–

330; translation in Ukr. Math. Bull. 1 (2004), no. 3, 291–334.[4] Michael Atiyah and Graeme Segal, Twisted K-theory and cohomology. Inspired by S. S.

Chern, 5–43, Nankai Tracts Math., 11, World Sci. Publ., Hackensack, NJ, 2006.[5] Paul Baum, Dirac operator and K-theory for discrete groups. to appear in proceedings of

conference in memory of Raoul Bott, Montreal, June 2008.[6] Paul Baum and Ronald G. Douglas, K homology and index theory. In Operator algebras and

applications, Part I (Kingston, Ont., 1980), volume 38 of Proc. Sympos. Pure Math., pages117–173. Amer. Math. Soc., Providence, R.I., 1982.

[7] Paul Baum, William Fulton, and Robert MacPherson, Riemann-Roch for singular varieties.Publ. Math. IHES 45 (1975), 101–167.

[8] Paul Baum, William Fulton, and Robert MacPherson, Riemann-Roch and topological K-theory for singular varieties, Acta Math 143 (1979), 155-192.

[9] Paul Baum, Nigel Higson, and Thomas Schick, On the equivalence of geometric and analyticK-homology. Pure Appl. Math. Q. 3 (2007), no. 1, part 3, 1–24.

[10] Paul Baum, Nigel Higson, and Thomas Schick, A geometric description of equivariant K-homology for proper actions, to appear in volume in honor of Alain Connes’ 60th birthday,Clay Mathematics Proceedings.

[11] Paul Baum, Herve Oyono-Oyono, and Thomas Schick, Equivariant geometric K-homology forcompact Lie groups actions, to appear in Abhandlungen aus dem Mathematischen Seminarder Universitat Hamburg.

[12] Paul Baum, Alan Carey, and Bai-Ling Wang, Geometric and analytic twisted K-homology.In preparation.

[13] Katrin Becker, Melanie Becker, and John H. Schwarz, String theory and M-theory : A modern

introduction, Cambridge University Press, Cambridge, 2007.[14] Armand Borel and Jean-Pierre Serre, Le theoreme de Riemann-Roch, Bull. Soc. math. France,

86 (1958), 97–136.[15] L. G. Brown, R. G. Douglas, and P. A. Fillmore, Extensions of C∗-algebras and K-homology.

Ann. of Math. (2) 105 (1977), no. 2, 265–324.[16] Quantum fields and strings: a course for mathematicians. Vol. 1, 2. Ed. Pierre Deligne,

Pavel Etingof, Daniel S. Freed, Lisa C. Jeffrey, David Kazhdan, John W. Morgan, David R.Morrison and Edward Witten. Ameri. Math. Soc., Providence, RI; Institute for AdvancedStudy (IAS), Princeton, NJ, 1999. Vol. 1: xxii+723 pp.; Vol. 2: pp. i–xxiv and 727–1501.

[17] L. Dixon, J. A. Harvey, C. Vafa, and Edward Witten, Strings on orbifolds, Nuclear Phys. B261 (1985), no. 4, 678–686.

[18] Daniel S. Freed, Edward Witten, Anomalies in string theory with D-branes. Asian J. Math.3 (1999), no. 4, 819–851.

[19] Klaus Janich. Vektorraumbundel und der Raum der Fredholm-Operatoren. Math. Ann. 1611965 129–142.

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[20] M. Karoubi, Twisted K-theory—old and new. K-theory and noncommutative geometry, 117–149, EMS Ser. Congr. Rep., Eur. Math. Soc., Zrich, 2008.

[21] G. G. Kasparov, Topological invariants of elliptic operators. I. K-homology. (Russian) (Rus-sian) ; translated from Izv. Akad. Nauk SSSR Ser. Mat. 39 (1975), no. 4, 796–838 Math.USSR-Izv. 9 (1975), no. 4, 751–792 (1976).

[22] Lynn H. Loomis, An introduction to Abstract Harmonic Analysis, D. Van Nostrand CompanyInc., New York, 1953.

[23] R. C. Myers, Dielectric-Branes. J. High Energy Phys. 9912, 022 (1999).[24] Joseph Polchinski, String theory. Vol I : An introduction to the bosonic string, and, String

theory. Vol II : Superstring theory and beyond. Cambridge Monographs on MathematicalPhysics, Cambridge University Press, Cambridge 2005 (reprint of the 2003 edition).

[25] Rui M. G. Reis, Richard J. Szabo, Geometric K-homology of flat D-branes. Comm. Math.Phys. 266 (2006), no. 1, 71–122.

[26] Jonathan Rosenberg, Topology, C∗-Algebras, and String Duality, CBMS Number 111, Amer.Math. Soc. 2009.

[27] Graeme Segal, Topological structures in string theory. Topological methods in the physicalsciences (London, 2000). R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci. 359 (2001),no. 1784, 1389–1398.

[28] Bai-Ling Wang, Geometric cycles, index theory and twisted K-homology. J. Noncommut.Geom. 2 (2008), no. 4, 497–552.

[29] N. E. Wegge-Olson, K-theory and C∗-algebras: A friendly approach, Oxford Science Publi-cations, Oxford Univ. Press, New York, 1993.

[30] Edward Witten, Bound states of strings and p-branes. Nuclear Phys. B 460 (1996), no. 2,335–350.

[31] Edward Witten, D-branes and K-theory. J. High Energy Phys. 1998, no. 12, Paper 19, 41pp. (electronic).

94

Proceedings of Symposia in Pure Mathematics

Riemann-Roch and Index Formulae in Twisted K-theory

Alan L. Carey and Bai-Ling Wang

ABSTRACT. In this paper, we establish the Riemann-Roch theorem in twisted K-theoryextending our earlier results. We also give a careful summary of twisted geometric cyclesexplaining in detail some subtle points in the theory. As an application, we prove a twistedindex formula and show that D-brane charges in Type I and Type II string theory are clas-sified by twisted KO-theory and twisted K-theory respectively in the presence of B-fieldsas proposed by Witten.

CONTENTS

1. Introduction2. Twisted K-theory: Preliminary Review3. Twisted K-homology4. The Chern Character in Twisted K-theory5. Thom Classes and Riemann-Roch Formula in Twisted K-theory6. The Twisted Index Formula7. Mathematical Definition of D-branes and D-brane ChargesReferences

1991 Mathematics Subject Classification. Primary 54C40, 14E20; Secondary 46E25, 20C20.Key words and phrases. Twisted K-theory, twisted K-homology, twisted Riemann-Roch, twisted index

theorem, D-brane charges.

c©0000 (copyright holder)

1

Proceedings of Symposia in Pure MathematicsVolume 81, 2010

c©2010 American Mathematical Society

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2 ALAN L. CAREY AND BAI-LING WANG

1. Introduction

1.1. String geometry. We begin by giving a short discussion of the physical back-ground. Readers uninterested in this motivation may move to the next subsection. In stringtheory D-branes were proposed as a mechanism for providing boundary conditions for thedynamics of open strings moving in space-time. Initially they were thought of as sub-manifolds. As D-branes themselves can evolve over time one needs to study equivalencerelations on the set of D-branes. An invariant of the equivalence class is the topologicalcharge of the D-brane which should be thought of as an analogue of the Dirac monopolecharge as these D-brane charges are associated with gauge fields (connections) on vectorbundles over the D-brane. These vector bundles are known as Chan-Paton bundles.

In [MM] Minasian and Moore made the proposal that D-brane charges should takevalues in K-groups and not in the cohomology of the space-time or the D-brane. However,they proposed a cohomological formula for these charges which might be thought of as akind of index theorem in the sense that, in general, index theory associates to a K-theoryclass a number which is given by an integral of a closed differential form. In string theorythere is an additional field on space-time known as the H-flux which may be thought of as aglobal closed three form. Locally it is given by a family of ‘two-form potentials’ known asthe B-field. Mathematically we think of these B-fields as defining a degree three integralCech class on the space-time, called a ‘twist’. Witten [Wit], extending [MM], gave aphysical argument for the idea that D-brane charges should be elements of K-groups and,in addition, proposed that the D-brane charges in the presence of a twist should take valuesin twisted K-theory (at least in the case where the twist is torsion). The mathematicalideas he relied on were due to Donovan and Karoubi [DK]. Subsequently Bouwknegt andMathai [BouMat] extended Witten’s proposal to the non-torsion case using ideas from[Ros]. A geometric model (that is, a ‘string geometry’ picture) for some of these stringtheory constructions and for twisted K-theory was proposed in [BCMMS] using the notionof bundle gerbes and bundle gerbe modules. Various refinements of twisted K-theory thatare suggested by these applications are also described in the article of Atiyah and Segal[AS1] and we will need to use their results here.

1.2. Mathematical results. From a mathematical perspective some immediate ques-tions arise from the physical input summarised above. When there is no twist it is wellknown that K-theory provides the main topological tool for the index theory of ellipticoperators. One version of the Atiyah-Singer index theorem due to Baum-Higson-Schick[BHS] establishes a relationship between the analytic viewpoint provided by elliptic dif-ferential operators and the geometric viewpoint provided by the notion of geometric cycleintroduced in the fundamental paper of Baum and Douglas [BD2]. The viewpoint thatgeometric cycles in the sense of [BD2] are a model for D-branes in the untwisted case isexpounded in [RS, RSV, Sz]. Note that in this viewpoint D-branes are no longer subman-ifolds but the images of manifolds under a smooth map.

It is thus tempting to conjecture that there is an analogous picture of D-branes as atype of geometric cycle in the twisted case as well. More precisely we ask the questionof whether there is a way to formulate the notion of ‘twisted geometric cycle’ (cf [BD1]and [BD2]) and to prove an index theorem in the spirit of [BHS] for twisted K-theory.This precise question was answered in the positive in [Wa]. It is important to emphasisethat string geometry ideas from [FreWit] played a key role in finding the correct way togeneralise [BD1].

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RIEMANN-ROCH AND INDEX FORMULAE IN TWISTED K-THEORY 3

Our purpose here in the present paper is threefold. First, we explain the results in[Wa] (see Section 3) in a fashion that is more aligned to the string geometry viewpoint.Second, we prove an analogue of the Atiyah-Hirzebruch Riemann-Roch formula in twistedK-theory by extending the results and approach of [CMW]. An interesting by-product ofour approach in Section 5 is a discussion of the Thom class in twisted K theory. Third,in Section 6 we prove an index formula using our twisted Riemann-Roch theorem. It willbe clear from our approach to this twisted index theory that our twisted geometric cyclesprovide a geometric model for D-branes and we give details in Section 7.

Our main new results are stated as two theorems, Theorem 5.3 (twisted Riemann-Roch) and Theorem 6.1 (the index pairing). We remark that the Minasian-Moore formula[MM] arises from the fact that the index pairing they discuss may be regarded as a qua-dratic form on K-theory. In the twisted index formula that we establish, the pairing isasymmetric and may be thought of as a bilinear form, from which there is no obvious wayto extract a twisted analogue of the Minasian-Moore formula. Nevertheless we interpretour results in terms of the physics language in Section 7 explaining the link to Witten’soriginal ideas on D-brane charges.Acknowledgements. We thank the Australian Research Council, the Hausdorff Institutefor Mathematics and the participants and organisers of the CBMS Conference on Topology,C*-algebras, and String Duality for their respective contributions to the writing of thispaper.

2. Twisted K-theory: Preliminary Review

2.1. Twisted K-theory: topological and analytic definitions. We begin by review-ing the notion of a ‘twisting’. Let H be an infinite dimensional, complex and separableHilbert space. We shall consider locally trivial principal PU(H)-bundles over a para-compact Hausdorff topological space X , the structure group PU(H) is equipped with thenorm topology. The projective unitary group PU(H) with the topology induced by thenorm topology on U(H) (Cf. [Kui]) has the homotopy type of an Eilenberg-MacLanespace K(Z, 2). The classifying space of PU(H), denoted BPU(H), is a K(Z, 3). Theset of isomorphism classes of principal PU(H)-bundles over X is given by (Proposition2.1 in [AS1]) homotopy classes of maps from X to any K(Z, 3) and there is a canonicalidentification

[X,BPU(H)] ∼= H3(X,Z).

A twisting of complex K-theory on X is given by a continuous map α : X →K(Z, 3). For such a twisting, we can associate a canonical principal PU(H)-bundle Pα

through the usual pull-back construction from the universal PU(H) bundle denoted byEK(Z, 2), as summarised by the diagram:

Pα

EK(Z, 2)

X α

K(Z, 3).

(2.1)

We will use PU(H) as a group model for a K(Z, 2). We write Fred(H) for the connectedcomponent of the identity of the space of Fredholm operators on H equipped with the normtopology. There is a base-point preserving action of PU(H) given by the conjugationaction of U(H) on Fred(H):

PU(H)× Fred(H) −→ Fred(H).(2.2)

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4 ALAN L. CAREY AND BAI-LING WANG

The action (2.2) defines an associated bundle over X which we denote by

Pα(Fred) = Pα ×PU(H) Fred(H)

We write ΩnXPα(Fred) = Pα ×PU(H) Ω

nFred for the fiber-wise iterated loop spaces.

DEFINITION 2.1. The (topological) twisted K-groups of (X,α) are defined to be

K−n(X,α) := π0

(Cc(X,Ωn

XPα(Fred))),

the set of homotopy classes of compactly supported sections (meaning they are the identityoperator in Fred off a compact set) of the bundle of Pα(Fred).

Due to Bott periodicity, we only have two different twisted K-groups K0(X,α) andK1(X,α). Given a closed subspace A of X , then (X,A) is a pair of topological spaces,and we define relative twisted K-groups to be

Kev/odd(X,A;α) := Kev/odd(X −A,α).

Take a pair of twistings α0, α1 : X → K(Z, 3), and a map η : X × [1, 0] → K(Z, 3)which is a homotopy between α0 and α1, represented diagrammatically by

X

α0

α1

K(Z, 3).η

Then there is a canonical isomorphism Pα0∼= Pα1

induced by η. This canonical isomor-phism determines a canonical isomorphism on twisted K-groups

η∗ : Kev/odd(X,α0)∼= Kev/odd(X,α1),(2.3)

This isomorphism η∗ depends only on the homotopy class of η. The set of homotopyclasses of maps between α0 and α1 is labelled by [X,K(Z, 2)]. Recall the first Chernclass isomorphism

Vect1(X) ∼= [X,K(Z, 2)] ∼= H2(X,Z)

where Vect1(X) is the set of equivalence classes of complex line bundles on X . Weremark that the isomorphisms induced by two different homotopies between α0 and α1 arerelated through an action of complex line bundles.

Let K be the C∗-algebra of compact operators on H. The isomorphism PU(H) ∼=Aut(K) via the conjugation action of the unitary group U(H) provides an action of aK(Z, 2) on the C∗-algebra K. Hence, any K(Z, 2)-principal bundle Pα defines a locallytrivial bundle of compact operators, denoted by Pα(K) = Pα ×PU(H) K.

Let C0(X,Pα(K)) be the C∗-algebra of sections of Pα(K) vanishing at infinity. ThenC0(X,Pα(K) is the (unique up to isomorphism) stable separable complex continuous-trace C∗-algebra over X with Dixmier-Douday class [α] ∈ H3(X,Z) (here we identifythe Cech cohomology of X with its singular cohomology, cf [Ros] and [AS1]).

THEOREM 2.2. ( [AS1] and [Ros]) The topological twisted K-groups Kev/odd(X,α)are canonically isomorphic to analytic K-theory of the C∗-algebra C0(X,Pα(K))

Kev/odd(X,α) ∼= Kev/odd(C0(X,Pα(K)))

where the latter group is the algebraic K-theory of C0(X,Pα(K)), defined to be

lim−→k→∞

π1

(GLk(C0(X,Pα(K)))

).

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RIEMANN-ROCH AND INDEX FORMULAE IN TWISTED K-THEORY 5

Note that the algebraic K-theory of C0(X,Pα(K)) is isomorphic to Kasparovs KK-theory ([Kas1] and [Kas2])

KKev/odd(C, C0(X,Pα(K)).

It is important to recognise that these groups are only defined up to isomorphism bythe Dixmier-Douady class [α] ∈ H3(X,Z). To distinguish these two equivalent definitionsof twisted K-theory if needed, we will write

Kev/oddtop (X,α) and Kev/odd

an (X,α)

for the topological and analytic twisted K-theories of (X,α) respectively. Twisted K-theory is a 2-periodic generalized cohomology theory: a contravariant functor on the cat-egory consisting of pairs (X,α), with the twisting α : X → K(Z, 3), to the category ofZ2-graded abelian groups. Note that a morphism between two pairs (X,α) and (Y, β) is acontinuous map f : X → Y such that β f = α.

2.2. Twisted K-theory for torsion twistings. There are some subtle issues in twistedK-theory and to handle these we have chosen to use the language of bundle gerbes, con-nections and curvings as explained in [Mur]. We explain first the so-called ‘lifting bundlegerbe’ Gα [Mur] associated to the principal PU(H)-bundle π : Pα → X and the centralextension

1 → U(1) −→ U(H) −→ PU(H) → 1.(2.4)

This is constructed by starting with π : Pα → X , forming the fibre product P [2]α which is

a groupoid

P [2]α = Pα ×X Pα

π1 π2

Pα

with source and range maps π1 : (y1, y2) → y1 and π2 : (y1, y2) → y2. There is an

obvious map from each fiber of P [2]α to PU(H) and so we can define the fiber of Gα over

a point in P [2]α by pulling back the fibration (2.4) using this map. This endows Gα with

a groupoid structure (from the multiplication in U(H)) and in fact it is a U(1)-groupoidextension of P [2]

α .A torsion twisting α is a map α : X → K(Z, 3) representing a torsion class in

H3(X,Z). Every torsion twisting arises from a principal PU(n)-bundle Pα(n) with itsclassifying map

X → BPU(n),

or a principal PU(H)-bundle with a reduction to PU(n) ⊂ PU(H). For a torsion twistingα : X → BPU(n) → BPU(H), the corresponding lifting bundle gerbe Ga

Gα

Pα(n)

[2]π1 π2

Pα(n)

π

M

(2.5)

is defined by Pα(n)[2] ∼= Pα(n) PU(n) ⇒ Pα(n) (as a groupoid) and the central

extension1 → U(1) −→ U(n) −→ PU(n) → 1.

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6 ALAN L. CAREY AND BAI-LING WANG

There is an Azumaya bundle associated to Pα(n) arising naturally from the PU(n)action on the n × n matrices. We denote this associated Azumaya bundle by Aα. AnAα-module is a complex vector bundle E over M with a fiberwise Aα action

Aσ ×M E −→ E .The C∗-algebra of continuous sections of Aα, vanishing at infinity if X is non-compact,is Morita equivalent to a continuous trace C∗-algebra C0(X,Pα(K)). Hence there is anisomorphism between K0(X,α) and the K-theory of the bundle modules of Aa.

There is an equivalent definition of twisted K-theory using bundle gerbe modules (Cf.[BCMMS] and [CW1]). A bundle gerbe module E of Gα is a complex vector bundle Eover Pα(n) with a groupoid action of Gα, i.e., an isomorphism

φ : Gα ×(π2,p) E −→ E

where Gα×(π2,π)E is the fiber product of the source π2 : Gα → Pα(n) and p : E → Pα(n)such that

(1) p φ(g, v) = π1(g) for (g, v) ∈ Gα ×(π2,p) E, and π1 is the target map of Gα.(2) φ is compatible with the bundle gerbe multiplication m : Ga ×(π2,π1) Gα → Gα,

which meansφ (id× φ) = φ (m× id).

Note that the natural representation of U(n) on Cn induces a Gα bundle gerbe module

Sn = Pα(n)× Cn.

Here we use the fact that Gα = Pα(n) U(n) ⇒ Pα(n) (as a groupoid). Similarly, thedual representation of U(n) on Cn induces a G−α bundle gerbe module S∗

n = Pα(n)×Cn.Note that S∗

n⊗Sn∼= π∗Aα descends to the Azumaya bundle Aα. Given a Gα bundle gerbe

module E of rank k, then as a PU(n)-equivariant vector bundle, S∗n ⊗ E descends to an

Aα-bundle over M . Conversely, given an Aα-bundle E over M , Sn ⊗π∗Aαπ∗E defines a

Gα bundle gerbe module. These two constructions are inverse to each other due to the factthat

S∗n ⊗ (Sn ⊗π∗Aα

π∗E) ∼= (S∗n ⊗ Sn)⊗π∗Aα

π∗E ∼= π∗Aα ⊗π∗Aαπ∗E ∼= π∗E .

Therefore, there is a natural equivalence between the category of Gα bundle gerbe modulesand the category of Aα bundle modules, as discussed in [CW1]. In summary, we have thefollowing proposition.

PROPOSITION 2.3. ([BCMMS][CW1]) For a torsion twisting α : X → BPU(n) →BPU(H), twisted K-theory K0(X,α) has another two equivalent descriptions:

(1) the Grothendieck group of the category of Gα bundle gerbe modules.(2) the Grothendieck group of the category of Aσ bundle modules.

One important example of torsion twistings comes from real oriented vector bundles.Consider an oriented real vector bundle E of even rank over X with a fixed fiberwise innerproduct. Denote by

νE : X → BSO(2k)

the classifying map of E. The following twisting

o(E) := W3 νE : X −→ BSO(2k) −→ K(Z, 3),

will be called the orientation twisting associated to E. Here W3 is the classifying map ofthe principal BU(1)-bundle BSpinc(2k) → BSO(2k). Note that the orientation twistingo(E) is null-homotopic if and only if E is K-oriented.

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RIEMANN-ROCH AND INDEX FORMULAE IN TWISTED K-THEORY 7

PROPOSITION 2.4. Given an oriented real vector bundle E of even rank over X withan orientation twisting o(E), then there is a canonical isomorphism

K0(X, o(E)) ∼= K0(X,W3(E))

where K0(X,W3(E)) is the K-theory of the Clifford modules associated to the bundleCliff(E) of Clifford algebras.

PROOF. Denote by Fr the frame bundle of V , the principal SO(2k)-bundle of posi-tively oriented orthonormal frames, i.e.,

E = Fr ×ρ2nR2k,

where ρn is the standard representation of SO(2k) on Rn. The lifting bundle gerbe asso-ciated to the frame bundle and the central extension

1 → U(1) −→ Spinc(2k) −→ SO(2k) → 1

is called the Spinc bundle gerbe GW3(E) of E, whose Dixmier-Douady invariant is givenby the integral third Stiefel-Whitney class W3(E) ∈ H3(X,Z). The canonical represen-tation of Spinc(2k) gives a natural inclusion

Spinc(2k) ⊂ U(2k)

which induces a commutative diagram

U(1)

=

Spinc(2k)

SO(2k)

U(1)

=

U(2k)

PU(2k)

U(1) U(H) PU(H).

This provides a reduction of the principal PU(H)-bundle Po(E). The associated bundleof Azumaya algebras is in fact the bundle of Clifford algebras, whose bundle modules arecalled Clifford modules ([BGV]). Hence, there exists a canonical isomorphism betweenK0(X, o(E)) and the K-theory of the Clifford modules associated to the bundle Cliff(E).

2.3. Twisted K-theory: general properties. Twisted K-theory satisfies the follow-ing properties whose proofs are rather standard for a 2-periodic generalized cohomologytheory ([AS1] [CW1] [Kar] [Wa]). (Note that when we write (X,A) for a pair of spaceswe assume A ⊂ X .)

(I) (The homotopy axiom) If two morphisms f, g : (Y,B) → (X,A) are homo-topic through a map η : (Y × [0, 1], B × [0, 1]) → (X,A), written in terms ofthe following homotopy commutative diagram

(Y,B)

g

f (X,A)

η

α

(X,A)

α K(Z, 3),

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8 ALAN L. CAREY AND BAI-LING WANG

then we have the following commutative diagram

Kev/odd(X,A;α)

g∗

f∗

Kev/odd(Y,B;α f)η∗ Kev/odd(Y,B;α g).

Here η∗ is the canonical isomorphism induced by the homotopy η.(II) (The exact axiom) For any pair (X,A) with a twisting α : X → K(Z, 3), there

exists the following six-term exact sequence

K0(X,A;α) K0(X,α) K0(A,α|A)

K1(A,α|A)

K1(X,α) K1(X,A;α)

here α|A is the composition of the inclusion and α.(III) (The excision axiom) Let (X,A) be a pair of spaces and let U ⊂ A be a sub-

space such that the closure U is contained in the interior of A. Then the inclusionι : (X−U,A−U) → (X,A) induces, for all α : X → K(Z, 3), an isomorphism

Kev/odd(X,A;α) −→ Kev/odd(X − U,A− U ;α ι).(IV) (Multiplicative property) Let α, β : X → K(Z, 3) be a pair of twistings on X .

Denote by α+ β the new twisting defined by the following map1

α+ β : X(α,β) K(Z, 3)×K(Z, 3)

m K(Z, 3),(2.6)

where m is defined as follows

BPU(H)×BPU(H) ∼= B(PU(H)× PU(H)) −→ BPU(H),

for a fixed isomorphism H⊗H ∼= H. Then there is a canonical multiplication

Kev/odd(X,α)×Kev/odd(X, β) −→ Kev/odd(X,α+ β),(2.7)

which defines a K0(X)-module structure on twisted K-groups Kev/odd(X,α).(V) (Thom isomorphism) Let π : E → X be an oriented real vector bundle of rank

k over X , then there is a canonical isomorphism, for any twisting α : X →K(Z, 3),

Kev/odd(X,α+ oE) ∼= Kev/odd(E,α π),(2.8)

with the grading shifted by k(mod 2).(VI) (The push-forward map) For any differentiable map f : X → Y between two

smooth manifolds X and Y , let α : Y → K(Z, 3) be a twisting. Then there is acanonical push-forward homomorphism

fK! : Kev/odd

(X, (α f) + of

)−→ Kev/odd(Y, α),(2.9)

with the grading shifted by n mod(2) for n = dim(X) + dim(Y ). Here of isthe orientation twisting corresponding to the bundle TX ⊕ f∗TY over X .

1In terms of bundles of projective Hilbert space, this operation corresponds to the Hilbert space tenrsorproduct, see [AS1].

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RIEMANN-ROCH AND INDEX FORMULAE IN TWISTED K-THEORY 9

(VII) (Mayer-Vietoris sequence) If X is covered by two open subsets U1 and U2 witha twisting α : X → K(Z, 3), then there is a Mayer-Vietoris exact sequence

K0(X,α) K1(U1 ∩ U2, α12) K1(U1, α1)⊕K1(U2, α2)

K0(U1, α1)⊕K0(U2, α2)

K0(U1 ∩ U2, α12) K1(X,α)

where α1, α2 and α12 are the restrictions of α to U1, U2 and U1∩U2 respectively.

3. Twisted K-homology

Complex K-theory, as a generalized cohomology theory on a CW complex, is devel-oped by Atiyah-Hirzebruch using complex vector bundles. It is representable in the sensethat there exists a classifying space Z×BU(∞), where BU(∞) = lim−→k

BU(k), such that

K0(X) = [X,Z×BU(∞)]

for any finite CW complex X . The classifying space for complex K-theory is referredto as the BU(∞)-spectrum with even term Z × BU(∞) and odd term U(∞). They arealso called the ‘complex K-spectra’ in the literature. The advantage of using spectra is thatthere is a natural definition of a homology theory associated to a classifying space of eachgeneralized cohomology theory. Hence, the topological K-homology of a CW complexX , dual to complex K-theory, is defined by the following stable homotopy groups

Ktopev (X) = lim−→

k→∞π2k(BU(∞) ∧X+)

andKtop

odd (X) = lim−→k→∞

π2k+1(BU(∞) ∧X+).

Here X+ is the space X with one point added as a based point, and the wedge product oftwo based CW complexes (X,x0) and (Y, y0) is defined to be

X ∧ Y =X × Y

(X × y0 ∪ x0 × Y ).

All the properties of K-homology, as a generalized homology theory, can be obtainedin a natural way see for example in [Swi]. There are two other equivalent definitionsof K-homology, called analytic K-homology developed by Kasparov, and geometric K-homology by Baum and Douglas. We now give a brief review of these two definitions.

Kasparov’s analytic K-homology KKev/odd(C(X),C) is generated by unitary equiv-alence classes of (graded) Fredholm modules over C(X) modulo an operator homotopyrelation ([Kas1] and [HigRoe]). For brevity we will use the notation Kan

ev/odd(X) for thisK-homology. A cycle for Kan

0 (X), also called a Z2-graded Fredholm module, consists ofa triple (φ0⊕, φ1,H0 ⊕H1, F ), where

• φi : C(X) → B(Hi) is a representation of C(X) on a separable Hilbert spaceHi;

• F : H0 → H1 is a bounded operator such that

φ1(a)F − Fφ0(a), φ0(a)(F∗F − Id) φ1(a)(FF ∗ − Id)

are compact operators for all a ∈ C(X).

A cycle for Kan1 (X), also called a trivially graded or odd Fredholm module, consists of a

pair (φ, F ), where

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10 ALAN L. CAREY AND BAI-LING WANG

• φ : C(X) → B(H) is a representation of C(X) on a separable Hilbert space H;• F is a bounded self-adjoint operator on H such that

φ(a)F − Fφ(a), φ(a)(F 2 − Id)

are compact operators for all a ∈ C(X).

In [BD1] and [BD2], Baum and Douglas gave a geometric definition of K-homologyusing what are now called geometric cycles. The basic cycles for Kgeo

ev (X) (respectivelyKgeo

odd (X)) are triples

(M, ι, E)

consisting of even-dimensional (resp. odd-dimensional) closed smooth manifolds M witha given Spinc structure on the tangent bundle of M together with a continuous map ι :M → X and a complex vector bundle E over M . The equivalence relation on the set ofall cycles is generated by the following three steps (see [BD1] for details):

(i) Bordism.(ii) Direct sum and disjoint union.

(iii) Vector bundle modification.

Addition in Kgeoev/odd(X) is given by the disjoint union operation of geometric cycles.

Baum-Douglas in [BD2] showed that the Atiyah-Singer index theorem is encoded inthe following commutative diagram

Ktopev/odd(X)

∼=

∼=

Kgeoev/odd(X)

μ Kanev/odd(X)

(3.1)

where μ is the assembly map assigning an abstract Dirac operator

ι∗([ /DEM ]) ∈ Kan

ev/odd(X)

to a geometric cycle (M, ι, E).For a paracompact Hausdorff space X with a twisting α : X → K(Z, 3), all these

three versions of twisted K-homology were studied in [Wa]. They are called there thetwisted topological, analytic and geometric K-homologies, and denoted respectively byKtop

ev/odd(X,α), Kanev/odd(X,α) and Kgeo

ev/odd(X,α). Our first task in this Section is toreview these three definitions, see [Wa] for greater detail.

3.1. Topological and analytic definitions of twisted K-homology. Let X be a CWcomplex (or paracompact Hausdorff space) with a twisting α : X → K(Z, 3). Let Pα

be the corresponding principal K(Z, 2)-bundle. Any base-point preserving action of aK(Z, 2) on a space defines an associated bundle by the standard construction. In particular,as a classifying space of complex line bundles, K(Z, 2) acts on the complex K-theoryspectrum K representing the tensor product by complex line bundles, where

Kev = Z×BU(∞), Kodd = U(∞).

Denote by Pα(K) = Pα ×K(Z,2) K the bundle of based K-theory spectra over X . Thereis a section of Pα(K) = Pα×K(Z,2)K defined by taking the base points of each fiber. Theimage of this section can be identified with X and we denote by Pα(K)/X the quotientspace of Pα(K) obtained by collapsing the image of this section.

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RIEMANN-ROCH AND INDEX FORMULAE IN TWISTED K-THEORY 11

The stable homotopy groups of Pα(K)/X by definition give the topological twistedK-homology groups Ktop

ev/odd(X,α). (There are only two due to Bott periodicity of K.)Thus we have

Ktopev (X,α) = lim−→

k→∞π2k

(Pα(BU(∞))/X

)

andKtop

odd (X,α) = lim−→k→∞

π2k+1

(Pα(BU(∞))/X

).

Here the direct limits are taken by the double suspension

πn+2k

(Pα(BU(∞))/X

)−→ πn+2k+2

(Pα(S

2 ∧BU(∞))/X)

and then followed by the standard map

πn+2k+2

(Pα(S

2 ∧BU(∞))/X) b∧1 πn+2k+2

(Pα(BU(∞) ∧BU(∞))/X

)

m πn+2k+2

(Pα(BU(∞))/X

)

where b : R2 → BU(∞) represents the Bott generator in K0(R2) ∼= Z, m is the basepoint preserving map inducing the ring structure on K-theory.

For a relative CW-complex (X,A) with a twisting α : X → K(Z, 3), the relativeversion of topological twisted K-homology, denoted Ktop

ev/odd(X,A, α), is defined to be

Ktopev/odd(X/A,α) where X/A is the quotient space of X obtained by collapsing A to a

point. Then we have the following exact sequence

Ktopodd (X,A;α) Ktop

ev (A,α|A) Ktopev (X,α)

Ktop

odd (X,α)

Ktopodd (A,α|A) Ktop

ev (X,A;α)

and the excision properties

Ktopev/odd(X,B;α) ∼= Ktop

ev/odd(A,A−B;α|A)for any CW-triad (X;A,B) with a twisting α : X → K(Z, 3). A triple (X;A,B) is ACW-triad if X is a CW-complex, and A, B are two subcomplexes of X such that A∪B =X .

For the analytic twisted K-homology, recall that Pα(K) is the associated bundle ofcompact operators on X . Analytic twisted K-homology, denoted by Kan

ev/odd(X,α), isdefined to be

Kanev/odd(X,α) := KKev/odd

(C0(X,Pα(K)),C

),

Kasparov’s Z2-graded K-homology of the C∗-algebra C0(X,Pα(K)).For a relative CW-complex (X,A) with a twisting α : X → K(Z, 3), the relative

version of analytic twisted K-homology Kanev/odd(X,A, α) is defined to be Kan

ev/odd(X −A,α). Then we have the following exact sequence

Kanodd(X,A;α) Kan

ev (A,α|A) Kanev (X,α)

Kan

odd(X,α)

Kanodd(A,α|A) Kan

ev (X,A;α)

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12 ALAN L. CAREY AND BAI-LING WANG

and the excision properties

Kanev/odd(X,B;α) ∼= Kan

ev/odd(A,A−B;α|A)for any CW-triad (X;A,B) with a twisting α : X → K(Z, 3).

THEOREM 3.1. (Theorem 5.1 in [Wa]) There is a natural isomorphism

Φ : Ktopev/odd(X,α) −→ Kan

ev/odd(X,α)

for any smooth manifold X with a twisting α : X → K(Z, 3).

The proof of this theorem requires Poincare duality between twisted K-theory andtwisted K-homology (we describe this duality in the next theorem), and the isomorphism(Theorem 2.2) between topological twisted K-theory and analytic twisted K-theory.

Fix an isomorphism H ⊗ H ∼= H which induces a group homomorphism U(H) ×U(H) −→ U(H) whose restriction to the center is the group multiplication on U(1). Sowe have a group homomorphism

PU(H)× PU(H) −→ PU(H)

which defines a continuous map, denoted m∗, of CW-complexes

BPU(H)×BPU(H) −→ BPU(H).

As BPU(H) is identified as K(Z, 3), we may think of this as a continuous map takingK(Z, 3)×K(Z, 3) to K(Z, 3), which can be used to define α+ oX .

There are natural isomorphisms from twisted K-homology (topological resp. analytic)to twisted K-theory (topological resp. analytic) of a smooth manifold X where the twistingis shifted by

α → α+ oX

where τ : X → BSO is the classifying map of the stable tangent space and α+oX denotesthe map X → K(Z, 3), representing the class [α] +W3(X) in H3(X,Z).

THEOREM 3.2. Let X be a smooth manifold with a twisting α : X → K(Z, 3). Thereexist isomorphisms

Ktopev/odd(X,α) ∼= K

ev/oddtop (X,α+ oX)

andKan

ev/odd(X,α) ∼= Kev/oddan (X,α+ oX)

with the degree shifted by dimX(mod 2).

Analytic Poincare duality was established in [EEK] and [Tu], and topological Poincareduality was established in [Wa]. Theorem 3.1 and the exact sequences for a pair (X,A)imply the following corollary.

COROLLARY 3.3. There is a natural isomorphism

Φ : Ktopev/odd(X,A, α) −→ Kan

ev/odd(X,A, α)

for any smooth manifold X with a twisting α : X → K(Z, 3) and a closed submanifoldA ⊂ X .

REMARK 3.4. In fact, Poincare duality as in Theorem 3.2 holds for any compactRiemannian manifold W with boundary ∂W and a twisting α : W → K(Z, 3). Thisduality takes the following form

Ktopev/odd(W,α) ∼= K

ev/oddtop (W,∂W,α+ oW )

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RIEMANN-ROCH AND INDEX FORMULAE IN TWISTED K-THEORY 13

andKan

ev/odd(W,α) ∼= Kev/oddan (X, ∂X, α+ oW )

with the degree shifted by dimW (mod 2). From this, we have a natural isomorphism

Φ : Ktopev/odd(X,A, α) −→ Kan

ev/odd(X,A, α)

for any CW pair (X,A) with a twisting α : X → K(Z, 3) using the Five Lemma.

3.2. Geometric cycles and geometric twisted K-homology. Let X be a paracom-pact Hausdorff space and let α : X −→ K(Z, 3) be a twisting over X .

DEFINITION 3.5. Given a smooth oriented manifold M with a classifying map ν ofits stable normal bundle then we say that M is an α-twisted Spinc manifold over X if Mis equipped with an α-twisted Spinc structure, that means, a continuous map ι : M → Xsuch that the following diagram

M

ι

ν BSOη

W3

X α

K(Z, 3),

commutes up to a fixed homotopy η from W3 ν and α ι. Such an α-twisted Spinc

manifold over X will be denoted by (M, ν, ι, η).

PROPOSITION 3.6. M admits an α-twisted Spinc structure if and only if there is acontinuous map ι : M → X such that

ι∗([α]) +W3(M) = 0.

If ι is an embedding, this is the anomaly cancellation condition obtained by Freed andWitten in [FreWit].

PROOF. This is clear.

A morphism between α-twisted Spinc manifolds (M1, ν1, ι1, η1) and (M2, ν2, ι2, η2)is a continuous map f : M1 → M2 where the following diagram

M1

ι1

ν1

f

M2

ι2

ν2 BSOη2

W3

X α

K(Z, 3)

(3.2)

is a homotopy commutative diagram such that

(1) ν1 is homotopic to ν2 f through a continuous map ν : M1 × [0, 1] → BSO;(2) ι2 f is homotopic to ι1 through continuous map ι : M1 × [0, 1] → X;(3) the composition of homotopies (α ι) ∗ (η2 (f × Id)) ∗ (W3 ν) is homotopic

to η1.

Two α-twisted Spinc manifolds (M1, ν1, ι1, η1) and (M2, ν2, ι2, η2) are called isomorphicif there exists a diffeomorphism f : M1 → M2 such that the above holds. If the identitymap on M induces an isomorphism between (M, ν1, ι1, η1) and (M, ν2, ι2, η2), then thesetwo α-twisted Spinc structures are called equivalent.

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14 ALAN L. CAREY AND BAI-LING WANG

Orientation reversal in the Grassmannian model defines an involution

r : BSO −→ BSO.

Choose a good cover Vi of M and hence a trivialisation of the universal bundle overBSO(n) with transition functions

gij : Vi ∩ Vj −→ SO(n).

Let gij : Vi ∩ Vj −→ Spinc(n) be a lifting of gij . Then cijk, obtained from

gij gjk = cijkgik,

defines [W3] ∈ H3(BSO,Z). Let h be the diagonal matrix with the first (n− 1) diagonalentries 1 and the last entry −1. Then hgijh−1 are the transition functions for the univer-sal bundle over BSO(n) with the opposite orientation. Note that hgijh−1 is a lifting ofhgijh−1, which leaves cijk unchanged. We have [W3] = [W3 r] ∈ H3(BSO,Z).Hence there is a homotopy connecting W3 and W3 r. (It is unique up to homotopy asH2(BSO,Z) = 0). Given an α-twisted Spinc manifold (M, ν, ι, η), let −M be the samemanifold with the orientation reversed. Then the homotopy commutative diagram

M

ι

ν BSOr

W3

η

BSO

W3X α

K(Z, 3)

determines a unique equivalence class of α-twisted Spinc structure on −M , called theopposite α-twisted Spinc structure, simply denoted by −(M, ν, ι, η).

DEFINITION 3.7. A geometric cycle for (X,α) is a quintuple (M, ι, ν, η, [E]) where[E] is a K-class in K0(M) and M is a smooth closed manifold equipped with an α-twistedSpinc structure (M, ι, ν, η).

Two geometric cycles (M1, ι1, ν1, η1, [E1]) and (M2, ι,2 ν2, η2, [E2]) are isomorphicif there is an isomorphism f : (M1, ι1, ν1, η1) → (M2, ι2, ν2, η2), as α-twisted Spinc

manifolds over X , such that f!([E1]) = [E2].

Let Γ(X,α) be the collection of all geometric cycles for (X,α). We now impose anequivalence relation ∼ on Γ(X,α), generated by the following three elementary relations:

(1) Direct sum - disjoint unionIf (M, ι, ν, η, [E1]) and (M, ι, ν, η, [E2]) are two geometric cycles with the sameα-twisted Spinc structure, then

(M, ι, ν, η, [E1]) ∪ (M, ι, ν, η, [E2]) ∼ (M, ι, ν, η, [E1] + [E2]).

(2) BordismGiven two geometric cycles (M1, ι1, ν1, η1, [E1]) and (M2, ι2, ν2, η2, [E2]), ifthere exists a α-twisted Spinc manifold (W, ι, ν, η) and [E] ∈ K0(W ) such that

∂(W, ι, ν, η) = −(M1, ι1, ν1, η1) ∪ (M2, ι2, ν2, η2)

and ∂([E]) = [E1]∪ [E2]. Here −(M1, ι1, ν1, η1) denotes the manifold M1 withthe opposite α-twisted Spinc structure.

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RIEMANN-ROCH AND INDEX FORMULAE IN TWISTED K-THEORY 15

(3) Spinc vector bundle modificationSuppose we are given a geometric cycle (M, ι, ν, η, [E]) and a Spinc vectorbundle V over M with even dimensional fibers. Denote by R the trivial rank onereal vector bundle. Choose a Riemannian metric on V ⊕ R, let

M = S(V ⊕ R)

be the sphere bundle of V ⊕ R. Then the vertical tangent bundle T v(M) ofM admits a natural Spinc structure with an associated Z2-graded spinor bundleS+V ⊕ S−

V . Denote by ρ : M → M the projection which is K-oriented. Then

(M, ι, ν, η, [E]) ∼ (M, ι ρ, ν ρ, η ρ, [ρ∗E ⊗ S+V ]).

DEFINITION 3.8. Denote by Kgeo∗ (X,α) = Γ(X,α)/ ∼ the geometric twisted K-

homology. Addition is given by disjoint union - direct sum relation. Note that the equiva-lence relation ∼ preserves the parity of the dimension of the underlying α-twisted Spinc

manifold. Let Kgeo0 (X,α) (resp. Kgeo

1 (X,α) ) the subgroup of Kgeo∗ (X,α) determined

by all geometric cycles with even (resp. odd) dimensional α-twisted Spinc manifolds.

REMARK 3.9. (1) If M , in a geometric cycle (M, ι, ν, η, [E]) for (X,α), is acompact manifold with boundary, then [E] has to be a class in K0(M,∂M).

(2) If f : X → Y is a continuous map and α : Y → K(Z, 3) is a twisting, thenthere is a natural homomorphism of abelian groups

f∗ : Kgeoev/odd(X,α f) −→ Kgeo

ev/odd(Y, α)

sending [M, ι, ν, η, E] to [M, f ι, ν, η, E].(3) Let A be a closed subspace of X , and α be a twisting on X . A relative geometric

cycle for (X,A;α) is a quintuple (M, ι, ν, η, [E]) such that(a) M is a smooth manifold (possibly with boundary), equipped with an α-

twisted Spinc structure (M, ι, ν, η);(b) if M has a non-empty boundary, then ι(∂M) ⊂ A;(c) [E] is a K-class in K0(M) represented by a Z2-graded vector bundle E over

M , or a continuous map M → BU(∞).

The relation ∼ generated by disjoint union - direct sum, bordism and Spinc vectorbundle modification is an equivalence relation. The collection of relative geometric cycles,modulo the equivalence relation is denoted by

Kgeoev/odd(X,A;α).

There exists a natural homomorphism, called the assembly map

μ : Kgeoev/odd(X,α) → Kan

ev/odd(X,α)

whose definition (which we will now explain) requires a careful study of geometric cycles.Given a geometric cycle (M, ι, ν, η, [E]), equip M with a Riemannian metric. Denote

by Cliff(TM) the bundle of complex Clifford algebras of TM over M . The algebra ofsections, C(M,Cliff(TM)), is Morita equivalent to C(M, τ∗BSpinc(K)). Hence, wehave a canonical isomorphism

Kanev/odd(M,W3 τ ) ∼= KKev/odd(C(M,Cliff(M)),C)

with the degree shift by dimM(mod 2). Applying Kasparov’s Poincare duality (Cf.[Kas2])

KKev/odd(C, C(M)) ∼= KKev/odd(C(M,Cliff(M)),C),

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16 ALAN L. CAREY AND BAI-LING WANG

we obtain a canonical isomorphism

PD : K0(M) ∼= Kanev/odd(M, oM ),

with the degree shift by dimM(mod 2). The fundamental class [M ] ∈ Kanev/odd(M, oM )

is the Poincare dual of the unit element in K0(M). Note that [M ] ∈ Kanev (M, oM )) if M

is even dimensional and [M ] ∈ Kanodd(M, oM ) if M is odd dimensional. The cap product

∩ : Kanev/odd(M, oM )⊗K0(M) −→ Kan

ev/odd(M, oM )

is defined by the Kasparov product. We remark that Poincare duality is given by the capproduct of the fundamental K-homology class [M ]

[M ]∩ : K0(M) ∼= Kanev/odd(M, oM ).

Choose an embedding ik : M → Rn+k and take the resulting normal bundle νM . Thenatural isomorphism

TM ⊕ νM ⊕ νM ∼= Rn+k ⊕ νM

and the canonical Spinc structure on νM ⊕ νM define a canonical homotopy between theorientation twisting oM of TM and the orientation twisting oνM

of νM . This canonicalhomotopy defines an isomorphism

I∗ : Kanev/odd(M, oM ) ∼= Kan

ev/odd(M, oνM).(3.3)

Given an α-twisted Spinc manifold (M, ν, ι, η) over X , the homotopy η induces anisomorphism ν∗BSpinc ∼= ι∗Pα as principal K(Z, 2)-bundles on M . Hence there is anisomorphism

ν∗BSpinc(K)η∗

∼= ι∗Pα(K)

as bundles of C∗-algebras on M . This isomorphism determines a canonical isomorphismbetween the corresponding continuous trace C∗-algebras

C(M, ν∗BSpinc(K)) ∼= C(M, ι∗Pα(K)).

Hence, we have a canonical isomorphism

η∗ : Kanev/odd(M, oνM

) ∼= Kanev/odd(M,α ι).(3.4)

Now we can define the assembly map as

μ(M, ι, ν, η, [E]) = ι∗ η∗ I∗([M ] ∩ [E])

in Kanev/odd(X,α). Here ι∗ is the natural push-forward map in analytic twisted K-homology.

THEOREM 3.10. (Theorem 6.4 in [Wa]) The assembly map μ : Kgeoev/odd(X,α) →

Kanev/odd(X,α) is an isomorphism for any smooth manifold X with a twisting α : X →

K(Z, 3).

The proof follows by establishing the existence of a natural map Ψ : Ktopev (X,α) →

Kgeo0 (X,α) such that the following diagram

Ktopev/odd(X,α)

Ψ∼=

∼=

Kgeoev/odd(X,α)

μ

∼= Kan

ev/odd(X,α)

commutes. All the maps in the diagram are isomorphisms.

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RIEMANN-ROCH AND INDEX FORMULAE IN TWISTED K-THEORY 17

REMARK 3.11. The equivalence between the geometric twisted K-homology and theanalytic twisted K-theory holds for any CW pair (X,A). We will return to this in a separatepaper.

COROLLARY 3.12. Kanev/odd(X,α) ∼= Kan

ev/odd(X,−α).

PROOF. By the Brown representation theorem ([Swi]), there is a continuous map i :K(Z, 3) → K(Z, 3) (unique up to homotopy as H2(K(Z, 3),Z) = 0) such that

[i α] = −[α] ∈ H3(X,Z)

for any map α : X → K(Z, 3). Then we have

[i W3] = −[W3] ∈ H3(BSO,Z).

As [W3] is 2-torsion, we know that [i W3] = −[W3] = [W3]. Therefore, there is a homo-topy η0 connecting iW3 and W3, that is, the following diagram is homotopy commutative

BSO

W3

W3

K(Z, 3)

η0

i K(Z, 3).

Note that the homotopy class of η0 as a homotopy connecting W3 and iW3 is unique dueto the fact that H2(BSO,Z) = 0.

Given an α-twisted Spinc manifold (M, ι, ν, η), then the following homotopy com-mutative diagram

M

ι

ν BSO

W3

η

W3

X α

K(Z, 3) η0

i K(Z, 3)

defines a unique (due to H2(BSO,Z) = 0) equivalence class of (−α)-twisted Spinc

structures. Here −α = iα. We denote by i(M, ι, ν, η) this (−α)-twisted Spinc manifold.Obviously,

i(i(M, ι, ν, η)

)= (M, ι, ν, η).

The isomorphism Kanev/odd(X,α) ∼= Kan

ev/odd(X,−α) is induced by the involution i ongeometric cycles.

4. The Chern Character in Twisted K-theory

In this Section, we will review the Chern character map in twisted K-theory on smoothmanifolds developed in [CMW] using gerbe connections and curvings. For the topolog-ical and analytic definitions, see [AS2] and [MatSte] respectively. Recently, Gomi andTerashima in [GoTe] gave another construction of a Chern character for twisted K-theoryusing a notion of connection on a finite-dimensional approximation of a twisted family ofFredholm operators developed by Gomi ([Gomi].

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18 ALAN L. CAREY AND BAI-LING WANG

4.1. Twisted Chern character. For a fibration π∗ : Y → X , let Y [p] denote the pthfibered product. There are projection maps πi : Y

[p] → Y [p−1] which omit the ith elementfor each i = 1 . . . p. These define a map

(4.1) δ : Ωq(Y [p−1]) → Ωq(Y [p])

by

(4.2) δ(ω) =

p∑i=1

(−1)iπ∗i (ω).

Clearly δ2 = 0. In fact, the δ-cohomology of this complex vanishes identically, hence, thesequence

0 Ωq(X)π∗

Ωq(Y ) · · · δ Ωq(Y [p−1])δ Ωq(Y [p]) · · ·

is exact.Returning now to our particular example, a bundle gerbe connection on Pα is a unitary

connection θ on the principal U(1)-bundle Gα over P [2]α which commutes with the bundle

gerbe product. A bundle gerbe connection θ has curvature

Fθ ∈ Ω2(P [2]α )

satisfying δ(Fθ) = 0. There exists a two-form ω on Pα such that

Fθ = π∗2(ω)− π∗

1(ω).

Such an ω is called a curving for the gerbe connection θ. The choice of a curving is notunique, the ambiguity in the choice is precisely the addition of the pull-back to Pα of atwo-form on X . Given a choice of curving ω, there is a unique closed three-form on β onX satisfying dω = π∗β. We denote by

α = (Gα, θ, ω)

the lifting bundle gerbe Gα with the connection θ and a curving ω. Moreover H =β

2π√−1

is a de Rham representative for the Dixmier-Douady class [α]. We shall call α the differ-ential twisting, as it is the twisting in differential twisted K-theory (Cf. [CMW]).

The following theorem is established in [CMW].

THEOREM 4.1. Let X be a smooth manifold, π : Pα → X be a principal PU(H)bundle over X whose classifying map is given by α : X −→ K(Z, 3). Let α = (Gα, θ, ω)be a bundle gerbe connection θ and a curving ω on the lifting bundle gerbe Gα. There is awell-defined twisted Chern character

Chα : K∗(X,α) −→ Hev/odd(X, d−H).

Here the groups Hev/odd(X, d−H) are the twisted cohomology groups of the complex ofdifferential forms on X with the coboundary operator given by d −H . The twisted Cherncharacter is functorial under the pull-back. Moreover, given another differential twistingα+ b = (Gα, θ, ω + π∗b) for a 2-form b on X ,

Chα+b = Chα · exp( b

2π√−1

).

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RIEMANN-ROCH AND INDEX FORMULAE IN TWISTED K-THEORY 19

PROOF. Choose a good open cover Vi of X such that Pα → X has trivializingsections φi over each Vi with transition functions gij : Vi ∩ Vj −→ PU(H) satisfyingφj = φigij . Define σijk by gij gjk = gikσijk for a lift of gij to gij : Vi ∩ Vj → U(H).

Note that the pair (φi, φj) defines a section of P [2]α over Vi ∩ Vj . The connection θ can

be pulled back by (φi, φj) to define a 1-form Aij on Vi ∩ Vj and the curving ω can bepulled-back by the φi to define two-forms Bi on Vi. Then the differential twisting definesthe triple

(σijk, Aij , Bi)(4.3)

which is a degree two smooth Deligne cocycle. Now we explain in some detail the twistedChern characters in both the odd and even case following [CMW].

The even case: As a model for the K0 classifying space, we choose Fred, the spaceof bounded self-adjoint Fredholm operators with essential spectrum ±1 and otherwisediscrete spectra, with a grading operator Γ which anticommutes with the given family ofFredholm operators.

A twisted K-class in K0(X,α) can be represented by f : Pα → Fred, a PU(H)-equivariant family of Fredholm operators. We can select an open cover Vi of X suchthat on each Vi there is a local section φi : Vi → Pα and for each i the Fredholm operatorsf(φi(x)), x ∈ Vi have a gap in the spectrum at both ±λi = 0. Then over Vi we have afinite rank vector bundle Ei defined by the spectral projections of the operators f(φi(x))corresponding to the interval [−λi, λ].

Passing to a finer cover Ui if necessary, we may assume that Ei is a trivial vectorbundle over Ui of rank ni. Choosing a trivialization of Ei gives a Z2 graded parametrixqi (an inverse up to finite rank operators) of the family f φi. In the index zero sector theoperator qi(x)−1 is defined as the direct sum of the restriction of f(φi(x)) to the orthogonalcomplement of Ei in H and an isomorphism between the vector bundles E+

i and E−i .

Clearly then f(φi(x))qi(x) = 1 modulo rank ni operators. In the case of nonzero indexone defines a parametrix as a graded invertible operator qi such that f(φi(x))qi(x) = snmodulo finite rank operators, with sn a fixed Fredholm operator of index n equal to theindex of f(φi(x)).

On the overlap Uij we have a pair of parametrices qi and qj of families of f φi andf φj respectively. These are related by an invertible operator fij which is of the form 1+a finite rank operator,

gijqj(x)g−1ij = qi(x)fij(x).

The conjugation on the left hand side by gij comes from the equivariance relation

f(φj(x)) = f(φi(x)gij(x)) = gij(x)−1f(φi(x))gij(x).

The system fij does not quite satisfy the Cech cocycle relation needed to define aprincipal bundle, because of the different local sections φi : Ui → Pσ involved. Instead,we have on Uijk

gjkqkg−1jk = qjfjk = (g−1

ij qifij gij)fjk = gjk(g−1ik qifikgik)g

−1jk .

Using the relation gjkg−1ik = σijkg

−1ij , we get

gjk(g−1ik qifikgik)g

−1jk = g−1

ij qifikgij

multiplying the last equation from right by g−1ij and from the left by q−1

i gij one gets thetwisted cocycle relation

fij(gijfjkg−1ij ) = fik,

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20 ALAN L. CAREY AND BAI-LING WANG

which is independent of the choice of the lifting gij . For simplicity, we will just write theabove twisted cocycle relation as

fij(gijfjkg−1ij ) = fik.(4.4)

This twisted cocycle relation (4.4) actually defines an untwisted cocycle relation for(gij , fij) in the twisted product

G = PU(H)GL(∞),

where the group PU(H) acts on the group GL(∞) of invertible 1+ finite rank operatorsby conjugation. Thus the product in G is given by

(g, f) · (g′, f ′) = (gg′, f(gf ′g−1)).

The cocycle relation for the pairs (gij , fij) then encodes both the cocycle relation for thetransition functions gij of the PU(H) bundle over X and the twisted cocycle relation(4.4). In summary, this cocycle (gij , fij) defines a principal G bundle over X.

The classifying space BG is a fiber bundle over K(Z, 3). The fiber at each point inK(Z, 3) is homeomorphic (but not canonically so) to the space Fred of graded Fredholmoperators; to set up the isomorphism one needs a choice of element in each fiber. Givena principal PU(H)-bundle Pα over X defined by α : X → K(Z, 3), the even twistedK-theory K0(X,Pα) is the set of homotopy classes of maps X → BG covering the mapα.

Next we construct the twisted Chern character from a connection ∇ on a principal Gbundle over X associated to the cocycle (gij , fij). Locally, on a good open cover Uiof X we can lift the connection to a connection taking values in the Lie algebra g of thecentral extension U(H) × GL(∞) of G. Denote by F∇ the curvature of this connection.On the overlaps Uij the curvature satisfies a twisted relation

F∇,j = Ad(gij ,fij)−1 F∇,i + g∗ijc,

where c is the curvature of the canonical connection θ on the principal U(1)-bundle U(H) →PU(H).

Since the Lie algebra u(∞) ⊕ C is an ideal in the Lie algebra of U(H) GL(∞),

the projection F ′∇,i of the curvature F∇,i onto this subalgebra transforms in the same way

as F under change of local trivialization. It follows that for a PU(H)-equivariant mapf : Pα → Fred, we can define a twisted Chern character form of f as

chα(f,∇) = eBi tr eF′∇,i/2πi,(4.5)

over Vi. Here the trace is well-defined on gl(∞) and on the center C it is defined as thecoefficient of the unit operator. Note that chα(f,∇) is globally defined and (d−H)-closed

(d−H)chα(f,∇) = 0,

and depends on the differential twisting

α = (Gα, θ, ω).

Let f0 and f1 be homotopic, and ∇0 and ∇1 be two connections on the principal G bundleover X , we have a Chern-Simons type form

CS((f0,∇0), (f1,∇1)),

well-defined modulo (d−H)-exact forms, such that

chα(f1,∇1)− chα(f0,∇0) = (d−H)CS((f0,∇0), (f1,∇1)).(4.6)

114

RIEMANN-ROCH AND INDEX FORMULAE IN TWISTED K-THEORY 21

The proof follows directly from the local computation using (4.5). Hence, the (d − H)-cohomology class of chα(f,∇) does not depend choices of a connection ∇ on the principalG bundle over X , and depends only on the homotopy class of f . We denote the (d−H)-cohomology class of chα(f,∇) by Chα([f1]) which is a natural homomorphism

Chα : K0(X,α) −→ Hev(X, d−H).

From (4.5), we have

Chα+b = Chα · exp( b

2π√−1

).

for a differential twisting α+ b = (Gα, θ, ω + π∗b).The odd case: The odd case is a little easier. First, as a model for the K1 classifying

space, we choose U(∞) = lim−→nU(n), the stabilized unitary group. Let Θ be the universal

odd character form on U(∞) defined by the canonical left invariant u(∞)-valued form onU(∞).

Let H = H+ ⊕H− be a polarized Hilbert space and let Ures = Ures(H) denotes thegroup of unitary operators in H with Hilbert-Schmidt off-diagonal blocks. The conjugationaction of U(H+)×U(H−) on Ures defines an action of PU0(H) = P (U(H+)×U(H−))on Ures. Note that the classifying space of Ures is U(∞).

Define

H = PU0(H) Ures.

Then given a principal PU0(H)-bundle Pα over X defined by α : X → K(Z, 3), the oddtwisted K-theory K1(X,α) is the set of homotopy classes of maps X → BH coveringthe map α. These are represented by PU0(H)-equivariant maps f : Pα → U(∞). Withrespect to trivializing sections φi over each Vi. Then

eBi(f φi)∗Θ

is a globally defined and (d − H)-closed differential form on X . This defines the oddversion of the twisted Chern character

Chα : K1(X,α) −→ Hodd(X, d−H).

4.2. Differential twisted K-theory. Recall that the Bockstein exact sequence in com-plex K-theory for any finite CW complex:

K0(X)ch Hev(X,R) K0

R/Z(X)

K1

R/Z(X)

Hodd(X,R) K1(X)ch

(4.7)

where K∗R/Z(X) is K-theory with R/Z-coefficients as in [Kar1] and [Ba].

Analogously, in twisted K-theory, given a smooth manifold X with a twisting α :X → K(Z, 3), upon a choice of a differential twisting

α = (Gα, θ, ω)

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22 ALAN L. CAREY AND BAI-LING WANG

lifting α, we have the corresponding Bockstein exact sequence in twisted K-theory

K0(X,α)Chα Hev(X, d−H) K0

R/Z(X,α)

K1

R/Z(X,α)

Hodd(X, d−H) K1(X,α)Chα

.(4.8)

Here K0R/Z(X,α) and K1

R/Z(X,α) are subgroups of differential twisted K-theory, respec-

tively K0(X, α) and K1(X, α) (see [CMW] for the detailed construction). Here we giveanother equivalent construction of differential twisted K-theory.

Fix a choice of a connection ∇ on a principal G bundle over X . Then K0(X, α) isthe abelian group generated by pairs

(f, η),

modulo an equivalence relation, where f : Pα → Fred is a PU(H)-equivariant map andη is an odd differential form modulo (d−H)-exact forms. Two pairs (f0, η0) and (f1, η1)are called equivalent if and only if

η1 − η0 = CS((f1,∇), (f0,∇)).

The differential Chern character form of f is given by

chα(f,∇)− (d−H)η

which defines a homomorphism

chα : K0(X, α) −→ Ωev0 (X, d−H),

where Ωev0 (X, d −H) is the image of chα : K0(X, α) → Ωev(X). The kernel of chα is

isomorphic to K1R/Z(X,α).

Similarly, we define the odd differential twisted K-theory K1(X, α) with the differ-ential Chern character form homomorphism

chα : K1(X, α) −→ Ωodd0 (X, d−H).

The kernel of chα is isomorphic to K0R/Z(X,α). The following commutative diagrams

were established in [CMW] relating differential twisted K-theory with twisted K-theory

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RIEMANN-ROCH AND INDEX FORMULAE IN TWISTED K-THEORY 23

and with the diagram (4.8)

0

Hodd(X, d−H)

K1R/Z(X, σ)

0 → Ωodd(X)

Ωodd0 (X, d−H)

d−HK0(X, σ)

chσ

K0(X,σ) → 0

Chσ

Ωev

0 (X, d−H)

Hev(X, d−H)

0

and

0

Hev(X, d−H)

K0R/Z(X, σ)

0 → Ωev(X)

Ωev0 (X, d−H)

d−HK1(X, σ)

chσ

K1(X,σ) → 0

Chσ

Ωodd

0 (X, d−H)

Hodd(X, d−H)

0

with exact horizontal and vertial sequences, and exact upper-right and exact lower-left 4-term sequences. We expect that these two commutative diagrams uniquely characterizedifferential twisted K-theory.

4.3. Twisted Chern character for torsion twistings. In this paper, we will only usethe twisted Chern character for a torsion twisting and in this case we will give an explicitconstruction. Let E be a real oriented vector bundle of rank 2k over X with its orientationtwisting denoted by

o(E) : X → K(Z, 3).

The associated lifting bundle Go(E) has a canonical reduction to the Spinc bundle gerbeGW3(E).

Choose a local trivialization of E over a good open cover Vi of X . Then the transi-tion functions

gij : Vi ∩ Vj −→ SO(2k).

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24 ALAN L. CAREY AND BAI-LING WANG

define an element in H1(X,SO(2k)) whose image under the Bockstein exact sequence

H1(X,Spin(2k)) → H1(X,SO(2k)) → H2(X,Z2)

is the second Stieffel-Whitney class w2(E) of E. Denote the differential twisting by

w2(E) = (GW3(E), θ, 0),

the Spinc bundle gerbe GW3(E) with a flat connection θ and a trivial curving. With re-spect to a good cover Vi of X the differential twisting w2(E) defines a Deligne cocy-cle (αijk, 0, 0) with trivial local B-fields, here αijk = gij gjkgki where gij : Uij →Spinc(2n) is a lift of gij .

By Proposition 2.4, a twisted K-class in K0(X, o(E)) can be represented by a Cliffordbundle, denoted E . Equip E with a Clifford connection, and E with a SO(2k)-connection.Locally, over each Vi we let E|Vi

∼= Si ⊗ Ei where Si is the local fundamental spinorbundle associated to E|Vi

with the standard Clifford action of Cliff(E|Vi) obtained from

the fundamental representation of Spin(2k). Then Ei is a complex vector bundle over Vi

with a connection ∇i such that on Vi ∩ Vj

Ch(Ei,∇i) = Ch(Ej,∇j).

Hence, the twisted Chern character

Chw2(E) : K0(X, o(E)) −→ Hev(X)

is given by [E ] → [ch(Ei,∇i)] = ch(Ei). The proof of the following proposition isstraightforward.

PROPOSITION 4.2. The twisted Chern character satisfies the following identities

(1) Chw2(E1⊕E2)([E1 ⊕ E2]) = Chw2(E1)([E1]) + Chw2(E2)([E2]).(2) Chw2(E1⊗E2)([E1 ⊗ E2]) = Chw2(E1)([E1])Chw2(E2)([E2]).

In the case that E has a Spinc structure whose determinant line bundle is L, there is acanonical isomorphism

K0(X) −→ K0(X, o(E)),

given by [V ] → [V ⊗ SE ] where SE is the associated spinor bundle of E. Then we have

Chw2(E)([V ⊗ SE ]) = ec1(L)

2 ch([V ]),

where ch([V ]) is the ordinary Chern character of [V ] ∈ K0(X).In particular, whenX is an even dimensional Riemannian manifold, and TX is equipped

with the Levi-Civita connection, under the identification of K0(X, o(E))with the Grothendieckgroup of Clifford modules. Then

Chw2(X)([E ]) = ch(E/S)(4.9)

where ch(E/S) is the relative Chern character of the Clifford module E constructed inSection 4.1 of [BGV].

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RIEMANN-ROCH AND INDEX FORMULAE IN TWISTED K-THEORY 25

5. Thom Classes and Riemann-Roch Formula in Twisted K-theory

5.1. The Thom class. Given any oriented real vector bundle π : E → X of rank 2k,E admits a Spinc structure if its classifying map τ : X → BSO(2k) admits a lift τ

BSpinc

X

τ

τ

BSO(2k).

As BSpinc → BSO(2k) is a BU(1)-principal bundle with the classifying map given by

W3 : BSO(2k) → K(Z, 3),

E admits a Spinc structure if W3 τ : X → K(Z, 3) is null homotopic, and a choice ofnull homotopy determines a Spinc structure on E. Associated to a Spinc structure s onE, there is canonical K-theoretical Thom class

UsE = [π∗S+, π∗S−, cl] ∈ K0

cv(E)

in the K-theory of E with vertical compact supports. Here S+ and S− are the positive andnegative spinor bundle over X defined by the Spinc structure on E, and cl is the bundlemap π∗S+ → π∗S− given by the Clifford action E on S±.

REMARK 5.1. (1) The restriction of UsE to each fiber is a generator of K0(R2k),

so a Spinc structure on E is equivalent to a K-orientation on E. Note that Thomclasses and K-orientation are functorial under pull-backs of Spinc vector bun-dles.

(2) Let s ⊗ L be another Spinc structure on E which differs from s by a complexline bundle p : L → X , then

Us′

E = UsE · p∗([L]).

(3) Let (E1, s1) and (E2, s2) be two Spinc vector bundles over X , p1 and p2 be theprojections from E1 ⊕ E2 to E1 and E2 respectively, then

Us1⊕s2E1⊕E2

= p∗1(Us1E1

) · p∗2(Us2E2

) ∈ K0cv(E1 ⊕ E2).

(4) The Thom isomorphism in K-theory for a Spinc vector bundle π : E → X ofrank 2k is given by

ΦKE : K0(X) −→ K0(E)

a → π∗(a)UsE.

Here for locally compact spaces, we shall consider only K-theory with compactsupports. When X is compact, Us

E ∈ K0(TX) and the Thom isomorphism ΦKX

is the inverse of the push-forward map

π! : K0(TX) → K0(X)

associated to the K-orientation of π defined the Spinc structure on E.

If an oriented vector bundle E of even rank over X does not admit a Spinc structure,W3 τ : X → K(Z, 3) is not null homotopic. Thus, W3 τ defines a twisting on Xfor K-theory, called the orientation twisting oE . In this Section we will define a canonicalThom class

UE ∈ K0(E, π∗oE)

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26 ALAN L. CAREY AND BAI-LING WANG

such that a → π∗(a) ∪ UE defines the Thom isomorphism K0(X, oE) ∼= K0(E). In fact,a → π∗(a) ∪ UE defines the Thom isomorphism (Cf. [CW1])

K0(X,α+ oE) ∼= K0(E,α π)

for any twisting α : X → K(Z, 3).Choose a good open cover Vi of X such that Ei = E|Vi

is trivialized by an isomor-phism

Ei∼= Vi × R2n.

This defines a canonical Spinc structure si on each Ei. Denote by UsiEi

the associatedThom class of (Ei, si). Then we have

Usj

Ej= Usi

Eiπ∗ij([Lij ]) ∈ K0

cv(Eij)

where Lij is the difference line bundle over Vij = Vi ∩ Vj defined by sj = si ⊗ Lij

on Eij = E|Vij. Recall that these local line bundles Lij define a bundle gerbe [Mur]

associated to the twisting oE = W3 τ : X → K(Z, 3) and a locally trivializing coverVi . By the definition of twisted K-theory, Usi

Ei defines a twisted K-theory class of E

with compact vertical supports and twisting given by

π∗(oE) = oE π : E → K(Z, 3).

We denote this canonical twisted K-theory class by

UE ∈ K0cv(E, π∗(oE)).

When X is compact, then UE ∈ K0(E, π∗(oE)). One can easily show that the Thom classUE does not depend on the choice of the trivializing cover.

Now we can list the properties of the Thom class in twisted K-theory.

PROPOSITION 5.2. (1) If E is equipped with a Spinc structure s, then s definesa canonical isomorphism

φs : K0cv(E, π∗(oE)) −→ K0

cv(E)

such that φs(UE) = UsE .

(2) Let f : X → Y be a continuous map and E be an oriented vector bundle of evenrank over Y , then

Uf∗E = f∗(UE).

(3) Let E1 and E2 be two oriented vector bundles of even rank over X , p1 and p2be the projections from E1 ⊕ E2 to E1 and E2 respectively, that is, we have thediagram

E1 ⊕ E2p2

p1

E2

π2

E1

π1 X,

then

UE1⊕E2= p∗1(UE1

) · p∗2(UE2).

(4) Let π : E → X be an oriented vector bundle of even rank over a compact spaceX , the Thom isomorphism in twisted K-theory ([CW1])

ΦKE : K0(X,α+ oE) ∼= K0(E, π∗(α))

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RIEMANN-ROCH AND INDEX FORMULAE IN TWISTED K-THEORY 27

is given by a → π∗(a)·UE . Moreover, the push-forward map in twisted K-theory([CW1])

π! : K0(E, π∗(α)) −→ K0(X,α+ oE)

satisfies π!(π∗(a) · UE) = a.

PROOF. (1) The Spinc structure s defines canonical isomorphism

φs : K0cv(E, π∗(oE)) −→ K0

cv(E)

as follows. Given a trivializing cover Vi and the canonical Spinc structure si

on Ei = E|Vi, we have

s|Ei= si ⊗ Li

for a complex line bundle πi : Li → Vi. This implies UsEi

= UsiEiπ∗([Li]). Note

that Lij = Li ⊗ L∗j .

Any twisted K-class a in K0cv(E, π∗(oE)) is given by a local K-class ai with

compact vertical support such that aj = aiπ∗ij([Lij ]), then

aiπ∗i ([Li]) = ajπ

∗([Lj ])

in K0cv(E|Vij

). This defines the homomorphism φ, which is obviously an iso-morphism sending UE to Us

E .(2) Choose a good open cover Vi of Y . By definition, the Thom class UE is

defined by UsiEi with

Usj

Ej= Usi

Eiπ∗ij([Lij ]).

Then f−1(Vi) is an open cover of X , and (f∗E)|f−1(Vi) = f∗Ei is trivializedwith the canonical Spinc structure f∗si, thus

Uf∗si

f∗Ei= f∗Usi

Ei.

This gives Uf∗E = f∗(UE).(3) The proof is similar to the proof of (2).(4) From ([CW1]), we know that the Thom isomorphism and the push-forward map

in twisted K-theory are both homomorphisms of K0(X,α)-modules. There ex-ists an oriented real vector bundle F of even rank such that

E ⊕ F = X × R2m

for some m ∈ N. Thus, we have

Ei

π

X × R2m

p

X

From the construction of the push-forward map in ([CW1]), we see that

π!(UE) = p! i!(UE) = p!(UE⊕F ) = 1.

As the Thom isomorphism and the push-forward map in twisted K-theory areboth homomorphisms of K0(X,α)-modules, we get π!(π

∗(a) · UE) = a.Note that the Thom isomorphism is inverse to the push-forward map π!,

hence, the Thom isomorphism in twisted K-theory

K0(X,α+ oE) ∼= K0(E, π∗(α))

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28 ALAN L. CAREY AND BAI-LING WANG

is given by a → π∗(a) · UE .

5.2. Twisted Riemann-Roch. By an application of the Thom class and Thom iso-morphism in twisted K-theory, we will now give a direct proof of a special case of theRiemann-Roch theorem for twisted K-theory. With some notational changes, the argumentcan be applied to establish the general Riemann-Roch theorem in twisted K-theory. Denoteby oX and oY the orientation twistings associated to the tangent bundles πX : TX → Xand πY : TY → Y respectively.

THEOREM 5.3. Given a smooth map f : X → Y between oriented manifolds, assumethat dimY − dimX = 0 mod 2. Then the Riemann-Roch formula is given by

Chw2(Y )

(fK! (a)

)A(Y ) = fH

∗(Chw2(X)(a)A(X)

).

for any a ∈ K0(X, oX). Here A(X) and A(Y ) are the A-hat classes of X and Y respec-tively.

PROOF. For simplicity, assume that both X and Y are of even dimension, say 2m and2n respectively, equipped with a Riemannian metric. We will consider Chern characterdefects in each of the following three squares

K0(X, oX)ΦK

TX

∼=

Chw2(X)

K0c (TX)

(df)K!

Ch

K0c (TY ) ∼=

(ΦKTY )−1

Ch

K0(Y, oY )

Chw2(Y )

Hev(X)

ΦHTX

∼= Hev

c (TX)(df)H∗ Hev

c (TY ) ∼=

(ΦHTY )−1

Hev(Y )

(5.1)

where ΦKTX and ΦK

TY are the Thom isomorphisms in twisted K-theory for TX and TY ,ΦH

TX and ΦHTY are the cohomology Thom isomorphisms for TX and TY . Then we have

(1) The push-forward map in twisted K-theory as established in [CW1]

fK! : K0(X, oX) → K0(Y, oY )

agrees with(ΦK

TY )−1 (df)K! ΦK

TX .

(2) The push-forward map in cohomology theory fH∗ : Hev(X) → Hev(Y ) is given

byfH∗ = (ΦH

TY )−1 (df)H∗ ΦH

TX .

Denote by UHTX and UH

TY the cohomological Thom classes for TX and TY . Thenunder the pull-back of the zero section, 0∗X(UH

TX) = e(TX) and 0∗Y (UHTY ) = e(TY ) are

the Euler classes for TX and TY respectively.Let the Pontrjagin classes of πX : TX → X be symmetric polynomials in x2

1, · · · , x2m,

then

A(X) =

m∏k=1

xk/2

sinh(xk/2).

The Chern character defect for the left square in (5.1) is given by

Ch(ΦK

TX(a))= ΦH

TX

(Chw2(X)(a)A

−1(X))

(5.2)

for any a ∈ K0(X, oX). Here A(X) is the A-hat class of TX .

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RIEMANN-ROCH AND INDEX FORMULAE IN TWISTED K-THEORY 29

To prove (5.2), note that

(ΦHTX)−1Ch

(ΦK

TX(a))

= (ΦHTX)−1

(Ch(π∗

X(a) · UTX))

Apply Prop. (4.2)

= (ΦHTX)−1

(Chπ∗

Xw2(X)(π∗X(a)) · Chπ∗

Xw2(X)(UTX))

= (ΦHTX)−1

(π∗X(Chw2(X)(a)) · Chπ∗

Xw2(X)(UTX))

Note that (ΦHTX)−1 = (πX)∗.

= Chw2(X)(a)(ΦHTX)−1

(Chπ∗

Xw2(X)(UTX))

By the projection formula.

So the Chern character defect for the square

K0(X, oX)ΦK

TX

∼=

Chw2(X)

K0c (TX)

Ch

Hev(X)

ΦHTX

∼= Hev

c (TX)

is given by

D(X) = (ΦHTX)−1

(Chπ∗

Xw2(X)(UTX))∈ Hev(X).

From the cohomology Thom isomorphism, we have

0∗X ΦHTX(D(X)) = D(X)e(TX),

under the pull-back of the zero section 0X of the tangent bundle TX .Therefore, we have

D(X) =0∗X

(Chπ∗

Xw2(X)(UTX))

e(TX).

By the construction of the Thom class UTX , under the pull-back of the zero section 0X ofTX , 0∗X(UTX) is a twisted K-class in K0(X, oX) and

0∗XChπ∗Xw2(X)(UTX)

= Chw2(X)(0∗X(UTX))

=∏m

k=1(exk/2 − e−xk/2).

Thus, (5.2) follows from

D(X) =

m∏k=1

(exk/2 − e−xk/2)

xk= A−1(X).

This implies that the following diagram commutes

K0(X, oX)ΦK

TX

∼=

Chw2(X)(−)·A(X)

K0c (TX)

Ch(−)·π∗XA2(X)

Hev(X)

ΦHTX

∼= Hev

c (TX).

(5.3)

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30 ALAN L. CAREY AND BAI-LING WANG

Similarly, the Chern character defect in

K0(Y, oY )ΦK

TY

∼=

Chw2(Y )

K0c (TY )

Ch

Hev(Y )

ΦHTY

∼= Hev

c (TY )

is given by

Ch(ΦK

TY (a))= ΦH

TY

(Chw2(Y )(a)A

−1(Y ))

for any a ∈ K0(Y, oY ). This implies that the Chern character defect for the right square in(5.1) is given by

Chw2(Y )

((ΦK

TY )−1(c)

)· A−1(Y ) = (ΦH

TY )−1

(Ch(c)

)(5.4)

for any c ∈ K0c (TY ). Hence, we have the following commutative diagram

K0c (TY )

Ch(−)·π∗Y A2(Y )

(ΦKTY )−1

∼= K0(Y, oY )

Chw2(Y )(−)·A(Y )

Hev

c (TY )(ΦH

TY )−1

∼= Hev(Y ).

(5.5)

The Chern character for the middle square in (5.1) follows from the Riemann-Rochtheorem in ordinary K-theory for the K-oriented map df : TX → TY with the orientationgiven by canonical Spinc manifolds TX and TY . Note that the Todd classes of Spinc

manifolds of TX and TY are given by π∗X(A2(X)) and π∗

Y (A2(Y )) respectively. This is

due to two facts, that T (TX) ∼= π∗X(TX ⊗ C) and that Td(TX ⊗ C) = A2(X). So we

have

Ch((df)K! (a)

)· π∗

Y

(A2(Y )

)= (df)H∗

(Ch(a)π∗

X(A2(X)))

(5.6)

for any a ∈ K0c (TX). Hence, the following diagram commutes

K0c (TX)

Ch(−)·π∗XA2(X)

(df!)K

K0c (TY )

Ch(−)·π∗Y A2(Y )

Hev

c (TX)(df)H∗ Hev

c (TY ).

(5.7)

Putting (5.3), (5.5) and (5.7) together, we get the following commutative diagram

K0(X, oX)fK!

Chw2(X)(−)·A(X)

K0(Y, oY )

Chw2(Y )(−)·A(Y )

Hev(X)

fH∗ Hev(Y )

which leads to

Chw2(Y )

(fK! (a)

)A(Y ) = fH

∗(Chw2(X)(a)A(X)

)

for any a ∈ K0(X, oX). This completes the proof of the Riemann-Roch theorem in twistedK-theory.

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RIEMANN-ROCH AND INDEX FORMULAE IN TWISTED K-THEORY 31

With some notational changes, the above argument can be applied to establish thegeneral Riemann-Roch theorem in twisted K-theory. Let f : X → Y a smooth mapbetween oriented manifolds with dimY − dimX = 0 mod 2. Let α = (Gα, θ, ω) bea differential twisting which lifts α : Y → K(Z, 3), f∗(α) is the pull-back differentialtwisting which lifts α f : X → K(Z, 3). Then we have the following Riemann-Rochformula

Chα

(fK! (a)

)A(Y ) = fH

∗(Chf∗α+w2(Y )+f∗w2(Y )(a)A(X)

)(5.8)

for any a ∈ K0(X,α f + oX + f∗(oY )). In particular, we have the following Riemann-Roch formula for a trivial twisting α : Y → K(Z, 3)

Ch(fK! (a)

)A(Y ) = fH

∗(Chw2(Y )+f∗w2(Y )(a)A(X)

)(5.9)

for any a ∈ K0(X, oX + f∗(oY )). When Y is a point, then the Riemann-Roch formula(5.9) agrees with the index formula obtained by Murray and Singer in [MS].

When f is K-oriented, and equipped with a Spinc structure whose determinant bundleis L, there is a canonical isomorphism

Ψ : K0(X) ∼= K0(X, oX + f∗(oY ))

such that Chw2(Y )+f∗w2(Y )(Ψ(a)) = ec1(L)/2Ch(a) for any a ∈ K0(X). Then thetwisted Riemann-Roch formula (5.9) agrees with the Riemann-Roch formula for K-orientedmaps as established in [AH] and [BEM] in the presence of H-flux.

6. The Twisted Index Formula

In this Section, we establish the index pairing for a closed smooth manifold with atwisting α

Kev/odd(X,α)×Kanev/odd(X,α) −→ Z

in terms of the local index formula for twisted geometric cycles.

THEOREM 6.1. Let X be a smooth closed manifold with a twisting α : X → K(Z, 3).The index pairing

K0(X,α)×K0(X,α) −→ Z

is given by

〈ξ, (M, ι, ν, η, [E])〉 =∫M

Chw2(M)

(η∗(ι

∗ξ ⊗ E))A(M)

where ξ ∈ K0(X,α), and the geometric cycle (M, ι, ν, η, [E]) defines a twisted K-homology class on (X,α). Here

η∗ : K∗(M, ι∗α) ∼= K∗(M, oM )

is an isomorphism, and Chw2(M) is the Chern character on K0(M, oM ).

PROOF. Recall that the index pairing K0(X,α) × K0(X,α) −→ Z can be definedby the internal Kasparov product (Cf. [Kas3] and [ConSka])

KK(C, C(X,Pα(K)))×KK(C(X,Pα(K)),C) −→ KK(C,C) ∼= Z,

and is functorial in the sense that if f : Y → X is a continuous map and Y is equippedwith a twisting α : X → Z then

〈f∗b, a〉 = 〈b, f∗(a)〉for any a ∈ K0(Y, f

∗α) and b ∈ K0(X,α).

125

32 ALAN L. CAREY AND BAI-LING WANG

Note that under the assembly map, the geometric cycle (M, ι, ν, η, [E]) is mapped toι∗ η∗([M ] ∩ [E]), for ξ ∈ K0(X,α). Hence, we have

〈ξ, (M, ι, ν, η, [E])〉= 〈ξ, ι∗ η∗([M ] ∩ [E])〉= 〈ι∗ξ, η∗([M ] ∩ [E])〉= 〈η∗(ι∗ξ ⊗ E), [M ]〉.

Here η∗(ι∗ξ ⊗ E) ∈ K0(M, oM ) and [M ] is the fundamental class in Kan

ev (M, oM )which is Poincare dual to the unit element C in K0(M). The index pairing betweenK0(M, oM )×Kan

ev (M, oM ) can be written as

K0(M, oM )×Kanev (M, oM ) → K0(M, oM )×K0(M) → K0(M, oM ) → Z

where the first map is given by the Poincare duality Kanev (M, oM ) ∼= K0(M), the middle

map is the action of K0(M) on K0(M, oM ), and the last map is the push-forward map ofε : M → pt. Therefore, we have

〈η∗(ι∗ξ ⊗ E), [M ]〉= εK!

(η∗(ι

∗ξ ⊗ E)⊗ C)

= εK!(η∗(ι

∗ξ ⊗ E)).

By twisted Riemann-Roch (Theorem 5.3),

ε!(η∗(ι

∗ξ ⊗ E))

= εH∗(Chw2(M)

(η∗(ι

∗ξ ⊗ E))A(M)

)

=

∫M

Chw2(M)

(η∗(ι

∗ξ ⊗ E))A(M).

This completes the proof of the twisted index formula.

Note that ε : M → pt can be written as ι εX : M → X → pt. Applying theRiemann-Roch Theorem 5.3, we can write the above index pairing as

< (M, ι, ν, η, [E]), ξ >

=

∫M

Chw2(M)

(η∗(ι

∗ξ ⊗ E))A(M)

=

∫X

Chw2(X)

(ι!(E)⊗ ξ

)A(X)

where ι! : K0(M) → K0(X,−α+ oX) is the push-forward map in twisted K-theory,

K0(X,α)×K0(X,−α+ oX) −→ K0(X, oX)

is the multiplication map (2.7), and

Chw2(X) : K0(X, oX) −→ Hev(X)

is the twisted Chern character (which agrees with the relative Chern character under theidentification K0(X, oX) ∼= K0(X,W3(X)), the K-theory of Clifford modules on X).

126

RIEMANN-ROCH AND INDEX FORMULAE IN TWISTED K-THEORY 33

7. Mathematical Definition of D-branes and D-brane Charges

Here we give a mathematical interpretation of D-branes in Type II string theory us-ing the twisted geometric cycles and use the index theorem in the previous Section tocompute charges of D-branes. In Type II superstring theory on a manifold X , a stringworldsheet is an oriented Riemann surface Σ, mapped into X with ∂Σ mapped to an ori-ented submanifold M (called a D-brane world-volume, a source of the Ramond-Ramondflux). The theory also has a Neveu-Schwarz B-field classified by a characteristic class[α] ∈ H3(X,Z).

In physics, the D-brane world volume M carries a gauge field on a complex vectorbundle (called the Chan-Paton bundle), so a D-brane is given by a submanifold M of Xwith a complex bundle E and a connection ∇E . This data actually defines a differentialK-class

[(E,∇E)]

in differential K-theory K(M).When the B-field is topologically trivial, that is [α] = 0, D-brane charge takes val-

ues in ordinary K-theory K0(X) or K1(X) for Type IIB or Type IIA string theory (asexplained in [MM][Wit]). For a D-brane M to define a class in the K-theory of X , itsnormal bundle νM must be endowed with a Spinc structure. Equivalently, the embedding

ι : M −→ X

is K-oriented so that the push-forward map in K-theory ([AH])

ιK! : K0(M) −→ Kev/odd(X)

is well-defined, (it takes values in even or odd K-groups depending on the dimension ofM ). So the D-brane charge of (ι : M → X,E) is

ιK! ([E]) ∈ Kev/odd(X).

It was proposed in [MM] that the cohomological Ramond-Ramond charge of the D-braneis given by

QRR(ι : M → X,E) = ch(fK! (E))

√A(X)

when X is a Spin manifold. A natural K-theoretic interpretation follows from the fact thatthe modified Chern character isomorphism

Kev/oddQ

(X) −→ Hev/odd(X,Q)

given by mapping a → ch(a)

√A(X) is an isometry with the natural bilinear parings on

K∗Q(X) = K∗(X) ⊗ Q and Hev/odd(X,Q). Here the pairing on K(X) is given by the

index of the Dirac operator

(a, b)K = Index( /Da⊗b) =

∫X

ch(a)ch(b)A(X) =(ch(a)

√A(X), (ch(b)

√A(X)

)H.

When the B-field is not topologically trivial, that is [α] = 0, then [α] defines a complexline bundle over the loop space LX , or a stable isomorphism class of bundle gerbes overX . Then in order to have a well-defined worldsheet path integral, Freed and Witten in[FreWit] showed that

ι∗[α] +W3(νM ) = 0.(7.1)

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34 ALAN L. CAREY AND BAI-LING WANG

When ι∗[α] = 0, that means ι is not K-oriented, then the push-forward map in K-theory([AH])

ιK! : K0(M) −→ K∗(X)

is not well-defined. Witten explained in [Wit] that D-brane charges should take values ina twisted form of K-theory, as supported further by evidence in [BouMat] and [Kap].

In [Wa], the mathematical meaning of (7.1) was discovered using the notion of α-twisted Spinc manifolds for a continuous map

α : X −→ K(Z, 3)

representing [α] ∈ H3(X,Z). When X is Spinc, the datum to describe a D-brane is ex-actly a geometric cycle for the twisted K-homology Kgeo

ev/odd(X,α). By Poincare duality,we have

Kgeoev/odd(X,α) ∼= K0(X,α+ oX)

with the orientation twisting oX : X → K(Z, 3) trivialized by a choice of a Spinc struc-ture. Hence,

K0(X,α+ oX) ∼= K0(X,α).

For a general manifold X , a submanifold ι : M → X with

ι∗([α]) +W3(νM ) = 0,

then there is a homotopy commutative diagram

M

ι

νM BSOη

W3

X α

K(Z, 3),

here νM also denotes a classifying map of the normal bundle, or a classifying map of thebundle TM ⊕ ι∗TX . This motivates the following definition (see also [CW2]).

DEFINITION 7.1. Given a smooth manifold X with a twisting α : X → K(Z, 3), aB-field of (X,α) is a differential twisting lifting α

α = (Gα, θ, ω),

which is a (lifting, or local) bundle gerbe Gα with a connection θ and a curving ω. Thefield strength of the B-field (Gα, θ, ω) is given by the curvature H of α.

A Type II (generalized) D-brane in (X,α) is a complex vector bundle E with a con-nection ∇E over a twisted Spinc manifold M . The twisted Spinc structure on M is givenby the following homotopy commutative diagram together with a choice of a homotopy η

M

ι

νι BSOη

W3

X α

K(Z, 3)

(7.2)

where νι is the classifying map of TM ⊕ ι∗TX .

REMARK 7.2. The twisted Spinc manifold M in Definition 7.1 is the D-brane worldvolume in Type II string theory. The twisted Spinc structure given in (7.2) implies thatD-brane world volume M ⊂ X in Type II string theory satisfies the Freed-Witten anomalycancellation condition

ι∗[α] +W3(νM ) = 0.

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RIEMANN-ROCH AND INDEX FORMULAE IN TWISTED K-THEORY 35

In particular, if the B-field of (X,α) is topologically trivial, then the normal bundle ofM ⊂ X is equipped with a Spinc structure given by (7.2).

Given a Type II D-brane (M, ι, νι, η, E,∇E), the homotopy η induces an isomorphism

η∗ : K0(M) → K0(M, ι∗α+ oι).

Here oι denotes the orientation twisting of the bundle TM ⊕ ι∗TX . Note that

ιK! : K0(M, ι∗α+ oι) −→ Ken/odd(X,α)

is the pushforward map (2.9) in twisted K-theory. Hence we have a canonical element inKen/odd(X,α) defined by

ιK! (η∗([E])),

called the D-brane charge of (M, ι, νι, η, E). We remark that a Type II D-brane

(M, ι, νι, η, E,∇E)

defines an element in differential twisted K-theory Ken/odd(X, α).From (7.2), we know that M is an (α + oX)-twisted Spinc manifold as we have the

following homotopy commutative diagram

M

ι

ν BSO

W3

X

α+oX K(Z, 3)

where ν is the classifying map of the stable normal bundle of M . Together with the follow-ing proposition, we conclude that the Type II D-brane charges, in the present of a B-field

α = (Gα, θ, ω),

are classified by twisted K-theory K0(X,α).

PROPOSITION 7.3. Given a twisting α : X → K(Z, 3) on a smooth manifold X ,every twisted K-class in Kev/odd(X,α) is represented by a geometric cycle supported onan (α+ oX)-twisted closed Spinc-manifold M and an ordinary K-class [E] ∈ K0(M).

For completeness, we also give a definition of Type I D-branes (Cf. [MMS], [RSV]and Section 8 in [Wa]).

DEFINITION 7.4. Given a smooth manifold X with a KO-twistingα : X → K(Z2, 2),a Type I (generalized) D-brane in (X,α) is a real vector bundle E with a connection ∇E

over a twisted Spin manifold M . The twisted Spin structure on M is given by the follow-ing homotopy commutative diagram together with a choice of a homotopy η

M

ι

νι BSOη

w2

X α

K(Z2, 2)

(7.3)

where w2 is the classifying map of the principal K(Z2, 1)-bundle BSpin → BSO asso-ciated to the second Stiefel-Whitney class, η is a homotopy between w2 νι and α ι. Hereνι is the classifying map of TM ⊕ ι∗TX .

129

36 ALAN L. CAREY AND BAI-LING WANG

REMARK 7.5. A Type I D-brane in (X,α) has its support on a manifold M if andonly if there is a differentiable map ι : M → X such that

ι∗([α]) + w2(νι) = 0.

Here νι denotes the bundle TM ⊕ ι∗TX .

Given a Type I D-brane in (X,α), the push-forward map in twisted KO-theory

KO∗(M)η∗

∼= KO∗(M,α ι+ oι)

ιKO! KO∗(X,α)

defines a canonical element in KO∗(X,α). Every class in KO∗(X,α) can be realizedby a Type I (generalized) D-brane in (X,α). Hence, we conclude that the Type I D-branecharges are classified by twisted KO-theory KOev/odd(X,α).

References

[ABS] M. Atiyah, R. Bott, A. Shapiro, Clifford modules, Topology Vol. 3, Suppl. 1, pp 3-38.[AH] M. Atiyah, F. Hirzebruch, Riemann-Roch theorems for differentiable manifolds. Bull. Amer. Math.

Society, pp. 276-281 (1959).[AS1] M. Atiyah, G. Segal, Twisted K-theory, Ukr. Mat. Visn. 1 (2004), no. 3, 287–330; translation in

Ukr. Math. Bull. 1 (2004), no. 3, 291–334. ArXiv:math/0407054[AS2] M. Atiyah, G. Segal, Twisted K-theory and cohomology. Inspired by S. S. Chern, 5-43, Nankai

Tracts Math., 11, World Sci. Publ., Hackensack, NJ, 2006.[AtiSin1] M. Atiyah, I. Singer, The index of elliptic operators. I. Ann. of Math. (2), 87, 1968, 484–530.[AtiSin3] M. Atiyah, I. Singer, The index of elliptic operators. III. Ann. of Math. (2), 87, 1968, 546-604.[AtiSin4] M. Atiyah, I. Singer, The index of elliptic operators. IV. Ann. of Math. (2), 93, 1971, 119–138.[BC] P. Baum, A. Connes, Geometric K-theory for Lie groups and foliations. Enseign. Math. (2) 46

(2000), no. 1-2, 3–42.[BD1] P. Baum, R. Douglas, K homology and index theory. Proceedings Symp. Pure Math 38, 117-173.

Amer. Math. Soc., Providence, 1982.[BD2] P. Baum, R. Douglas, Index theory, bordism, and K-homology. Operator Algebras and K-Theory,

Contemporary Math. 10, 1-31. Amer. Math. Soc., Providence, 1982.[BHS] P. Baum, N. Higson, T. Schick, On the equivalence of geometric and analytic K-homology. Pure

and Applied Mathematics Quarterly 3(1):1-24, 2007. ArXiv:math/0701484[Ba] D. Basu, K-theory with R/Z coefficients and von Neumann algebras. K-Theory 36 (2005), no.

3-4, 327–343.[BGV] N. Berline, E. Getzler and M. Vergne, Heat kernels and Dirac operators, Grundlehren Math. Wiss.,

vol. 298, Springer-Verlag, New York, 1992.[BEM] P. Bouwknegt, J. Evslin and V. Mathai, T -duality: topology change from H-flux. Comm. Math.

Phys. 249 (2004), no. 2, 383–415.[BouMat] P. Bouwknegt, V. Mathai, D-branes, B-fields and twisted K-theory, J. High Energy Phys. 03 (2000)

007, hep-th/0002023.[BCMMS] P. Bouwknegt, A. Carey, V. Mathai, M. Murray, D. Stevenson, Twisted K-theory and K-theory of

bundle gerbes. Comm. Math. Phys. 228 (2002), no. 1, 17–45.[BMRS] J. Brodzki, V. Mathai, J. Rosenberg, R. Szabo, D-branes, RR-fields and duality on noncommutative

manifolds. Comm. Math. Phys. 277 (2008), no. 3, 643–706.[CMW] A. Carey, J. Mickelsson, B.-L. Wang, Differential Twisted K-theory and Applications. Jour. of

Geom. Phys., Vol. 59, no. 5, 632-653, (2009), arXiv:0708.3114.[CW1] A. Carey, B.-L. Wang, Thom isomorphism and Push-forward map in twisted K-theory. Journal of

K-Theory, Volume 1, Issue 02, April 2008, pp 357-393.[CW2] A. Carey, B.-L. Wang, Fusion of symmetric D-branes and Verlinde rings. Comm. Math. Phys. 277

(2008), no. 3, 577–625.[ConSka] A. Connes, G. Skandalis, The longitudinal index theorem for foliations. Publ. Res. Inst. Math. Sci.

Kyoto 20 (1984), 1139-1183.[DK] P. Donavan, M. Karoubi, Graded Brauer groups and K-theory with lcoal coefficients, Publ. Math.

de IHES, Vol 38, 5-25, 1970.

130

RIEMANN-ROCH AND INDEX FORMULAE IN TWISTED K-THEORY 37

[ES] Evslin, Jarah; Sati, Hisham Can D-branes wrap nonrepresentable cycles? J. High Energy Phys.2006, no. 10.

[EEK] S. Echterhoff, H. Emerson, H. J. Kim, KK-theoretical duality for proper twisted action. Math.Ann. 340 (2008), no. 4, 839–873.

[FH] D. Freed, M. J. Hopkins, On Ramond-Ramond fields and K-theory. J. High Energy Phys. no.5,2000.

[FreWit] D. Freed and E. Witten, Anomalies in String Theory with D-Branes, Asian J. Math. 3 (1999), no.4, 819–851.

[Gomi] K. Gomi, Twisted K-theory and finite-dimensional approximation. arXiv:0803.2327.[GoTe] K. Gomi, Y. Terashima, Chern-Weil construction for twisted K-theory. Preprint.[HigRoe] N. Higson, J. Roe, Analytic K-homology. Oxford University Press. 2000.[HopSin] M. Hopkins, I. Singer, Quadratic functions in geometry, topology, and M-theory. J. Differential

Geom. 70 (2005), no. 3, 329–452.[Kar] M. Karoubi, Twisted K-theory, old and new. Preprint, math.KT/0701789.[Kar1] M. Karoubi, Homologie cyclique et K-theorie. Astrisque No. 149 (1987), 147 pp.[Kap] A. Kapustin, D-branes in a topologically non-trivial B-field, Adv. Theor. Math. Phys. 4 (2001)

127, hep-th/9909089.[Kas1] G. Kasparov, Topological invariants of elliptic operators I: K-homology. Math. USSR Izvestija 9,

1975, 751-792.[Kas2] G. Kasparov, Equivariant KK -theory and the Novikov conjecture. Invent. Math. 91 (1988) 147201.[Kas3] G. Kasparov, The operator K-functor and extensions of C∗-algebras. Izv. Akad. Nauk. SSSR Ser.

Mat. 44 (1980), 571-636.[Kui] N. Kuiper, The homotopy type of the unitary group of Hilbert space. Topology, 3 (1965), 19-30.[LM] B. Lawson, M.-L. Michelsohn, Spin Geometry, Princeton University Press, 1989.[MMS] V. Mathai, M. Murray, D. Stevenson, Type-I D-branes in an H-flux and twisted KO-theory. J.

High Energy Phys. 2003, no. 11, 053, 23[MatSte] V. Mathai and D. Stevenson, Chern character in twisted K-theory: equivariant and holomorphic

cases, Commun.Math.Phys. 228, 17-49, 2002.[MM] R. Minasian and G. W. Moore, K-theory and Ramond-Ramond charge. JHEP 9711:002 (1997)

(arXiv:hep-th/9710230)[Mur] M. K. Murray Bundle gerbes, J. London Math. Soc. (2) 54, 403–416, 1996.[MS] Michael K. Murray and Michael A. Singer, Gerbes, Clifford modules and the index theorem. Ann.

Global Anal. Geom. 26 (2004), no. 4, 355–367.[Pol] J. Polchinski, Dirichlet branes and Ramond-Ramond charges. Phys. Rev. Lett. 75 (1995) 4724-

4727, arXiv:hep-th/9510017[Ros] J. Rosenberg, Continuous-trace algebras from the bundle-theoretic point of view. Jour. of Australian

Math. Soc. Vol A 47, 368-381, 1989.[RS] R. M. G. Reis, R.J. Szabo, Geometric K-Homology of Flat D-Branes, Commun. Math. Phys. 266

(2006), 71122.[RSV] R. M. G. Reis, R.J. Szabo, A. Valentino, KO-homology and type I string theory arXiv:hep-

th/0610177[Swi] R. Switzer, Algebraic topology - homotopy and homology. Springer-Verlag, Berlin, 1975[Sz] R. J. Szabo, D-branes and bivariant K-theory, arXiv:0809.3029.[Tu] J.-L. Tu, Twisted K-theory and Poincare duality, Trans. Amer. Math. Soc. 361 (2009), no. 3, 1269–

1278.[Wa] B.-L. Wang, Geometric cycles, Index theory and twisted K-homology. Journal of Noncommutative

Geometry 2 (2008), no. 4, 497552.[Wit] E. Witten, D-branes and K-theory J. High Energy Phys. 1998, no. 12, Paper 19.

MATHEMATICAL SCIENCES INSTITUTE, AUSTRALIAN NATIONAL UNIVERSITY, CANBERRA ACT 0200,AUSTRALIA

E-mail address: acarey@maths.anu.edu.au

DEPARTMENT OF MATHEMATICS, AUSTRALIAN NATIONAL UNIVERSITY, CANBERRA ACT 0200, AUS-TRALIA

E-mail address: wangb@maths.anu.edu.au

131

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Proceedings of Symposia in Pure Mathematics

Noncommutative Principal Torus Bundles viaParametrised Strict Deformation Quantization

Keith C. Hannabuss and Varghese Mathai

Abstract. In this paper, we initiate the study of a parametrised version of

Rieffel’s strict deformation quantization. We apply it to give a classificationof noncommutative principal torus bundles, in terms of parametrised strictdeformation quantization of ordinary principal torus bundles. The paper alsocontains a putative definition of noncommutative non-principal torus bundles.

Introduction

Operator theoretic deformation quantization appeared in quantum physics along time ago, but was put on a firm footing relatively recently by Rieffel [15] (seethe references therein), who called it strict deformation quantization, mainly to dis-tinguish it from formal deformation quantization, where convergence isn’t an issue.His theory has been remarkably successful, giving rise to many examples of noncom-mutative manifolds, which have become extremely useful both in mathematics andmathematical physics. In a recent paper [3] Echterhoff, Nest, and Oyono-Oyonodefined noncommutative principal torus bundles, inspired by fundamental resultsin [17], as well as the T-duals of certain continuous trace algebras [13, 14]. Theyalso classified all noncommutative principal torus bundles in terms of (noncommu-tative) fibre products of principal torus bundles and group C∗-algebras of latticesin simply-connected 2-step nilpotent Lie groups, cf. §5. In this paper, we show thattheir classification can be neatly understood in terms of a generalization of Rieffel’sstrict deformation quantization [15, 16], to the parametrised case that is developedhere. More precisely, we generalize to the parametrised case, the recent version ofRieffel’s strict deformation quantization given by Kasprzak [9] based on work ofLandstad [11, 12]. More precisely, we give a classification of noncommutativeprincipal torus bundles, in terms of parametrised strict deformation quantizationof ordinary principal torus bundles.

Strict deformation quantization theory works with smooth subalgebras, so westart with a section, §1, on smooth subalgebras of C∗-algebras and a smooth version

2000 Mathematics Subject Classification. 58B34 (81S10, 46L87, 16D90, 53D55).Key words and phrases. Parametrised strict deformation quantization, Noncommutative

principal torus bundles, T-duality.

Acknowledgements. V.M. thanks the Australian Research Council for support.

c©0000 (copyright holder)

1

Proceedings of Symposia in Pure MathematicsVolume 81, 2010

c©2010 American Mathematical Society

133

2 KC HANNABUSS AND V MATHAI

of the noncommutative torus bundle theory. That is followed in §2 by a summaryrecalling the ideas of Rieffel’s strict deformation quantization, and then a section,§3, generalising that to a parametrised version. Then we explain in §4, Kasprzak’srecent account of Rieffel’s strict deformation quantization theory based on ideas ofLandstad, and extend it to a parametrised version. In §5, after a summary of therelevant parts of the Echterhoff, Nest, and Oyono-Oyono classification of noncom-mutative principal torus bundles, we explain the connection to parametrised strictdeformation quantization theory. We end with a section, §6, containing examples ofparametrised strict deformation quantization including the case of principal torusbundles. It also contains a putative definition of noncommutative non-principaltorus bundles. There is an appendix containing a discussion about factors of auto-morphy which is used in the paper.

1. Fibrewise Smooth ∗-bundlesWe begin by recalling the notion of C∗-bundles over X and the special case

of noncommutative principal bundles. Then we discuss the fibrewise smoothing ofthese, which is used in parametrised Rieffel deformation later on.

Let X be a locally compact Hausdorff space and let C0(X) denote the C∗-algebra of continuous functions on X that vanish at infinity. A C∗-bundle A(X)over X in the sense of [3] is exactly a C0(X)-algebra in the sense of Kasparov [8].That is, A(X) is a C∗-algebra together with a non-degenerate ∗-homomorphism

ΦA : C0(X) → ZM(A(X)),

called the structure map, where ZM(A) denotes the center of the multiplier algebraM(A) of A. The fibre over x ∈ X is then A(X)x = A(X)/Ix, where

Ix = Φ(f) · a; a ∈ A(X) and f ∈ C0(X) such that f(x) = 0,and the canonical quotient map qx : A(X) → A(X)x is called the evaluation mapat x.

Note that this definition does not require local triviality of the bundle, or evenfor the fibres of the bundle to be isomorphic to one another.

Let G be a locally compact group. One says that there is a fibrewise action ofG on a C∗-bundle A(X) if there is a homomorphism α : G −→ Aut(A(X)) whichis C0(X)-linear in the sense that

αg(Φ(f)a) = Φ(f)(αg(a)), ∀g ∈ G, a ∈ A(X), f ∈ C0(X).

This means that α induces an action αx on the fibre A(X)x for all x ∈ X.The first observation is that if A(X) is a C∗-algebra bundle over X with a

fibrewise action α of a Lie group G, then there is a canonical smooth ∗-algebrabundle over X. We recall its definition from [2]. A vector y ∈ A(X) is said to be asmooth vector if the map

G g −→ αg(y) ∈ A(X)

is a smooth map from G to the normed vector space A(X). Then

A∞(X) = y ∈ A(X) | y is a smooth vectoris a ∗-subalgebra of A(X) which is norm dense in A(X). Since G acts fibrewise onA(X), it follows that A∞(X) is again a C0(X)-algebra which is fibrewise smooth.

Let T denote the torus of dimension n. The authors of [3] define a noncommu-tative principal T -bundle (or NCP T -bundle) over X to be a separable C∗-bundle

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NCP TORUS BUNDLES VIA PARAMETRISED DEFORMATION QUANTIZATION 3

A(X) together with a fibrewise action α : T → Aut(A(X)) such that there is aMorita equivalence,

A(X)α T ∼= C0(X,K),

as C∗-bundles over X, where K denotes the C∗-algebra of compact operators.The motivation for calling such C∗-bundles A(X) NCP T -bundles arises from

a special case of a theorem of Rieffel [17], which states that if q : Y −→ X is aprincipal T -bundle, then C0(Y ) T is Morita equivalent to C0(X,K).

If A(X) is a NCP T -bundle over X, then we call A∞(X) a fibrewise smoothnoncommutative principal T -bundle (or fibrewise smooth NCP T -bundle) over X.In this paper, we are able to give a complete classification of fibrewise smooth NCPT -bundles overX via a parametrised version of Rieffel’s theory of strict deformationquantization.

2. Rieffel Deformation

Unlike Rieffel’s deformation theory [15, 16], the version which we shall use [11,12, 9] starts with multipliers, so in this section we shall recapitulate some standardresults but in a formulation which suits the later extension to a parametrised theoryand the Landstad–Kasprzak approach. In what follows, a Poisson bracket , onA is a bilinear form from A to itself, which is a Hochschild 2-cocycle satisfyinga couple of additional technical conditions that will not be repeated here, but werefer the reader to §5 in [15].

Definition 2.1 (§5, [15]). Let A be a dense ∗-subalgebra of a C∗ algebra,equipped with a Poisson bracket , . A strict deformation quantisation of A inthe direction of , means an open interval I containing 0 in R, together withassociative products , ∈ I, involutions and C∗-norms on A which for = 0 arethe original product, involution and norm on A, such that:

(1) The corresponding field of C∗-algebras with continuity structure given bythe elements of A as constant fields, is a continuous field of C∗-algebras.

(2) For all a, b ∈ A, as → 0 one has ‖(a b− ab)/(i)− a, b‖ → 0.

Typically, one tries to find strict deformation quantizations of Poisson mani-folds, thus obtaining interesting noncommutative manifolds.

Rieffel’s definition and construction are motivated by Moyal’s product but tolink it with Kasprzak’s work it is useful to give the background.

Suppose that A is a pre- C∗-algebra with an action α of a locally compactabelian group V (written additively), and let σ be a multiplier on its Pontryagin

dual V , that σ : V × V → T is a borel map, satisfying the cocycle identity

σ(ξ, η)σ(ξ + η, ζ) = σ(ξ, η + ζ)σ(η, ζ),

for all ξ, η, ζ ∈ V . The group of all such cocycles (or multipliers) is denoted by

Z2(V ,T). Two multipliers σ1 and σ2 are equivalent (or cohomologous) if and only

if there is a borel map ρ : V → T such that

σ1(ξ, η)ρ(ξ + η) = σ2(ξ, η)ρ(ξ)ρ(η),

and the equivalence classes form the cohomology group H2(V ,T). A cocycle equiv-

alent to the constant cocycle V × V → 1 is said to be trivial.

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4 KC HANNABUSS AND V MATHAI

Recalling that a bicharacter β : V × V → T defines characters β1ξ : η → β(ξ, η)

for each fixed ξ, and β2η : ξ → β(ξ, η) for fixed η, we see that bicharacters always

define cocycles, because

β(ξ, η)β(ξ + η, ζ) = β(ξ, η)β(ξ, ζ)β(η, ζ) = β(ξ, η + ζ)β(η, ζ).

Theorem 2.2 ([10, 5]). Every multiplier on an abelian group V is equivalentto a bicharacter (and so is continuous). Two bicharacters β1 and β2 are equivalentif and only if β = β1β

−12 a symmetric bicharacter , that is β(ξ, η) = β(η, ξ). If

V = 2V then each cohomology class can be represented by a unique antisymmetricbicharacter β, that is β(ξ, η) = β(η, ξ)−1.

We can therefore assume that σ is a bicharacter, and this means that it is actu-

ally continuous in each variable. When V = 2V , the element σ(ξ, 12η)/σ(η,12ξ) gives

the canonical antisymmetric bicharacter representative of the class containing σ.

We note that vector groups V = Rn = 2V , so that each cocycle can be representedby a continuous antisymmetric antisymmetric bicharacter, which must be the expo-nential exp[iπs(ξ, η)] of a skew-symmetric bilinear form s. These can be identified

with∧2 V . There is a similar analysis for lattices L ∼= Zn, where the bicharacters

are given by the torus∧2

(V/L) =∧2

V/∧2

L (restrictions modulo those with triv-ial restriction), but a torus Tn has only trivial bicharacters β(ξ, η) = 1, due to thefollowing observation.

Corollary 2.3. There are no non-trivial bicharacters on a connected compactgroup V .

This follows because the non-trivial multipliers on the dual of an infinite con-nected compact group (such as V = T2n) are never invertible, since the dual (e.g.

V = Z2n) is discrete and the two groups are not isomorphic.

Theorem 2.4. Given a continuous bicharacter cocycle σ on V and a pre-C∗

algebra A we may form the ∗-algebra of functions f, g : V → A smooth with respectto the translation automorphisms τu[f ] = f(u + v), with the twisted convolutionproduct

(f ∗ g)(ξ) =∫V

σ(η, ξ − η)f(η)g(ξ − η) dη,

and involution f∗(ξ) = σ(ξ, ξ)f(−ξ). Up to isomorphism this algebra depends onlyon the cohomology class of σ.

Proof. The cocycle identity on σ ensures associativity. When σ is an anti-symmetric bicharacter the involution reduces to f∗(ξ) = f(−ξ). Changing σ to

σ(ξ, η)ρ(ξ)ρ(η)ρ(ξ + η)−1

gives the algebra isomorphism f → ρ.f (the pointwise product).

These functions can be Fourier transformed to functions on V

f(v) =

∫V

ξ(v)f(ξ) dξ,

where we assume that the Haar measure is normalised to make the transform uni-tary, and one has the usual inverse transform.

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NCP TORUS BUNDLES VIA PARAMETRISED DEFORMATION QUANTIZATION 5

Theorem 2.5. The transformed product is

(f ∗ g)(v) =∫

σ(η, ξ)η(u)ξ(w)f(v + u)g(v + w) dudwdξdη.

Proof. We calculate that

(f ∗ g)(v) =∫V

ξ(v)(f g)(ξ) dξ

=

∫V×V

ξ(v)σ(η, ξη−1)f(η)g(ξη−1) dξdη

=

∫V×V

σ(η, ξ)η(v)f(η)ξ(v)g(ξ) dξdη

=

∫σ(η, ξ)η(v)f(u)η(u)ξ(v)g(w)ξ(w) dudwdξdη

=

∫σ(η, ξ)η(u)ξ(w)f(v + u)g(v + w) dudwdξdη,

where we replaced u and w by v + u and v + w in the last step.

We now want to connect this transformed product with Rieffel’s deformation.To this end we introduce a bicharacter e on V , which defines a homomorphism

e1 : V → V . Rieffel works with a vector group V and e(u,w) = exp(i(u · w)) for

some inner product on V . When σ is non-degenerate (that is, σ1 : V → V is anisomorphism) we can choose e so that e1 is the inverse of σ1, but in general we havean automorphism T = σ1 e1 : V → V . As a final piece of notation we introducethe adjoint T ∗ with respect to e: e(T ∗u,w) = e(u, Tw).

Proposition 2.6. The bicharacters σ and e are related by σ(e1u, e1v) = e1v(Tu) =

e(Tu, v) for all u, v ∈ V . Suppose that σ is an antisymmetric bicharacter. Then ife is symmetric T = −T ∗, and if e is antisymmetric T = T ∗.

Proof. By definition we have

σ(e1u, e1v) = e1v(Tu) = e(Tu, v).

Since σ is skew symmetric this gives

e(Tu, v) = σ(e1u, e1v) = σ(e1v, e

1u)

−1 = e(Tv, u)−1 = e(−Tv, u).

When e is symmetric this shows that e(Tu, v) = e(u,−Tv), so that T = −T ∗, andwhen e is antisymmetric T = T ∗.

Theorem 2.7. Given a non-degenerate bicharacter e on V , set T = σ1 e1 :V → V . and e(T ∗u,w) = e(u, Tw). Then

(f ∗ g)(v) =∫

e(u,w)f(v + T ∗u)g(v + w) dudw.

We change the order of integration in our earlier expression for f ∗ g and con-

centrate on the integrals over V :∫V×V

σ(η, ξ)η(u)ξ(w) dξdη.

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6 KC HANNABUSS AND V MATHAI

Since e is nondegenerate we may set η = ev, and then, by definition, we have

σ(ev, ξ)ev(u)ξ(w) = ξ(Tv)e(v, u)ξ(w).

By the Fourier inversion theorem, integration over ξ gives a delta function δ(Tv−w).Replacing u by T ∗u and integrating over v, we now get

∫σ(ev, ξ)e(v, T ∗u)ξ(w) dξdv =

∫δ(w − Tv)e(Tv, u) dv = e(w, u).

Up to a multiple, integration over η and v are the same, and with appropriatechoices of measure we can ensure that they agree precisely. Then inserting this intothe original formula for the product we have

(f ∗ g)(v) =∫

e(u,w)f(v + T ∗u)g(v + w) dudw.

Theorem 2.8. The Fourier transformed product (f ∗ g) = f g where

(f g)(v) =

∫e(u,w)f(v + T ∗u)g(v + w) dudw.

In terms of the translation automorphisms τw[g](v) = g(v + w), we have

(f g)(v) =

∫e(u,w)τT∗u[f ](v)τw[g](v) dudw.

Evaluating at the identity v = 0 gives

(f g)(0) =

∫e(u,w)τT∗u[f ](0)τw[g](0) dudw.

Rieffel noticed that this formula can now be interpreted whenever α defines auto-morphisms of A, so that one can define

a b =

∫e(u,w)αT∗u[a]αw[b] dudw,

for a and b in the algebra. (Our T ∗ is Rieffel’s J .) When both bicharacters σ ande are nondegenerate we can also write this as

a b =

∫det[T ∗]−1e(T ∗−1u,w)αu[a]αw[b] dudw.

The above arguments are formal and one must check that the integrals converge.In the standard Moyal theory this is done by working only with Schwarz functionsand in the general case one uses the smooth vectors A∞ for the action α, whichform a dense Frechet subalgebra of A.

For vector groups this works particularly smoothly, and one obtains a strictdeformation quantisation [15], Theorem 9.3. However, there are technical problemswhen V = T2n since, as we have seen, there are no nontrivial bicharacters e on V .There are two ways of dealing with this problem. One is by the Kasprzak–Landstadapproach of working with the dual crossed product algebra, [12, 9], and the otheris Rieffel’s approach of lifting the action of the torus T = V/L (with L a lattice),to the vector group V , [15] Ch 2.

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NCP TORUS BUNDLES VIA PARAMETRISED DEFORMATION QUANTIZATION 7

3. Parametrised Rieffel Deformations

An interesting generalisation comes from inserting a parameter. More precisely,we work with a C0(X)-algebra A, where X is a locally compact Hausdorff space.

(That is there is a map C0(X) → ZMA.) Consider a function σ ∈ Cb(X,Z2(V ,T))taking values in the bicharacter cocycles. At each point x ∈ X this defines a

multiplier σx, and a map σ1x : V → V . We then form Tx = σ1

x e1 and its adoint T ∗x

with respect to e, where e, e1 are defined just prior to Proposition 2.6. If the imageof σ lies in the non-degenerate cocycles we can then form the continuous functionx → e(T ∗

x−1u, v), which acts on A.

Theorem 3.1. Given a C0(X)-algebra A, where X is a locally compact Haus-

dorff space, and a function σ ∈ Cb(X,Z2(V ,T)) taking values in the nondegeneratebicharacter cocycles , let Tx = σ1

x e1 and e(T ∗xu,w) = e(u, Txw). Then, if T has

an inverse in a subalgebra of Cb(X) whose action preserves the (fibrewise) smoothsubalgebra A∞, one has a product

a b =

∫det[T ∗]−1e(T ∗−1u,w)αu[a]αw[b] dudw,

defined by the actions of the continuous functions det[T ∗]−1 and e(T ∗−1u,w) onA. This gives an algebra Aσ with an involution, which inherits a C0(X)-algebrastructure. (The C0(X)-structure on the algebra is such that for F ∈ C0(X) we haveF.(a b) = (F.a) b = a (F.b).)

For vector groups it follows from the definition that the iterated parametrisedstrict deformation quantization (Aσ1

)σ2∼= Aσ1σ2

, with the isomorphism definedby the obvious identification map. This follows on writing down the repeateddeformation product and evaluating a double integral using Parseval’s formula orthe Fourier inversion formula. Alternatively we can note that the bicharacter σ−1

can always be written in Rieffel form, and then the result follows from his. Yetanother approach would be to use the equivalence with Kasprzak’s formulationgiven below, and then to deduce it from his result. In particular, we can undeformAσ using σ.

In this more general context we can generalise Rieffel’s discussion of the actionof continuous automorphisms of the group V (which give GL(V ) when V is a vectorgroup), to allow functions S ∈ C∞(X,Aut(V )) and using

a S b =

∫V×V

σ−1(Su, Sw).(αu[a]αw[b]) dudw.

Note that the original automorphisms of V on A are also automorphisms ofthe deformed algebra, since

αv[a] σ αv[b] =

∫V×V

det[T ∗]−1e(T ∗−1u, v).(αu+v[a]αw+v[b]) dudw

=

∫V×V

det[T ∗]−1e(T ∗−1u, v).αv[(αu[a]αw[b]) dudw

= αv[a σ b],

since αw commutes with the C0(X) action.We constructed the deformation as the dual of a twisted crossed product, and

the reverse is also true. Given an algebra A with an action of V one can take

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8 KC HANNABUSS AND V MATHAI

the crossed product A V with a dual action of V . Looking first at the un-parametrised case, when σ is non-degenerate there is a dual multiplier σ on V

defined by σ(u, σ1η) = η(u), and similarly for e(ξ, e1v) = ξ(v), and T = σ1 e1.These definitions effectively mean that e1 is the inverse of e1 and similarly for σ.We can now deform the crossed product.

(a σ b)(v) =

∫e(ξ, η)α

Tξ[a]αη[b] dξdη

=

∫e(ξ, η)α

Tξ[a](u)αu[αη[b](v − u)] dξdηdu

=

∫e(ξ, η)(T ξ)(u)η(v − u)a(u)αu[b(v − u)] dξdηdu

=

∫η(e1ξ)T ξ)(u)η(v − u)a(u)αu[b(v − u)] dξdηdu.

The integral over η gives a delta function concentrated on e1ξ = u− v, or equiva-lently where ξ = e1(u− v), so that the ξ integral then gives

(a σ b)(v) =

∫(T e1(u− v))(u)a(u)αu[b(v − u)] , du.

By definition, we have

T e1 = σ1e1e1 = σ1,

which leads to the reduction

(aσb)(v) =

∫(σ1(u−v))(u)a(u)αu[b(v−u)] , du =

∫σ(u−v, u)a(u)αu[b(v−u)] , du.

This is a twisted crossed product with multiplier. There is a similar parametrisedversion.

4. Landstad–Kasprzak and Rieffel Deformation

Building on work of Landstad [11, 12], Kasprzak [9] gives an alternative dualpicture of deformation theory. It is useful to give the equivalence with Rieffeldeformation explicitly, as Kasprzak omits the details. (The correspondence is notobvious since the algebra elements in Rieffel’s deformation are the same and only theproduct changes, whereas in Kasprzak’s formulation the deformed and undeformedalgebras are distinct fixed point subalgebras of the multiplier algebra of the crossedproduct, with different actions of V . Smoothness or some equivalent is also needed;Landstad suggests in [12] that it is sufficient to use the Fourier algebra instead ofsmooth subalgebras.) In the following account we use Rieffel’s notation of α ratherthan ρ for the automorphisms.

Landstad showed in [11] that when a group V acts on an algebra A, the crossedproduct B = A V has a coaction which is defined by a homomorphism λ : V →UMB (the unitary multiplier algebra). By integration λ extends to C(V ) → MB.

When V is abelian there is also the dual Takai–Takesaki action α of V , and these

interact by αξ[λv] = ξ(v)λv. By Takai–Takesaki duality B α V is isomorphic toA⊗K, reconstructing A up to stable equivalence. When B has the Landstad λ aswell we can deduce a stronger duality that there is an algebra A with V -action αsuch that B = Aα V . Kasprzak’s idea is that αξ can be deformed by a cocycle σ

for V to a new action ασξ .

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NCP TORUS BUNDLES VIA PARAMETRISED DEFORMATION QUANTIZATION 9

Theorem 4.1. [9]. Let (B, λ, α) be as above, and σ a continuous cocycle for

V . Setting Uξ = λ(σ1ξ ) there is an action of V on A given by

ασξ : b → Uxi

∗αξ[b]Uξ,

which also satisfies ασξ [λv] = ξ(v)λv.

Corollary 4.2. There is an algebra Aσ and V -action ασ such that the crossedproduct Aσ ασ V ∼= B = Aα V .

The deformed and undeformed algebras can be identified with the subalgebras

of MB fixed by the action α of V .In particular, the undeformed algebra is fixed under the dual group action on

the crossed product given by αξ[a](v) = ξ(v)a(v). The fixed points of this actionare distributions concentrated on the group identity v = 0, which make sense aselements of the multiplier algebra. They give an algebra isomorphic to A, and thisis just Rieffel’s construction as defined above.

For the algebra deformed by σ, Uξ = δσ1ξ, and by the covariance property of

crossed products the adjoint action of Uξ is the same as the action of ασ1ξ. We

therefore have

ασξ [a](v) = α−1

σ1ξ[αξ)[a(v)]] = ξ(x)α−1

σ1ξ[a(v)].

Changing variable, the fixed subalgebra, where ασξ [a] = a, therefore consists of

elements a satisfying

ασ1ξ[a(v)] = ξ(v)a(v).

In the notation of previous sections we set ξ = e1u so that σ1ξ = Tu, and then the

condition becomes

αTu[a(v)] = e(u, v)a(v).

Thus the value of a(v) always lies in a particular eigenspace of the action α. (Inparticular, when e is an antisymmetric bicharacter a(0) must be in the fixed pointalgebra of αT .) In other words we can characterise the fixed point algebra elementsas the elements whose value at v lies in the relevant spectral subspace ker[αTu −e(u, v)] of the action of α.

To get all the eigenspaces we must do a direct integral, or, for suitably well-behaved functions (the smooth subalgebra), we set I(a) =

∫a(v) dv.

Theorem 4.3. When T is invertible, the product of fixed point algebra elementsa and b satisfies

I(a ∗ b) = I(a) I(b).

Proof. When T is invertible, the product of fixed point algebra elements isgiven by

I(a ∗ b) =∫

a(u)αu[b(v − u)] dudv

=

∫a(u)αu[b(v)] dudv

=

∫e(T−1u, v)a(u)b(v) dudv.

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10 KC HANNABUSS AND V MATHAI

On the other hand, under the same conditions, and with e symmetric, so thatT = −T ∗

(I(a) I(b)) =

∫det[T ∗]−1e(T ∗−1u, v)αu[I(a)]αv[I(b)] dudv

=

∫det[T ∗]−1e(−T−1u, v)αu[a(y)]αv[b(x)] dxdydudv

=

∫det[T ∗]−1e(T−1u,−v)e(T−1u, y)a(y)e(T−1v, x)b(x) dxdydudv

=

∫det[T ∗]−1e(T−1u, y − v)a(y)e(T−1v, x)b(x) dxdydudv.

The integral of e(T ∗−1u, v − y) over u produces a delta function concentrated onv = −y, and then the v integral gives

(I(a) I(b)) =

∫e(T−1y, x)a(y)b(x) dudv

=

∫e(T−1x, y)a(y)b(x) dudv

= I(a ∗ b),showing that I defines a homomorphism from the Kasprzak deformation to theRieffel deformation.

Standard harmonic analysis shows that this is formally an isomorphism onsuitably defined smooth subalgebras. (The inverse map takes an algebra elementa and does harmonic analysis of α action setting a(x) to be the component of asuch that αy[a(x)] = σ−1(x, y)a(x).) The same constructions can be carried outfor C0(X)-algebras.

5. Classifying Noncommutative Principal Torus Bundles

The noncommutative principal torus bundles of Echterhoff, Nest, and Oyono-Oyono, whose definition was recalled in Section 1, were classified in [3] and willbe outlined in this section. We also give a classification of fibrewise smooth non-commutative principal torus bundles in terms of parametrized strict deformationquantization of ordinary principal torus bundles.

By Takai–Takesaki duality A(X) is Morita equivalent to C0(X,K) T , sothe authors in [3] note that the NCPT-bundles can be classified by up to Morita

equivalence by the outer equivalence classes ET (X) of T -actions, and one has the

sequence

0 −→ H1(X,T ) −→ ET (X) −→ C(X,H2(T ,T)) −→ 0.

This leads to a classification in terms of a principal torus bundle q : Y → X, from

H1(X,T ), and a map σ ∈ Cb(X,H2(T ,T)), the equivalence classes of multipliers

on the dual group T . These data define a noncommutative torus bundle by formingthe fixed point algebra

[C0(Y )⊗C0(Z) C∗(Hσ))]

T

with C∗(Hσ) being the bundle of group C∗-algebras of the central extensions of T

by Z := H2(T ,T) defined by σ(x) at x, the action of C0(Z) on C0(Y ) coming from

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NCP TORUS BUNDLES VIA PARAMETRISED DEFORMATION QUANTIZATION 11

the composition σ q : Y → X → Z and that on C∗(Hσ) from the natural actionof a subgroup algebra. The bundle is a classical principal bundle when σ is theconstant map to the trivial multiplier 1 (or indeed is homotopic to any constantmap).

A key observation is that the datum σ, or more practically an equivalence class

of σ ∈ C(X,Z2(T ,T)) can be identified with the similar map in the parametriseddeformation theory, and that the Landstad–Kasprzak dual deformation theory con-veniently matches the duality in the definition of NCPT-bundles with the groupV = T and the dual algebra B = C0(X,K). The analysis in [3] starts with thecase of X a point, where the algebra is shown to be the twisted C∗ group algebra

of T defined by the multiplier σ, or, equivalently, the deformed algebra definedby σ. The same construction can be carried out in the case of general X usingour parametrised deformation constructions, and this can then be twisted using anordinary principal T -bundle. Now given a fibrewise smooth NCPT-bundle A∞(X)the defining deformation σ can be removed by a further deformation by σ sincethen one has a total deformation σσ = 1, and a constant map 1 gives an ordi-nary principal torus bundle up to T -equivariant Morita equivalence over C0(X). Inother words one can recover the principal torus bundle q : Y → X in this way upto T -equivariant Morita equivalence over C0(X) via an iterated parametrized strictdeformation quantization. To summarize, we have the following main result, whichfollows from Theorem 3.1, §4, Example 6.2, and the observations above.

Theorem 5.1. Given a fibrewise smooth NCPT-bundle A∞(X), there is a

defining deformation σ ∈ Cb(X,Z2(T ,T)) and a principal torus bundle q : Y → Xsuch that A∞(X) is T -equivariant Morita equivalent over C0(X), to the parametrisedstrict deformation quantization of C∞

fibre(Y ) (continuous functions on Y that are fi-brewise smooth) with respect to σ, that is,

A∞(X) ∼= C∞fibre(Y )σ.

Conversely, by Example 6.2, the parametrised strict deformation quantization ofC∞

fibre(Y ) is the noncommutative principal torus bundle C∞fibre(Y )σ.

6. Fine Structure of Parametrised Strict Deformation Quantization

We have seen in Theorem 5.1 that all fibrewise smooth NCPT-bundles are justparametrised strict deformation quantizations of ordinary principal torus bundles.We will use this to write out the fine structure of fibrewise smooth NCPT-bundles.

Example 6.1. We begin by recalling the construction by Rieffel [15] realizingthe smooth noncommutative torus as a deformation quantization of the smoothfunctions on a torus T = Rn/Zn of dimension equal to n.

Recall that any translation invariant Poisson bracket on T is just

a, b =∑

θij∂a

∂xi

∂b

∂xj,

for a, b ∈ C∞(T ), where (θij) is a skew symmetric matrix. The action of T on itself isgiven by translation. The Fourier transform is an isomorphism between C∞(T ) and

S(T ), taking the pointwise product on C∞(T ) to the convolution product on S(T )and taking differentiation with respect to a coordinate function to multiplication

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12 KC HANNABUSS AND V MATHAI

by the dual coordinate. In particular, the Fourier transform of the Poisson bracketgives rise to an operation on S(T ) denoted the same. For φ, ψ ∈ S(T ), define

ψ, φ(p) = −4π2∑

p1+p2=p

ψ(p1)φ(p2)γ(p1, p2)

where γ is the skew symmetric form on T defined by

γ(p1, p2) =∑

θij p1,i p2,j .

For ∈ R, define a skew bicharacter σ on T by

σ(p1, p2) = exp(−πγ(p1, p2)).

Using this, define a new associative product on S(T ),

(ψ φ)(p) =∑

p1+p2=p

ψ(p1)φ(p2)σ(p1, p2).

This is precisely the smooth noncommutative torus A∞σ.

The norm || · || is defined to be the operator norm for the action of S(T ) on

L2(T ) given by . Via the Fourier transform, carry this structure back to C∞(T ),to obtain the smooth noncommutative torus as a strict deformation quantizationof C∞(T ), [15] with respect to the translation action of T .

Example 6.2. We next generalize the above to the case of principal torusbundles q : Y → X of rank equal to n. Note that fibrewise smooth functions on Ydecompose as a direct sum,

C∞fibre(Y ) =

⊕α∈T

C∞fibre(X,Lα)

φ =∑

α∈T

φα

where C∞fibre(X,Lα) is defined as the subspace of C∞

fibre(Y ) consisting of functions

which transform under the character α ∈ T , and where Lα denotes the associatedline bundle Y ×α C over X. That is, φα(yt) = α(t)φα(y), ∀ y ∈ Y, t ∈ T . The

direct sum is completed in such a way that the function T α → ||φα||∞ ∈ R

is in S(T ). In this interpretation of C∞fibre(Y ), it is easy to extend to this case,

the explicit deformation quantization given in the previous example, which wenow briefly outline. For φ, ψ ∈ C∞

fibre(Y ), define a new associative product on C∞

fibre(Y ) as follows. For y ∈ Y , α, α1, α2 ∈ T , let

(ψ φ)(y, α) =∑

α1α2=α

ψ(y, α1)φ(y, α2)σ(q(y);α1, α2),

using the notation ψ(y, α1) = ψα1(y) etc., and where σ ∈ Cb(X,Z2(T ,T)) is a

continuous family of bicharacters of T such that σ0 = 1, which is part of the datathat we start out with. We remark that one way to get such a σ is to choosea continuous family skew-symmetric forms on T , γ : X −→ Z2(T ,R), and defineσ = exp(−πγ). In the case of the principal torus bundle Y , we note that the

vertical tangent bundle of Y has a Poisson structure, i.e. γ ∈∧2 T vertY , which can

be naturally interpreted as a continuous family of symplectic structures along thefibre, that is, γ is of the sort considered just previously. We denote the deformedalgebra by C∞

fibre(Y ), and using §3, we can realize it as a parametrised strict

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NCP TORUS BUNDLES VIA PARAMETRISED DEFORMATION QUANTIZATION 13

deformation quantization of C∞fibre(Y ). Since the construction is T -equivariant,

C∞fibre(Y ) has a T -action that is induced from the given T -action on C∞

fibre(Y ).

Example 6.3. We next consider the example which was one of the inspirationsfor Theorem 5.1. Although it is a special case of the previous example, and isprobably also treated elsewhere, nevertheless we think that it is worthwhile to treatin our context. Consider a 3-dimensional torus, which we write as S1×T , where Tis a two dimensional torus. Let ·, · denote the Poisson bracket on S1 × T comingfrom T and trivial on S1. Then this Poisson bracket is invariant under the T actionon S1×T , where T acts trivially on S1 and via translation on itself. Here the fibresare T . As in the example above, we construct a strict deformation quantization ofC∞

fibre(S1 × T ). Taking the partial Fourier transform in the T -variables, we obtain

an isomorphism between C∞fibre(S

1 × T ) and Sfibre(S1 × T ). In the notation of the

previous example, for φ, ψ ∈ Sfibre(S1 × T ), define

(ψ φ)(y, p) =∑

p1+p2=p

ψ(y, p1)φ(y, p2)σ(y; p1, p2).

where σ : S1 = R/Z −→ H2(T ,T) ∼= T is the family of bicharacters of T givenby

σ(y; p1, p2) = exp(−πyγ(p1, p2)).

Here γ is defined as in Example 6.1. This gives us a family of smooth noncommu-tative tori, that is,

Sfibre(S1 × T ) =

∫y∈S1

A∞σ(y)

which in turn can be identified with (when = 1) the fibrewise Schwartz subalgebraof the 3-dimensional integer Heisenberg group, HeisZ. That is, the norm closure ofSfibre(S

1 × T )=1 is isomorphic to C∗(HeisZ).

On the other hand, using the results of §3, we see that Sfibre(S1 × T ) is a

parametrised strict deformation quantization of C∞fibre(S

1 × T ).

Example 6.4. Motivated by Theorem 5.1 and an example in [15], we definenoncommutative non-principal torus bundles as follows. Let ρ : π1(X) → Sp(2n,Z)be a representation of the fundamental group, T = R2n/Z2n be the torus and

q : Yρ → X be the non-principal torus bundle given by Yρ = (X×T )/π1(X), wherewe observe that the symplectic group is a subgroup of the automorphism group of

T and Γ = π1(X) acts on T via ρ and on the universal cover r : X −→ X via decktransformations.

Let ·, · denote the Poisson bracket on X × T coming from T and trivial on

X. Then this Poisson bracket is invariant under the T action on X × T , therefore

descending to a Poisson bracket on X, denoted by the same symbol. As in the

example above, we construct a strict deformation quantization of C∞fibre(X × T ),

which is the algebra of continuous functions on X × T that are smooth alongthe fibres. Taking the partial Fourier transform in the T -variables, we obtain an

isomorphism between C∞fibre(X × T ) and Sfibre(X × T ). In the notation of the

previous example, for φ, ψ ∈ Sfibre(X × T ), define

(ψ φ)(y, p) =∑

p1+p2=p

ψ(y, p1)φ(y, p2)σ(r(y); p1, p2).

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14 KC HANNABUSS AND V MATHAI

where σ : X −→ Z2(T ,T) is a continuous family of bicharacters of T , which ispart of the data that we start out with.

Then transporting this structure back to C∞fibre(X×T ) gives a strict deformation

quantization such that Γ = π1(X) acts properly on it, cf. [15]. We denote the

deformed algebra as C∞fibre(X×T ). The fixed point subalgebra C∞

fibre(X×T )Γof the

deformed algebra is then the desired parametrised strict deformation quantization,

C∞fibre(X×T )Γ

= C∞

fibre(Yρ)σ, where we note that C∞

fibre(X×T )Γ = C∞fibre(Yρ). This

is our definition of a noncommutative non-principal torus bundle. To summarize,it is determined by two pieces of data:

• ρ ∈ Hom(π1(X), Sp(2n,Z));

• σ ∈ C(X,Z2(T ,T)), that is, a continuous family of bicharacters of T .

Appendix A. Factors of Automorphy

Appendix C to [1] introduced a method for lifting algebra bundles to a con-tractible universal cover and encoding information about the Dixmier–Douady classin a factor of automorphy j. This also fits into a parametrised deformation picture,but with the further generalisation that the group Γ now acts on the parameterspace X. The cocycle j for a lifting can be reconstructed from the Dixmier–Douadyclass δ ∈ H3(X,Z) ∼ H3(Γ,Z), by first finding τ(k1, k2, x) (k1, k2 ∈ Γ, x ∈ X) with

dτ = δ, defining τ = exp(2πiτ), and then finding j(k, x) satisfying

j(k1, k2x)j(k2, x) = τ (k1, k2, x)j(k1k2, x),

which can be achieved by a modified τ -inducing construction, which gives j in termsof τ .

We know that τ is a C0(X)-valued cocycle satisfying the cocycle condition

τ (k1k2, k3)α−1k3

[τ (k1, k2)] = τ (k1, k2k3)τ (k2, k3)

where α just gives the translation action on C0(X), and similarly suppressing the

X-dependence in j allows us to rewrite its cocycle condition as

α−1k2

[j(k1)]j(k2) = τ (k1, k2)j(k1k2).

The cocycle condition on τ can also be written as

αk3[τ (k1k2, k3)]τ (k1, k2) = αk3

[τ (k1, k2k3)]αk3[τ (k2, k3)]

so setting U(k1) : k → αk[τ (k1, k)] we get

U(k1k2)(k3)τ (k1, k2) = α−1k2

[U(k1)(k2k3)]U(k2)(k3).

We can lift the automorphism αk2to

α−1k2

[U(k1)](k3)] = α−1k2

[U(k1)(k2k3)]

and thenU(k1k2)τ (k1, k2) = α−1

k2[U(k1)]U(k2),

the type of cocycle condition to be satisfied by j.

To compare these with the Landstad–Kasprzak construction we take Γ = V ,ρ the left translation (Lkf)(x) = f(k−1x). Now we think of j as a map from K

to unitary multipliers on C0(X), and take U(k) : x → j(k, k−1x), noting that the

cocycle condition j(k1, k2x)j(k2, x) = τ (k1, k2, x)j(k1k2, x) gives

U(k1)ρ(k1)[U(k2)] = τ (k1, k2)U(k1k2),

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NCP TORUS BUNDLES VIA PARAMETRISED DEFORMATION QUANTIZATION 15

precisely the condition arising in deformation (though the new ingredient is that Kacts on the X argument of τ ).

We note also that the predual algebra in the Landstad theory is the generalisedfixed point algebra A = Bρ.

When this paper was completed, Marc Rieffel pointed out to us that [18] dealswith an interesting bundle situation with varying deformation form, but where hewas able to transform it in that special case into a constant-form situation. This isof course not possible to arrange in general.

References

[1] P. Bouwknegt, K. C. Hannabuss, and V. Mathai, C∗-algebras in tensor categories, ClayMathematics Proceedings. 12 (2009) 39 pages, (in press). [math.QA/0702802]

[2] A. Connes, C∗-algebres et geometrie differentielle, C.R. Acad. Sci. Paris, Ser. A-B, 290,(1980) no. 13, 599–604.

[3] S. Echterhoff, R. Nest, and H. Oyono-Oyono, Principal non-commutative torus bundles, Proc.London Math. Soc. (3) 99, (2009) 1–31.

[4] S. Echterhoff and D. P. Williams, Crossed products by C0(X)-actions, J. Funct. Anal. 158(1998) no. 1, 113–151.

[5] K.C. Hannabuss, Representations of nilpotent locally compact groups, J. Funct. Anal. 34(1979) no. 1, 146–165.

[6] A. an Huef, I. Raeburn, and D.P. Williams, Functoriality of Rieffel’s generalised fixed pointalgebras for proper actions, [arXiv:0909.2860].

[7] S. Kaliszewski and J. Quigg, Categorical Landstad duality for actions, Indiana Math. J. 58(2009) 415–441.

[8] G. Kasparov, Equivariant K-theory and the Novikov conjecture, Invent. Math. 91, (1988)

147–201.[9] P. Kasprzak, Rieffel deformation via crossed products, J. Funct. Anal. 257 (2009) 1288–1332.

[10] A. Kleppner, Multipliers on abelian groups, Math. Ann. 158 (1965) 11–34.[11] M. B. Landstad, Duality theory for covariant systems, Trans. Amer. Math. Soc. 248 (1979)

223–267.[12] M. B. Landstad, Quantization arising from abelian subgroups, Internat. J. Math. 5 (1994)

897–936.[13] V. Mathai and J. Rosenberg, T-duality for torus bundles via noncommutative topology, Com-

mun. Math. Phys., 253 no. 3 (2005) 705–721. [hep-th/0401168][14] V. Mathai and J. Rosenberg, T-duality for torus bundles with H-fluxes via noncommutative

topology, II: the high-dimensional case and the T-duality group, Adv. Theor. Math. Phys.,10 no. 1 (2006) 123–158. [hep-th/0508084]

[15] M. A. Rieffel, Deformation quantization for actions of Rd , Memoirs of the Amer. Math. Soc.106 (1993), no. 506, 93 pp.

[16] M. A. Rieffel, Quantization and C∗-algebras, Contemporary Math. 167, (1994), 67–97.[17] M. A. Rieffel, Applications of strong Morita equivalence to transformation group C∗-algebras,

Proceedings of Symposia in Pure Mathematics, 38 (1982) Part I, 299–310.[18] M. A. Rieffel, On the operator algebra for the space-time uncertainty relations, Operator

algebras and quantum field theory (Rome, 1996), 374–382, Internat. Press, Cambridge, MA,1997.

(Keith Hannabuss) Mathematical Institute, 24-29 St. Giles’, Oxford, OX1 3LB, and

Balliol College, Oxford, OX1 3BJ, England

E-mail address: kch@balliol.ox.ac.uk

(Varghese Mathai) Department of Pure Mathematics, University of Adelaide, Ade-

laide, SA 5005, Australia

E-mail address: mathai.varghese@adelaide.edu.au

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A Survey of Noncommutative Yang-Mills Theory forQuantum Heisenberg Manifolds

Sooran Kang

Abstract. In this paper, we give a short overview of noncommutative Yang-

Mills theory developed by Connes and Rieffel and discuss Yang-Mills theoryfor quantum Heisenberg manifolds.

Introduction

It has been over 50 years since the publication of the fundamental paper of[YM], when Yang and Mills introduced a new mathematical framework to describethe interactions among elementary particles. At the time, the importance of theiridea was not fully recognized, because a Yang-Mills field is massless and, such atheory was incompatible to experiments. In the late 1960s, these problems weresolved in [Hig], and now their idea is at the heart of theoretical physics includingquantum field theory, string theory and the theory of gravitation.

As mentioned elsewhere in this volume, the interplay between mathematics andphysics has provided many interesting results since the birth of quantum physics. Infact, it was recognized soon by mathematicians that Yang-Mills non-abelian gaugetheory is related to connections on fiber bundles, and the paper [WuY] providinga dictionary between the two different languages of physics and mathematics forthe same subject was published by Wu and Yang shortly after. This recognitioncaused the rapid development of new realms of mathematics, especially in the studyof three- and four-dimensional manifolds, and conversely, the existing mathematicsprovided new insights and rigorous mathematical formulations for this new areaof physics. There has been a great deal of effort to extend Yang-Mills theory tovarious areas of mathematics in last twenty years, and some remarkable works canbe found in [AtB], [CR], [Donal1], [SW], [Tau], [UY].

Historically, the motivation behind noncommutative geometry can be found inthe development of quantum theory in the 1920s, when a noncommutative quan-tum mechanical system was constructed. However, the modern origins of noncom-mutative geometry first appeared in the algebraic version of differential geometryconstructed by J. K. Koszul around 1960 in [Kosz], and the subject now known asnoncommutative differential geometry was developed extensively by French mathe-matician Alain Connes in the early 1980s in [C1], [C2]. Objects in noncommutative

1991 Mathematics Subject Classification. Primary 46L87 ; Secondary 58B34.Key words and phrases. The Yang-Mills functional, quantum Heisenberg manifolds, Laplace’s

equation, Morita equivalence.1

Proceedings of Symposia in Pure MathematicsVolume 81, 2010

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2 SOORAN KANG

geometry are usually noncommutative deformations, sometimes called deformationquantizations, of commutative counterparts, and one studies the geometric the-ory of noncommutative algebraic structures, such as connections and curvaturesof noncommutative spaces. A typical example of a noncommutative space is thenoncommutative torus, which can be viewed as the deformation quantization ofthe function algebra of the ordinary two-torus. This example has been studiedextensively and used as a test case for more complicated situations. Readers canfind more about noncommutative tori in [Rf2], [Rf5], [Rf8], [RfSch], and someapplications in [CR], [CDS], [KoSch], [KoSch2], [Ros].

As we find in the papers [C2], [C3], [CDS], [LLS], [Lnm], [SW], noncommuta-tive geometry has naturally arisen in the development of recent quantum physics.In particular, the framework of noncommutative geometry can be found in theframeworks for open string theory and M(atrix) theory. More references can befound in [CDS], [KoSch] and [SW]. What should be noted here is that the coreidea in these papers is based on Connes’ and Rieffel’s noncommutative Yang-Millstheory on the noncommutative torus. As mentioned in [SW],

“the framework of noncommutative Yang-Mills theory seems verypowerful since the T-duality acts within the noncommutative Yang-Mills framework”.

In other words, Morita equivalence of the noncommutative torus plays a key rolein these theories. After the fundamental work of Connes and Rieffel in the mid-1980s, no example of noncommutative Yang-Mills theory on any other differentnoncommutative manifold than the noncommutative torus within the same non-commutative framework has been completely understood to date. This may bebecause the proofs in [CR] depend highly on two specific properties of the noncom-mutative torus: a symmetry of the projective modules over the noncommutativetorus, and a well-known relation between its generators.

Using the same framework developed in [CR], the author in her Ph.D thesis[K1] has attempted to extend Yang-Mills theory to quantum Heisenberg manifolds,which are different type of noncommutative C∗-algebras first constructed by MarcRieffel [Rf3]. The main theorem in this thesis what will appear in [K2] describesa certain family of connections on a projective module over the quantum Heisen-berg manifold that give rise to critical points of the Yang-Mills functional on thequantum Heisenberg manifold, and that are also related to Laplace’s equation onquantum Heisenberg manifolds. Readers can find the full details in [K2]. Moritaequivalence for quantum Heisenberg manifolds and construction of (finitely gen-erated) projective modules over quantum Heisenberg manifolds have been studiedby Abadie and her collaborators in [Ab1]–[Ab4] and [AEE]. Some other worksrelated to quantum Heisenberg manifolds can be found in [Ch], [ChS], [ConDu1],[ConDu2], [Li], [W]. Also, Yang-Mills for commutative Heisenberg manifolds canbe found in [Ur], and an explicit example of compatible connections on a quantumHeisenberg manifold is given in [L].

The main difference between the study of Yang-Mills theory for quantumHeisenberg manifolds and that of the noncommutative torus is the following. Firstof all, the projective modules over the quantum Heisenberg manifolds, denoted byDc,

μν , were constructed by realizing the algebra Dc,μν as generalized fixed point al-

gebras of certain crossed product C∗-algebras in [Ab2], while projective modulesover the noncommutative torus were constructed from a foliation of the ordinary

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YANG-MILLS THEORY FOR QUANTUM HEISENBERG MANIFOLDS 3

two-torus. Also, in [K2] a particular Grassmannian connection is used to producea compatible connection on the projective module, and the method of finding sucha nontrivial connection is related to finding Rieffel projections in noncommutativetori, which is a different approach from that used by Connes and Rieffel in [CR]and [Rf7]. The last step in finding actual solutions to the Yang-Mills equation isrelated to solving an elliptic partial differential equation, which is also very differ-ent from the methods of [CR] and [Rf7]. On the other hand, the theory of Moritaequivalence still plays an important role in our study, in particular in establishinga relationship between a particular family of critical points of the Yang-Mills func-tional on quantum Heisenberg manifolds and a set of solutions to Laplace’s equationon quantum Heisenberg manifolds. Recall that the Laplacian is the leading termfor the coupled set of equations making up the Yang-Mills equation.

This paper is organized as follows. In Section 1, we discuss noncommutativeYang-Mills theory as originally developed by Connes and Rieffel, based on theirpaper [CR]. In Section 2, we describe Rieffel’s quantum Heisenberg manifolds andMorita equivalence for quantum Heisenberg manifolds. In Section 3, we discuss theYang-Mills functional and Laplace’s equation on quantum Heisenberg manifoldsdescribed in [K2].

1. Connes’ and Rieffel’s Noncommutative Yang-Mills Theory

To formulate Yang-Mills theory on a noncommutative manifold, we need afinitely generated projective module over a noncommutative counterpart of themanifold, usually a deformation quantization of the manifold. We also need a notionof compatible connection and curvature on the finitely generated projective module,which will provide geometric properties. (For brevity, we will use “projective” tomean “finitely generated projective” from now on). We then define the Yang-Millsfunctional on the set of compatible connections. This functional corresponds tothe classical Yang-Mills functional, which can be also interpreted as the Energyfunctional in physics. We then analyze the nature of the set of critical points of theYang-Mills functional, YM . We are particularly interested in the critical pointswhere YM attains its minimum.

We describe Connes’ and Rieffel’s noncommutative Yang-Mills theory intro-duced in [CR] as follows. Let A be a unital C∗-algebra. Let α be an action of Liegroup G on A, and let g be the corresponding Lie algebra with basis Z1, . . . , Zn.Then we can give a smooth structure on A using the Lie group action α, defined byA∞ = a ∈ A | g → αg(a) is smooth in norm. The infinitesimal form of α givesan action, δ, of the Lie algebra g of G, as a Lie algebra of derivations on A∞.

Remark 1.1. When we consider C∞(M) as a commutative algebra correspond-ing to a manifold M , it is known that there is a natural isomorphism between theset of all vector fields on M and the set of all derivations of C∞(M) as Lie algebras.Thus, for given noncommutative C∗-algebra A, we can consider the Lie algebra ofderivations on A∞ as a proper analogue of vector field on M .

Let Ξ be a (right) projective module over a unital C∗-algebra A. As shown inLemma 1 of [C1], given a finitely generated projective A-module Ξ, there is a denseA∞-submodule Ξ∞ ⊂ Ξ, such that Ξ∞ is finitely generated and projective over A∞

and Ξ is isomorphic to Ξ∞ ⊗A∞ A. Furthermore, Ξ∞ is unique up to isomorphismas an A∞-module. We will denote Ξ∞ and A∞ by Ξ and A for notational simplicityfrom now on.

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4 SOORAN KANG

Also, we can always equip Ξ with an A-valued positive definite inner product〈·, ·〉A, called a Hermitian metric, such that 〈ξ, η〉∗A = 〈η, ξ〉A , 〈ξ, ηa〉A = 〈ξ, η〉Aa,for ξ, η ∈ Ξ and a ∈ A. See the details of this construction for C∗-modules in [Rf1]and [Lnc].

Definition 1.2. [C1] Let Ξ, A and g be as above. A connection ∇ is a linearmap from Ξ to Ξ⊗ g∗ such that

∇X(ξa) = (∇X(ξ))a+ ξ(δX(a)),

for all X ∈ g, ξ ∈ Ξ and a ∈ A. We say that the connections are compatible withthe Hermitian metric if

δX(〈ξ, η〉A) = 〈∇Xξ, η〉A + 〈ξ,∇Xη〉A.

We denote the set of compatible connections by CC(Ξ). Note that the con-nection ∇ in the above definition is defined on Ξ that has a value in Ξ ⊗ g∗. i.e.∇(ξ) : g → Ξ so that ∇(ξ)(X) ∈ Ξ for ξ ∈ Ξ and X ∈ g. We then denote ∇(ξ)(X)by ∇X(ξ).

According to Connes’ theory, we can always define a compatible connectionon a projective module over A as follows. For a given unital C∗-algebra A anda projection Q ∈ A, QA is a projective right A-module in an obvious way. Asdescribed in [C1], we define a connection ∇0 on QA, called the “Grassmannianconnection”, by

∇0X(ξ) = QδX(ξ) ∈ QA, for all ξ ∈ QA and X ∈ g.

Obviously, this is a compatible connection with the canonical Hermitian metric onQA, such that 〈ξ, η〉 = ξ∗η for ξ, η ∈ QA.

For given right A-module Ξ, let E = EndA(Ξ). Then the following facts areshown in [CR]. If ∇ and ∇′ are any two connections, then ∇X −∇′

X is an elementof E, for each X ∈ g. If ∇ and ∇′ are both compatible with the Hermitian metric,then ∇X −∇′

X is a skew-symmetric element of E for each X ∈ g. Thus, once wehave a compatible connection ∇, every other compatible connection ∇′ is of theform ∇ + μ, where μ is a linear map from g into Es, the set of skew-symmetricelement of E, such that μX

∗ = −μX for X ∈ g.The curvature of a connection ∇ is defined to be the alternating bilinear form

Θ∇ on g, given by

Θ∇(X,Y ) = ∇X∇Y −∇Y ∇X −∇[X,Y ],

for X,Y ∈ g. It is not hard to check that the values of Θ are in E for a givenconnection ∇, and the values of Θ are in Es if a connection ∇ is compatible withthe Hermitian metric.

For given A-valued inner product 〈·, ·〉A, we can define an E-valued inner prod-uct 〈·, ·〉E by

〈ξ, η〉Eζ = ξ〈η, ζ〉A,for ξ, η, ζ ∈ Ξ. So there is a natural bimodule structure (left E-right A) on Ξ.

Now we assume that A has a faithful α-invariant trace, i.e. τ (δX(a)) = 0 fora ∈ A and X ∈ g. Then τ determines a faithful trace, τE, on E = EndA(Ξ), definedby

τE(〈ξ, η〉E) = τ (〈η, ξ〉A).

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YANG-MILLS THEORY FOR QUANTUM HEISENBERG MANIFOLDS 5

To define the Yang-Mills functional on CC(Ξ), we need a bilinear form1 on thespace of alternating 2-forms with values in E. Let Z1, · · · , Zn be a basis for g asstated before. We define a bilinear form ·, ·E by

Φ,ΨE =∑i<j

Φ(Zi ∧ Zj)Ψ(Zi ∧ Zj),

for alternating E-valued 2-forms Φ,Ψ. Clearly, its values are in E.Then the Yang-Mills functional, YM , is defined on CC(Ξ) by

(1.1) YM(∇) = −τE(Θ∇,Θ∇E).

To describe the moduli space for Ξ, we need an analogue of the gauge groupacting on CC(Ξ). As defined in [CR], the gauge group here is just the group UEof unitary element of E = EndA(Ξ), acting on CC(Ξ) by conjugation, i.e. foru ∈ UE, ∇ ∈ CC(Ξ), we define γu(∇) by

(γu(∇))Xξ = u(∇X(u∗ξ)),

for ξ ∈ Ξ and X ∈ g. It is easily verified that γu(∇) ∈ CC(Ξ). Also it is not tohard to show that Θγu(∇)(X,Y ) = uΘ∇(X,Y )u∗ for X,Y ∈ g, and that the moduliΘγu(∇),Θγu(∇) = uΘ∇,Θ∇u∗. Thus, it follows that

YM(γu(∇)) = YM(∇)

for all u ∈ UE and ∇ ∈ CC(Ξ). Thus YM is a well-defined functional on thequotient space CC(Ξ)/UE. Since we are interested in finding the minimizing con-nections for YM , we call MC(Ξ)/UE the moduli space for Ξ, where MC(Ξ) isthe set of compatible connections where YM attains its minimum. More generally,the moduli space is the quotient of the set of critical points for YM by prescribedgauge groups.

As mentioned earlier, the Yang-Mills problem is about determining the natureof the set of the critical points for YM . According to the differential calculus, ∇ isa critical point of YM if D(YM(∇)) = 0, i.e. if the derivative of YM at ∇ is zero.Thus, we have

d

dt

∣∣∣t=0

YM(∇+ tμ) = D(YM(∇)) · μ,

where D is the derivative. So ∇ is a critical point of YM if we have, for all linearmaps μ : g → Es,

d

dt

∣∣∣t=0

YM(∇+ tμ) = 0.

According to [Rf7], this leads to the following proposition.

Proposition 1.3. [Rf7] ∇ is a critical point of YM if for all Zig,

(1.2)∑j

[∇Zi,Θ∇(Zi, Zj)]−

∑j<k

cijkΘ∇(Zj , Zk) = 0,

where cijk are structure constants of g.

1In classical Yang-Mills theory, YM measures the “strength” of the curvature of a connection.Thus, we need a proper analogue for the Riemannian metric on the “tangent space” of A, whichcan be given by an ordinary positive definite inner product on g, taking values in E.

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6 SOORAN KANG

Thus, if the Lie algebra g is abelian and of dimension two, then ∇ is a criticalpoint for YM if and only if Θ∇(X,Y ) commutes with ∇Z for all X,Y, Z ∈ g.Therefore, in the case of the noncommutative torus, ∇ is a critical point if eitherΘ∇ = 0 (so that ∇ is a flat connection, clearly minimizing YM) or the rangeof ∇ generates a 3-dimensional Heisenberg Lie algebra, which also explains theterminology “Heisenberg modules” for the noncommutative torus used in [Rf5] and[Rf7]. This proposition is somewhat compatible with the main results in [CR].

Theorem 1.4. [CR] Let A be the non-commutative two torus. Assume that gis abelian, and that Ξ admits compatible connections with constant curvature. Thenthe set MC(Ξ) of compatible connections where YM attains its minimum, consistsexactly of all compatible connections with constant curvature. Furthermore, thecurvature of all these minimizing connections will be the same.

Note that the proof of this theorem does not depend on the previous propo-sition. Also, the moduli space for the case of the noncommutative torus can bedescribed as follows.

Theorem 1.5. [CR] Let Ξ be the Heisenberg modules for the noncommuta-tive torus. The moduli space MC(Ξ)/UE for compatible connections on Ξ whichminimize the Yang-Mills functional, is homeomorphic to T2.

The definition of the Heisenberg modules is given in [CR], but the reader canfind a complete description of the construction of Heisenberg (equivalence) modulesin [Rf1]. Also, note that the proof of this theorem depends highly on the structureof the noncommutative torus. See more details in [CR].

2. Quantum Heisenberg Manifolds

We start with the definition of the quantum Heisenberg manifolds introducedby Rieffel in [Rf3] as follows.

Let G be the Heisenberg group, parametrized by

(x, y, z) =

⎛⎝1 y z0 1 x0 0 1

⎞⎠ ,

so that when we identify G with R3, the product is given by

(2.1) (x, y, z)(x′, y′, z′) = (x+ x′, y + y′, z + z′ + yx′).

For any positive integer c, let Dc denote the subgroup of G consisting of those(x, y, z) such that x, y, and cz are integers. Then the Heisenberg manifold, Mc,is the quotient G/Dc, on which G acts on the left. In [Rf3], Rieffel constructeddeformation quantizations2 of Mc, D∈R, in the direction of a Poisson bracket

2Since the first development of quantum mechanics, physicists have looked for the propermathematical structure to explain both classical physics and quantum physics. Quantizationis mostly an attempt to form quantum mechanical systems from classical mechanical systems,and deformation quantization is related to the classical limiting process ( → 0, where is thePlanck’s constant), which can be understood as an inverse process of contraction in physics. Thetypical way to construct a deformation quantization is by using power series in the variable ,called formal deformation quantization. This method does not guarantee convergence. Rieffel’sstrict deformation quantization takes care of the convergence problem and the result of the strictdeformation quantization provides a continuous field of noncommutative C∗-algebras [Rf3]. Onecan read more about deformation quantizations within the setting of operator algebras in [Rf9],[Rf10], and [Rf11].

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YANG-MILLS THEORY FOR QUANTUM HEISENBERG MANIFOLDS 7

Λμν determined by two real parameters μ and ν, where μ2 + ν2 = 0 and is thePlanck constant. Moreover, he recognized that for fixed , D can be described asa generalized fixed-point algebra of a crossed product C∗-algebra under a properaction3.

B. Abadie described how one can construct a projective bimodule Ξ over twogeneralized fixed-point algebras denoted by Ec,

μν and Dc,μν under appropriate con-

ditions in [Ab1] and [Ab2]. Here Dc,μν is the quantum Heisenberg manifold cor-

responding to D for fixed . In other words, the quantum Heisenberg manifoldis a non-commutative unital C∗-algebra which can be realized as a generalizedfixed point algebra of a certain crossed product C∗-algebra under a proper action.Thus, we know the formula of ∗-product and involution on the quantum Heisenbergmanifold, and there is a natural Heisenberg group action that is ergodic on this C∗-algebra, which also gives a smooth structure on it. To be specific, we describe thequantum Heisenberg manifold as follows.

Let M = R× T and λ and σ be the commuting actions of Z on M defined by

λp(x, y) = (x+ 2pμ, y + 2pν) and σp(x, y) = (x− p, y),

where is Planck’s constant, μ, ν ∈ R, and p ∈ Z.Construct the crossed product Cb(R×T)×λZ in the usual way, where Cb(R×T)

is a set of bounded functions on R×T. This means that the ∗-product and involutionof Cb(R× T)×λ Z are given by

(Φ ∗Ψ)(x, y, p) =∑q∈Z

Φ(x, y, q)Ψ(x− 2qμ, y − 2qν, p− q),

Φ∗(x, y, p) = Φ(x− 2pμ, y − 2pν,−p),

for Φ, and Ψ ∈ Cc(R× T× Z).To describe the generalized fixed-point algebra, we need a proper action, in the

sense of [Rf6], on the crossed product C∗-algebra, Cb(R × T) ×λ Z. Let ρ denotethe action of Z on Cb(R× T)×λ Z. Then ρ is given by

(ρkΦ)(x, y, p) = e(ckp(y − pν))Φ(x+ k, y, p),

where k,p ∈ Z, Φ ∈ Cc(R× T× Z), for any real number x, e(x) = exp(2πix).

Remark 2.1. The above formula ρ can be obtained from the proper action ρthat gives rise to the deformation quantization of Heisenberg manifolds in Theorem5.5, [Rf3].

Then the generalized fixed point algebra of Cb(R × T) ×λ Z by the action ρ,denoted by Dc,

μν , is the closure in the multiplier algebra of Cb(R × T)×λ Z of the∗-subalgebra D0 consisting of functions Φ ∈ Cc(R × T × Z) which have compactsupport on Z and satisfy

ρk(Φ) = Φ for all k ∈ Z.

3If a compact group G acts on a manifold M , obviously the action of G is proper using

the standard definition. Also, in that case, we can consider the set of functions on the space oforbits as the fixed point algebra of C0(M), the set of continuous complex-valued functions on Mwhich vanish at infinity. In other words, C0(M/α) can be identified with the fixed point algebraf ∈ C0(M) | αg(f) = f for all g ∈ G, where M/α is the space of orbits, and the action α of G

on C0(M) is defined by αg(f)(m) = f(α−1g (m)). However, if the group is noncompact, we can

no longer identify C0(M/α) with the fixed point algebra. Rieffel developed an analogue of properactions for noncompact group G in [Rf6].

155

8 SOORAN KANG

The corresponding action of the Heisenberg group on Dc,μν is given by

(L(r,s,t)Φ)(x, y, p) = e(p(t+ cs(x− r − pμ)))Φ(x− r, y − s, p),

where Φ ∈ D0.The construction of the bimodule Ξ over two generalized fixed points algebras

is based on Situation 2 in [Rf4], which says that “a transformation group C∗-algebra C∗(G,M) is strongly Morita equivalent to C∗(M/G) if the action of G isfree.” See pp. 300–301. However, our case is not exactly the situation 2 since weare dealing with a generalized fixed point algebra of a crossed product C∗-algebranot the crossed product C∗-algebra itself. So here we provide a description of themethod first used in [Ab2] to find a bimodule between two generalized fixed pointalgebras.

For given a proper action λ on M = R × T, it is well known that Cc(M) is aleft and right Cc(M × Z)-rigged module with formulas for the left and right innerproducts, λ〈·, ·〉 and 〈·, ·〉λ given in [Rf1]. Also, since the generalized fixed pointalgebra Dc,

μν contains Cc(M × Z) as a dense subalgebra, we can give an explicitformula for the map from Cc(M ×Z) into the multiplier algebra M(Cb(M)×λ Z),denoted by Pσ,u in Proposition 2.1 in [Ab2], whose image generates Dc,

μν . Then we

take the right inner product 〈·, ·〉λ by Pσ,u to produce the Dc,μν -valued inner product

at the level of the generalized fixed point algebra. Once we have an inner producton the right then we can determine the right action and induce the inner product onthe left with an appropriate choice of left action. Of course, the procedure shown in[Ab2] is more technical. First we need the correct analogue of a proper, free actionα on the crossed product Cb(M) ×λ Z that is induced from the free and properaction of Z on M , and the inner product we obtain from the previous procedureshould be compatible with the generalized fixed point algebra of Cb(M)×λZ underthe action α.

Using this construction, we can describe the corresponding Ec,μν -Dc,

μν bimodule

Ξ as the completion of Cc(R×T) with respect to the norm induced from the Dc,μν -

valued inner product, given by

〈f, g〉D(x, y, p) =∑k∈Z

e(ckp(y − pν))f(x+ k, y)g(x− 2pμ+ k, y − 2pν) ,

where f, g ∈ Cc(R× T) and k, p ∈ Z.Before closing this section, we would like to list some useful results obtained re-

garding the classification of the quantum Heisenberg manifolds up to ∗-isomorphismand Moreta equivalence. It was shown in [Ab2] and [Ab3] thatK0(D

c,μν )

∼= Z3⊕Zc,

K1(Dc,μν )

∼= Z3 and the range of traces on Dc,μν is Z+2μZ+2νZ. In addition, Dc,

μν

and Dc,μ′ν′ are isomorphic when (2μ, 2ν) and (2μ′, 2ν′) belong to the same orbit

under the usual action of GL2(Z) on T2. Also, the quantum Heisenberg manifoldDc,

μν was described in [AEE] as a crossed product by Hilbert C∗-bimodules, and

it is shown in [Ab4] that Dc,μν is Morita equivalent to Dc,

14μ , ν

2μ

for μ = 0 using the

method of crossed product by Hilbert C∗-bimodules. The main idea of a crossedproduct by a Hilbert C∗-bimodule is the following. The generalized fixed-pointalgebra of a certain crossed C∗-algebra constructed by Rieffel can be generated by

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YANG-MILLS THEORY FOR QUANTUM HEISENBERG MANIFOLDS 9

the fixed-point algebra and the first spectral subspace for the action of T4 on thecrossed product C∗-algebra. Also, it is known that the first spectral subspace has anatural bimodule structure over the fixed-point algebra. Thus, we can study gener-alized fixed-point algebras by examining each fixed-point algebra, its first spectralsubspace, and the bimodule structure. Using the same technique introduced in thepapers [Ab4] and [AEE], we can prove that Ec,

μν is ∗-isomorphic to a quantum

Heisenberg manifold with parameters(

14μ ,

ν2μ

). The proof is given in [K2]. It was

not claimed in [Ab4] or [AEE] that the generalized fixed-point algebra Ec,μν can

be identified with Dc,14μ

ν2μ

.

3. The Yang-Mills Functional and Laplace’s Equation on QuantumHeisenberg Manifolds

In [Ros], J. Rosenberg developed a noncommutative theory of nonlinear ellip-tic PDEs motivated by noncommutative geometry, and, as an initial example, heshowed that there was a close relationship between Laplace’s equation and PDEson the noncommutative two-torus. More precisely, he defined the energy functionalin terms of the infinitesimal generators of the usual action of T2 on the noncom-mutative two-torus, and he described the Euler-Lagrangian equation for criticalpoints of the energy functional on self adjoint elements a of the noncommutativetwo-torus as Laplace’s equation Δa = 0. Inspired by this work [Ros] and the workin [KoSch], in this section, we show that there is a relationship between a certainfamily of critical points of the Yang-Mills functional and Laplace’s equation on“multiplication-type”, skew-symmetric elements of quantum Heisenberg manifolds,which will be defined in Definition 3.3.

First we give an explicit formula for the Laplacian on quantum Heisenbergmanifolds first obtained by N. Weaver in [W] as follows. For Φ ∈ (Dc,

μν )∞,

Δ(Φ)(x, y, p) = (δX)2(Φ)(x, y, p) + (δY )2(Φ)(x, y, p) = −4πip2(x− pμ)2Φ(x, y, p)

−2πip(x−pμ)∂Φ

∂y(x, y, p)−2πip(x−pμ)

∂Φ

∂y(x, y, p)+

∂2Φ

∂y2(x, y, p)+

∂2Φ

∂x2(x, y, p).

When p = 0, then the above Laplacian becomes the following :

Δ(Φ)(x, y, 0) =∂2Φ

∂y2(x, y, 0) +

∂2Φ

∂x2(x, y, 0),

for Φ ∈ (Dc,μν )

∞. Thus, if we restrict the Laplacian on a smooth element H of Dc,μν

that is only supported at p = 0, i.e. H(x, y, p) = H(x, y)δ0(p), H ∈ C∞(T2), thenwe have

(3.1) Δ(H)(x, y, p) =∂2H

∂y2(x, y, p) +

∂2H

∂x2(x, y, p).

Now let Ξ be a left-Ec,μν and right-Dc,

μν bimodule as described before. To finda compatible connection on Ξ∞, we use the canonical Grassmannian connection∇0 on Q(Dc,

μν )∞ for a projection Q ∈ (Dc,

μν )∞. As we know, QDc,

μν and Ξ are

projective modules over the same C∗-algebra Dc,μν on the right. Since Ec,

μν and

4As seen in the previous section, the quantum Heisenberg manifold Dc,μν is the generalized

fixed point algebra of Cb(R× T)×λ Z under a proper action ρ. So there is a natural dual actionof T on the crossed product C∗-algebra Cb(R× T)×λ Z.

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10 SOORAN KANG

Dc,μν both have identity elements and are Morita equivalent5, there is a function

R ∈ Ξ such that 〈R,R〉E = IdE and Q = 〈R,R〉D is a projection of Dc,μν as shown

in Proposition 2.1 in [Rf2] (using Rieffel’s notation, n = 1 in our case). It is shownin [K2] that we can choose R to be smooth, so that letting Q = 〈R,R〉D, we havethe following lemma.

Lemma 3.1. Let Ξ and Q be as above. Define a module map φ : Ξ −→ QDc,μν

by φ(f) = 〈R, f〉D. Then φ is a module isomorphism between Ξ and QDc,μν .

Proof. Since Ξ is an Ec,μν - Dc,

μν bimodule, for any f, g and h ∈ Ξ we have〈f, g〉Eh = f〈g, h〉D. Then for f ∈ Ξ,

φ(f) = 〈R, f〉D = 〈〈R,R〉E ·R, f〉D = 〈R · 〈R,R〉D, f〉= 〈R,R〉∗D〈R, f〉D = 〈R,R〉D〈R, f〉D = Q〈R, f〉D.

Thus, φ(f) = 〈R, f〉D ∈ QDc,μν .

An inverse of φ, φ−1 is given by φ−1(d) = R · d, where · is the right action ofDc,

μν on Ξ. Then we have, for d ∈ QDc,μν

(φ φ−1)(d) = φ(R · d) = 〈R,R · d〉D = 〈R,R〉D · d.Since d ∈ QDc,

μν , we can write d = 〈R,R〉D · d′ for some d′ ∈ Dc,μν . So

(φ φ−1)(d) = 〈R,R〉D〈R,R〉D · d′ = 〈R,R〉D · d′ = d.

Similarly, we have (φ−1 φ)(f) = f for f ∈ Ξ. Therefore, φ is an isomorphism.

Remark 3.2. Notice that the trace of the projection Q = 〈R,R〉D is 2μ since

τD(〈R,R〉D) = τE(〈R,R〉E) = τE(IdE) =

∫ 2μ

0

∫ 1

0

1 dx dy = 2μ.

The explicit formula for τE on Ec,μν was first given in [Ab1].

To define a linear connection on Ξ∞, we take the image of the Grassmannianconnection on Q(Dc,

μν )∞ using the above map φ, which looks like

(3.2) ∇0X(f) = R · δX(〈R, f〉D),

for the given R ∈ Ξ∞ stated in Lemma 3.1 and X ∈ g, f ∈ Ξ∞. For notationalsimplicity, we denote Ξ∞, (Dc,

μν )∞ and (Ec,

μν )∞ by Ξ, Dc,

μν and Ec,μν in the rest of

this section.The method of constructing a smooth function R in (3.2) is related to the

mollification technique in PDEs and also it has similarities with the method offinding a scaling function in wavelet theory, in particular for the Meyer-type wavelet.The specific example of such a function R is given in [K2]. It turns out that theGrassmannian connection with the special function R on Ξ has nontrivial curvature,and ∇0 is not a critical point of the Yang-Mills functional. For further discussion,we introduce the notion of multiplication-type elements of Ec,

μν as follows.

Definition 3.3. For an element G ∈ C∞(T2), define the multiplication-typeelement G of Ec,

μν associated to G by

G(x, y, p) = G(x, y)δ0(p).

5The existence of the projective bimodule Ξ over two C∗-algebras, Ec,μν and Dc,

μν , means

that Ec,μν and Dc,

μν are Morita equivalent.

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YANG-MILLS THEORY FOR QUANTUM HEISENBERG MANIFOLDS 11

Remark 3.4. Recall that Ec,μν is a generalized fixed point algebra of a certain

crossed product C∗-algebra. So, G ∈ Ec,μν means that G has to satisfy the fixed

point condition imposed on Ec,μν , which implies that the corresponding function G

has to be defined on T2 to produce an element G of Ec,μν . Also, any multiplication-

type element G is a smooth element of Ec,μν since the corresponding function G is

smooth.

We list several properties of multiplication-type elements ofEc,μν without proofs,

as follows.

Lemma 3.5 ([K2], Lemma 6). Let G be a multiplication-type element of Ec,μν .

Then G is skew-symmetric, i.e G∗ = −G if and only if the corresponding functionG is also skew-symmetric. i.e. G(x, y) = −G(x, y).

Proposition 3.6 ([K2], Proposition 7). Let G be a multiplication-type elementof Ec,

μν with corresponding function G ∈ C∞(T2). Then G is skew-symmetric if and

only if G acts on the given Ec,μν -D

c,μν bimodule Ξ as a skew-symmetric multiplication

operator, i.e.(G · f)(x, y) = −G(x, y)f(x, y) for f ∈ Ξ.

In fact, the special function R that we construct in [K2] is such that thecurvature Θ0

∇ of ∇0, Θ0∇(X,Y ), is a multiplication-type element of Ec,

μν for allX,Y ∈ g. Note that g is the Heisenberg Lie algebra with basis X,Y, Z satisfying[X,Y ] = cZ for a positive integer c. (See [Rf3] for details). More precisely, it isgiven as follows.

Θ0∇(X,Y )(x, y, p) = f1(x)δ0(p),

Θ0∇(X,Z)(x, y, p) = 0,

Θ0∇(Y, Z)(x, y, p) = f2(x)δ0(p),

where f1 and f2 are smooth skew-symmetric periodic functions in the sense thatf1(x) = −f1(x) and f2(x) = −f2(x). Details of derivations of these formulas canbe found in [K2].

Also, it turns out that a multiplication-type element has a simple Lie bracketrelation with the Grassmannian connection ∇0, as follows.

Proposition 3.7. Let ∇0 be the Grassmannian connection on Ξ with the spe-cial function R given in (3.2) and let X,Y, Z be the basis of the Heisenberg Liealgebra g with [X,Y ] = cZ before. Let G be any multiplication-type skew-symmetricelement of Ec,

μν as defined above. Then, we have

[∇0X ,G](x, y, p) = − ∂

∂yG(x, y, p),

[∇0Y ,G](x, y, p) = − ∂

∂xG(x, y, p),

[∇0Z ,G] = 0.

Since the Grassmannian curvature Θ0∇ produced by R does not vanish under

the partial derivative∂

∂x, the above proposition gives the following.

Proposition 3.8 ([K2], Proposition 10). The Grassmannian connection ∇0

generates an infinite dimensional Lie algebra, in the sense that the Lie algebra ofoperators generated by ∇0

X ,∇0Y ,∇0

Z is infinite dimensional.

159

12 SOORAN KANG

Proof. For notational simplicity we write the Grassmannian curvature as fol-lows.

Θ0∇(X,Y ) = f1 , Θ0

∇(X,Z) = 0 , Θ0∇(Y, Z) = f2,

where f1 and f2 are the elements of Ec,μν with the corresponding skew-symmetric

functions f1 and f2. i.e. f1(x, y, p) = f1(x)δ0(p), f2(x, y, p) = f2(x)δ0(p) andf1(x) = −f1(x), f2(x) = −f2(x). Also, the above Lie bracket equations can beexpressed as follows. For a skew-symmetric multiplication-type element G ∈ Ec,

μν ,

[∇0X ,G] = − ∂

∂yG , [∇0

Y ,G] = − ∂

∂xG , [∇0

Z ,G] = 0.

Now using the formula for the curvature, we obtain

[∇0X ,∇0

Y ] = Θ0∇(X,Y ) +∇0

Z = f1 +∇0Z ,

[∇0X ,∇0

Z ] = Θ0∇(X,Z) = 0,

[∇0Y ,∇0

Z ] = Θ0∇(Y, Z) = f2.

Also, by the previous Proposition we have

[∇0X , f1] = 0, [∇0

X , f2] = 0,

[∇0Y , f1] = − ∂

∂xf1 = 0, [∇0

Y , f2] = − ∂

∂xf2 = 0,

[∇0Z , f1] = 0, [∇0

Z , f2] = 0.

The proof of the Lemma 5 in [K2] shows that − ∂

∂xf1 = 0 and − ∂

∂xf2 = 0, in

general. Thus ∇0X ,∇0

Y ,∇0Z generate an infinite dimensional Lie algebra.

In order to find the set of critical points of YM , recall that g is the HeisenbergLie algebra with basis X,Y, Z satisfying [X,Y ] = cZ for a positive integer c.Then Proposition 1.3 gives the following. The connection ∇ will be a critical pointif and only if

(3.3) [∇Y ,Θ∇(X,Y )] + [∇Z ,Θ∇(X,Z)] = 0,

(3.4) [∇X ,Θ∇(Y,X)] + [∇Z ,Θ∇(Y, Z)] = 0,

(3.5) [∇X ,Θ∇(Z,X)] + [∇Y ,Θ∇(Z, Y )]− c ·Θ∇(X,Y ) = 0.

It is now easy to see that the Grassmannian connection ∇0 given in (3.2) is not acritical point of YM . But we know that any other compatible connection can beobtained from the Grassmannian connection ∇0 by adding a linear map μ from g

into (Ec,μν )

s, the set of skew-symmetric element of Ec,μν . Thus, we can construct

another set of compatible connections on Ξ that are of the form ∇0 +μ for a linearmap μ : g → (Ec,

μν )s whose range lies in the set of multiplication-type elements

of Ec,μν . So, consider the case where μX is a multiplication-type, skew-symmetric

element. Let μX = GX , μY = GY and μZ = GZ for GX ∈ Ec,μν for X ∈ g with

the corresponding Gi ∈ C∞(T2) such that Gi(x, y) = −Gi(x, y) for i = 1, 2, 3.Using the formulas in the proof of Proposition 3.8, we can write the curvature Θ∇in terms of the Grassmannian curvature Θ0

∇ and the multiplication-type elementsGX, GY and GZ. Then the proposed conditions for obtaining critical points, (3.3),(3.4) and (3.5) give the following two equations.

(3.6)∂

∂xGX − ∂

∂yGX = c ·GZ − f1,

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YANG-MILLS THEORY FOR QUANTUM HEISENBERG MANIFOLDS 13

(3.7)∂2

∂y2GZ +

∂2

∂x2GZ =

∂

∂xf2 + c · a0,

where a0 =∫Tf1(x) dx, f1 = f1 − a0.

The existence of solutions of the form GX, GY and GZ as in Equations (3.6)and (3.7) is proven in [K2]. Also, notice that Equation (3.7) is an elliptic partialdifferential equation over the two-torus, while the operator on the left side of theequation looks exactly like the Laplacian in (3.1). In fact, Equations (3.6) and(3.7) can be expressed in terms of the Laplacian in (3.1) and derivations on quan-tum Heisenberg manifolds, since Ec,

μν is isomorphic (in a manner preserving the

bimodule structure) to a quantum Heisenberg manifold with parameters(

14μ ,

ν2μ

)as mentioned before. Thus, we obtain the following theorem.

Theorem 3.9. [K2] Let g, Ξ and Dc,μν be as before, and let c be a fixed positive

integer. Let δ be the infinitesimal form of the Heisenberg group action on Dc,μν

given as before and Δ = δ2X + δ2Y for X,Y ∈ g, where [X,Y ] = cZ. Let ∇0 bethe Grassmannian connection on Ξ, and let Θ0

∇ be the corresponding Grassman-nian curvature such that Θ0

∇(X,Y ) and Θ0∇(Y, Z) are non-trivial multiplication-type

skew-symmetric elements of Ec,μν , and Θ0

∇(X,Z) = 0. Let G be a linear map ong whose range lies in a set of the multiplication-type skew-symmetric elements ofEndDc,

μν(Ξ) = Ec,

μν . Then, ∇ = ∇0 + G is a critical point for the Yang-Mills

functional if and only if GX, GY and GZ satisfy the following equations.

δX(GY)− δY (GX) = c ·GZ −Θ0∇(X,Y ) + a0,

Δ(GZ) = −δY (Θ0∇(Y, Z)) + c · a0,

where a0 =∫Tf1(x)dx, and Θ0

∇(X,Y )(x, y, p) = f1(x)δ0(p) for a smooth periodicfunction f1(x).

Remark 3.10. We note that these critical points in the above theorem are notminima, but inflection points, which means that we can find a set of connectionsgiving smaller values of the Yang-Mills functional.

Acknowledgements. The author would like to use this opportunity to expressher gratitude to her thesis advisor, Judith Packer, for her constant patience andencouragement as well as her guidance and support. Also, she wishes to thankRobert S. Doran and Greg Friedman for organizing the CBMS conference in TexasChristian University in May 2009.

References

[Ab1] B. Abadie, “Vector Bundles” over quantum Heisenberg manifolds, Algebraic Methods inOperator Theory, (1994), 307–315.

[Ab2] B. Abadie, Generalized fixed-point algebras of certain actions on crossed products, PacificJ. Math. 171 (1995), 1-21.

[Ab3] B. Abadie, The range of traces on quantum Heisenberg manifolds, Trans. Amer. Math. Soc.352, no. 12, (2000), 5767–5780.

[Ab4] B. Abadie, Morita equivalence for quantum Heisenberg manifolds, Proc. Amer. Math. Soc.133, no. 12, (2005), 3515-3523.

[AEE] B. Abadie, S. Eilers, R. Exel, Morita equivalence for crossed products by Hilbert C∗-bimodules, Trans. Ameri. Math. Soc. 350 (1998), 3043–3054.

[AtB] M. F. Atiyah, R. Bott, The Yang-Mills equations over Riemann surfaces, Philos. Trans.Roy. Soc. London Ser. A, 308, no. 1505, (1983), 523–615.

161

14 SOORAN KANG

[C1] A. Connes, C∗-algebres et geometrie differentielle, C. R. Acad. Sc. Paris, 290 (1980), 559–604.

[C2] A. Connes, Noncommutative Geometry, Academic Press, 1994.[C3] A. Connes, Noncommutative geometry and the standard model with neutrino mixing, J. High

Energy Phys. 11 (2006), 081, 19pp. (electronic).[CDS] A. Connes, M. Douglas, A. Schwarz, Noncommutative geometry and matrix theory: Com-

pactification on tori, J. High Energy Phys. (2), (1998)

[Ch] P. S. Chakrabority, Metrics on the quantum Heisenberg manifold, J. Operator Theory, 54,no. 1, (2005), 93–100.

[ChS] P. S. Chakrabority, K. B. Sinha, Geometry on the quantum Heisenberg manifold, J. Funct.Anal. 203, no.2, (2003), 425–452.

[ConDu1] A. Connes, M. Dubois-Violette, Moduli space and structure of noncommutative 3-spheres, Lett. Math. Phys. 66, no.1–2, (2003), 91–121.

[ConDu2] A. Connes, M. Dubois-Violette, Noncommutative finite dimensional manifolds. II.Moduli space and structure of noncommutative 3-spheres, Comm. Math. Phys. 281, no.1,(2008), 23–127.

[CR] A. Connes, M. Rieffel, Yang-Mills for non-commutative two-tori, Contemp. Math. 62 (1987),237–266.

[Donal1] S. K. Donaldson, Anti-self-dual Yang-Mills connections on complex algebraic surfacesand stable vector bundles, Proc. London Math Soc. 3 (1985), 1–26.

[DonalKr] S. K. Donaldson, P. B. Kronheimer, The geometry of four-manifolds, Oxford U. P.(1990), 1–26.

[EAS] R. Exel, B. Abadie, S. Eilers, Morita equivalence for crossed products by Hilbert C∗-bimodules, Trans. Amer. Math. Soc. 350 (1998), 3043–3054.

[Hig] P. W. Higgs, Broken symmetries and the masses of gauge bosons, Phys. Rev. Lett. 13(1964), 508–509.

[K1] S. Kang The Yang-Mills functional and Laplace’s equation on quantum Heisenberg mani-folds, Ph.D. thesis, (2009).

[K2] S. Kang, The Yang-Mills functional and Laplace’s equation on quantum Heisenberg mani-folds, J. Funct. Anal. 258 (2010), 307–327.

[KoSch] A. Konechny, A. Schwarz, Introduction to M(atrix) theory and noncommutative geome-try, Phys. Rep. 360 (2002), 354–465.

[KoSch2] A. Konechny, A. Schwarz, BPS states on noncommutative tori and duality, Nucl. Phys.B(550) (1999), 561–584.

[Kosz] J. L. Koszul, Fiber Bundles and Differential Geometry, Tata Institute of FundamentalResearch, Bombay, (1960).

[L] H. Lee, Yang-Mills for quantum Heisenberg manifolds, Preprint, arXiv: 0907.0465, (2009).[Li] H. Li, Metric aspects of non-commutative homogeneous spaces, Preprint, arXiv:0810.4694,

(2008).[LLS] G. Landi, F. Lizzi, R. J. Szabo, String geometry and the noncommutative torus, Comm.

Math. Phys. 206, no. 3, (1999), 603–637.[Lnc] E. C. Lance, Hilbert C∗-modules - A toolkit for operator algebraists, London Math. Soc.

Lec. Ser. 210 (1995).[Lnm] N. P. Landsman, Lie groupoids and Lie algebroids in physics and noncommutative geom-

etry, J. Geom. Phys. 56, no. 1, (2006), 24–54.[Rf1] M. Rieffel, Induced representations of C∗-algebra, Adv. Math. 13 (1974), 176–257.[Rf2] M. Rieffel, C∗-algebras associated with irrational rotation, Pacific J. Math., 93, no.2 (1981),

415–429.[Rf3] M. Rieffel, Deformation quantization of Heisenberg manifold, Comm. Math. Phys. 122

(1981), 531–562.[Rf4] M. Rieffel, Applications of strong Morita equivalence to transformation group C∗-algebras,

Proc. Symp. Pure. Math. 38, Part I, (1982), 299–310.[Rf5] M. Rieffel, Projective modules over higher-dimensional non-commutative tori, Can. J. Math.

XL. 2 (1988), 257–338.[Rf6] M. Rieffel, Proper actions of groups on C∗-algebras, Mapping of Operator Algebras, Proc.

Janpan-US joint seminar, Birkhauser (1990), 141–182.[Rf7] M. Rieffel, Critical points of Yang-Mills for non-commutative two-tori, J. Diff. Geom. 31

(1990), 535–546.

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YANG-MILLS THEORY FOR QUANTUM HEISENBERG MANIFOLDS 15

[Rf8] M. Rieffel, Non-commutative tori - a case study of non-commutative differentiabl manifolds,Contemp. Math. 105 (1990), 191–211.

[Rf9] M. Rieffel, Deformation quantization and operator algebras, Proc. Symp. Pure. Math. 51(1990), 160–174.

[Rf10] M. Rieffel, Quantization and C∗-algebras, Contemp. Math. 167 (1994), 67–96.[Rf11] M. Rieffel, Questions on quantization, Contemp. Math., vol. 228, Amer. Math.Soc. Provi-

dence RI. (1998), 315–326.

[RfSch] M. Rieffel, A. Schwarz, Morita equivalence of multidimensional noncommutative tori,Intl. J. Math. 10(2) (1999), 257–338.

[Ros] J. Rosenberg, Noncommutative variations on Laplace’s equation, Annal. PDE. 1 (2008),95–114.

[SW] N. Seiberg, E. Witten, String theory and noncommutative geometry, J. High Energy Phys.9 (1999), 32–93

[Tau] C. H. Taubes, A framework for Morse theory for the Yang-Mills functional, InventionesMath. 94 (1988), 327–402.

[Ur] H. Urakawa, Yang-Mills connections over compact strongly pseudoconvex CR manifolds,Mathematische Zeitschrift, 216 (1993), 541–573.

[UY] K. K. Uhlenbeck, S-T, Yau, On the existence of hermitian Yang-Mills connections on stablebundles over compact kahler manifolds, Comm. Pure. Applied Math. 39 (1986), 257–293.

[W] N. Weaver, Sub-Riemannian metrics for quantum Heisenberg manifold, J. Operator Theory,43, no. 2 (2000), 93–100.

[Wn] A. Weinstein, The local structure of Poisson manifolds, J. Diff. Geom. 18 (1983), 523–557.[WuY] T. T. Wu, C. N. Yang, Concept of nonintegrable phase factors and global formulation of

gauge fields, Phys. Rev. D12, 3845 (1975)[YM] C. N. Yang, R. L. Mills, Conservation of isotopic spin and isotopic gauge invariance, Phys.

Rev. (2), 96 (1954), 191–195.

Department of Mathematics, University of Colorado, Boulder, CO 80303, USA,

School of Mathematics and Applied Statistics, University of Wollongong, NSW 2522,

Australia.

E-mail address: sooran@colorado.edu

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Proceedings of Symposia in Pure Mathematics

From Rational Homotopy to K-Theoryfor Continuous Trace Algebras

John R. Klein, Claude L. Schochet, and Samuel B. Smith

Abstract. The K-theory of continuous trace C∗-algebras has been known for

some years. We recently computed the rational homotopy groups of the unitalcontinuous trace C∗-algebras. In this note we discuss the relationship between

these results and, in particular, the additional insight gained by regarding thethe natural map

π∗(UA)⊗ Q −→ K∗+1(A)⊗ Q

as a Z+-graded map.

1. Introduction

Let A be a unital C∗-algebra. Its unitary group, UA, contains a wealth oftopological information about A. However, complete knowledge of the homotopygroups of UA is out of reach even in the simplest cases. For example, if A = M2(C)then UA S1 × S3 as topological spaces. The groups πj(S

3) are known for onlythe first hundred or so values of j.

There are two simplifications which have been considered. The first, well-traveled road, is to pass to π∗(U(A⊗K)) which is isomorphic (with a degree shift)to K∗(A). This approach has led to spectacular success in many arenas.

A different approach is to consider π∗(UA)⊗Q, the rational homotopy of UA.In joint work with G. Lupton and N. C. Phillips we have calculated this functor forthe cases

(1) A = C(X)⊗Mn(C) and(2) A a unital continuous trace C∗-algebra.

In this note we state our principal results and look at some concrete examplesof this calculation. We are particularly interested in the interplay between theseresults and the K-theory results, and so we focus upon the Z+-graded map

π∗(UA)⊗Q −→ K∗+1(A)⊗Q.

2000 Mathematics Subject Classification. 46J05, 46L85, 55P62, 54C35, 55P15, 55P45.Key words and phrases. continuous trace C∗-algebra, space of sections, unitary group of

C∗-algebra, rational H-space, topological group, localization.

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2 KLEIN, SCHOCHET, AND SMITH

2. Statement of the Main Theorem

Let Mn = Mn(C) be the complex matrices, Un its group of unitaries, and letPUn be the quotient of Un by its center. Let ζ :T → X be a principal PUn-bundleover a compact space X, let PUn act on Mn by conjugation and let

T ×PUnMn → X

be the associated n-dimensional complex matrix bundle. Define Aζ to be the set ofcontinuous sections of the latter. These sections have natural pointwise addition,multiplication, and ∗-operations and give Aζ the structure of a unital C∗-algebra.The algebra Aζ is the most general unital continuous trace C∗-algebra. Let UAζ

denote the topological group of unitaries of Aζ . We have determined the rationalhomotopy type of UAζ .

To state our calculation of the rational homotopy groups of UAζ , we introducesome notation. Given Z-graded vector spaces V andW , we grade the tensor productV ⊗ W by declaring that v ⊗ w has grading |v| + |w|. Let V ⊗W be the effect ofconsidering only tensors with non-negative grading.

Given elements x1, . . . , xn, each of homogeneous degree, write 〈x1, . . . , xn〉 forthe graded vector space with basis x1, . . . , xn. Given a topological group G, writeG for the path component of the identity in G.

The following is the principal result of our recent paper [4].

Theorem A. [4] Let ζ be a principal PUn bundle over a compact metric spaceX. Let Aζ be the associated continuous trace C∗-algebra, and let UAζ its groupof unitaries. Then the rationalization of (UAζ) is H-equivalent to a product ofrational Eilenberg-Mac Lane spaces with the standard multiplication, with degreesand dimensions corresponding to an isomorphism of graded vector spaces

π∗ ((UAζ))⊗Q ∼= H∗(X;Q) ⊗ 〈s1, s3, . . . , s2n−1〉 .

(In the above, H∗(X;Q) denotes the Cech cohomology of X with rational coeffi-cients1 and we follow the convention that cohomology is graded in degrees ≤ 0.The basis element s2j−1 has degree 2j − 1.)

3. Stabilization and the first example: A = Mn

We wish to explore the relationship between this result and the classical resultson the K-theory of continuous trace C∗-algebras. Recall first that homotopy andK-theory are related by stabilization. K-theory for unital C∗-algebras is definedby taking K0(A) to be the Grothendieck group of finitely generated projective leftA-modules. For j > 0, write U∞A = lim−→UnA with the weak topology and thendefine K-theory for j > 0 by

Kj(A) = πj−1(U∞A).

Thus there is a natural stabilization map

πj(UA) −→ πj(U∞A) ∼= Kj+1(A)

1Note that Cech and singular cohomology theories agree on finite complexes but not on allcompact metric spaces. This was well-known in the 1940’s (Spanier mentions it in passing ina 1948 paper). The strong wedge of a countable number of copies of S2 provides a dramaticexample; see Barratt and Milnor. [2]

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FROM RATIONAL HOMOTOPY TO K-THEORY 3

which we denote

σintegral : π∗(UA) −→ K∗+1(A).

At this point the functor Kj(A) is defined for j ≥ 0. Bott periodicity implies that

K∗(A) ∼= K∗+2(A)

for ∗ > 0. So it is natural to regardK∗(A) as a Z/2-graded theory (and we confess tohaving done so in the past.) However, for our present purpose it will be vital NOTto do so. That is, in this note we regard K-theory as defined for all non-negativeintegers. This allows us, for instance to distinguish between Morita-equivalent C∗-algebras.2

Ideally we would like to compute the image of σintegral. However, this is prob-ably as difficult as computing πj(UA), which is out of reach, as noted. Hence wehave been focusing on the rationalization of this group which, while far simpler andhence carrying less information, has the advantage of being computable. Thus wefocus on the rational stabilization map

σ : π∗(UA)⊗Q −→ K∗+1(A)⊗Q.

The first case to consider is A = Mn. Then we are looking at

σ : π∗(Un)⊗Q −→ K∗+1(Mn)⊗Q.

These are graded rational vector spaces which are abstractly isomorphic (afterdegree shift, of course) in degrees 1, . . . , 2n. Now

π∗(Un)⊗Q ∼= 〈s1, s3, . . . , s2n−1〉

with |s2j−1| = 2j − 1 and of course σ is linear. So σ is determined by its image onthe basis vectors, and indeed σ(s2j−1) = 0 for each j. So there is an isomorphism

σ : π2j−1(Un)⊗Q∼=−→ K2j(Mn)⊗Q

for j = 1, . . . n and hence the range of σ is exactlyK2(Mn)⊗Q, K4(Mn)⊗Q, . . . , K2n(Mn)⊗Q

.

In particular, if we regard the range of σ as an invariant of A with valuesin Z+-graded K-theory, then this invariant distinguishes between Mn and Mk forn = k.

Note that we are NOT saying that the classes σ(s2j−1) are actually in K∗(Mn).Consider the diagram

π∗(Un) −−−−→ π∗(Un)⊗Qσ−−−−→ K∗+1(Mn)⊗Q

h′ h

H∗(Un;Z) −−−−→ H∗(Un;Q)

2J. F. Adams [1] does the same thing for a somewhat different reason. Having proved thatthe Adams operations do not commute with the Bott periodicity maps in Corollary 5.3, he noteson page 619: “Owing to the state of affairs revealed by Corollary 5.3, we shall be most careful not

to identify K−n−2C

(X,Y ) with K−nC

(X,Y ). . .We therefore regard K−nC

(X,Y ) as graded over Z,

not over Z/2 . . . ”

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4 KLEIN, SCHOCHET, AND SMITH

where the vertical maps are the Hurewicz maps. Filling in three well-understoodgroups, this diagram becomes

π∗(Un) −−−−→ 〈s1, s3, . . . , s2n−1〉 σ−−−−→ K∗+1(Mn)⊗Q

h′ h

ΛZ(g1, g3, . . . g2n−1) −−−−→ ΛQ(g1, g3, . . . g2n−1)

where ΛR denotes the exterior algebra over the ring R with given generators andh(sj) = gj for each j. The map h′ is not onto the generators.

4. Second Example: X = S3

In order to make Theorem A concrete, focus upon the special case where X =S3 and contrast our results with those of Jonathan Rosenberg [5].

Theorem A asserts that the space of unitaries UAζ is rationally equivalent tothe space of functions F (S3, Un). Actually, though, we know much more in thiscase. Any finite-dimensional PU(n)-bundle ζ has Dixmier-Douady invariant offinite order, and since H3(S3;Z) ∼= Z this implies that the invariant vanishes forζ over S3. Thus ζ ∼= End(V ) for some complex vector bundle V over S3. Thisbundle must also be trivial, since it is classified by an element of π3(BUn) = 0.Thus ζ must be a trivial bundle, and so UAζ is homeomorphic to F (S3, Un) evenbefore rationalization. So let us examine this space carefully.

Note first that its path components are interesting. Fix a base point x0 for S3.There is a standard fibration

F•(S3, Un) −→ F (S3, Un)

p−→ Un

where p(f) = f(xo) and F• denotes base point preserving maps. This fibration hasa section (send a point u ∈ Un to the constant map S3 → Un that takes everyelement of S3 to u) and hence there are split short exact sequences in each degree

0 → π∗(F•(S3, Un)) −→ π∗(F (S3, Un))

p∗−→ π∗(Un) → 0.

In particular, since Un is connected,

π0(F (S3, Un) ) ∼= π0(F•(S3, Un)) ∼= π3(Un) ∼= Z.

The generator of π3(Un) ∼= Z is given by the natural composition

S3 ∼= SU2 → U2 → Un

where U2 is included in Un via u → u⊕ 1.In higher degrees we obtain for each k the split short exact sequence

0 → πk(F•(S3, Un)) −→ πk(F (S3, Un))

p∗−→ πk(Un) → 0.

This helps us understand the result for Theorem A, which states (in this case) that

π∗(F (S3, Un) )⊗Q ∼= H∗(S3;Q)⊗ 〈s1, s3, . . . , s2n−1〉.Write

H∗(S3;Q) = 〈 1, x3 〉with x3 denoting the generator in dimension 3. Then π∗(F (S3, Un))⊗Q is spannedby two types of classes. There are the classes 1 ⊗ s2j−1 of degree 2j − 1 and theclasses x3 ⊗ s2j−1 of degree 2j − 1− 3 = 2j − 4. The short exact sequence

0 → π∗(F•(S3, Un))⊗Q −→ π∗(F (S3, Un))⊗Q

p∗−→ π∗(Un)⊗Q → 0

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FROM RATIONAL HOMOTOPY TO K-THEORY 5

becomes

0 → 〈x3⊗s3, . . . x3⊗s2n−1〉 −→ π∗(F (S3, Un))⊗Q −→ 〈1⊗s1, . . . 1⊗s2n−1〉 → 0

Note that the class x3 ⊗ s1 is not present since it would have negative degree.Now, what happens when we map to K-theory? First, the fact that UAζ is

homeomorphic to F (S3, Un) implies that

K∗(Aζ) ∼= K∗(C(S3)) ∼= K∗(S3) ∼= Z

in every degree. The generator in even degree is simply the class of the trivial linebundle (i.e. the one dimensional trivial projection) and the generator in odd degreecorresponds to the Bott generator in that degree. An easy naturality argumentusing the result of the previous section implies that the class 1⊗ s2j−1 maps to theclass in K2j(Aζ)⊗Q that corresponds to a multiple of the one-dimensional trivialprojection in K0(Aζ) under the Bott map as is the case for A = Mn.

The other classes are more interesting. The class x3⊗s2j−1 has degree 2j−4 andhence must map to K2j−3(Aζ). The first example is x3⊗s3 mapping to K1(Aζ)⊗Q

and the last is x3 ⊗ s2n−1 mapping to K2n−3(Aζ)⊗Q.To summarize: if X = S3 then, independent of the (finite-dimensional) bundle

ζ, the image of the stabilization map

σ : π∗(UAζ)⊗Q −→ K∗+1(Aζ)⊗Q

has basis elements

σ(1⊗ s2j−1) ∈ K2j(Aζ)⊗Q j = 1, . . . , n

andσ(x3 ⊗ s2j−1) ∈ K2j−3(Aζ)⊗Q j = 3, . . . n.

Thus the image of the stabilization map σ consists of the groupsK0(Aζ)⊗Q, K1(Aζ)⊗Q, . . . , K2n−3(Aζ)⊗Q, K2n−2(Aζ)⊗Q, K2n(Aζ)⊗Q

and no others.Turning to the infinite-dimensional situation, the first thing to note is that Aζ

is no longer unital; in fact it is stable. We add a unit to Aζ in the canonical fashionto obtain a short exact sequence

0 → Aζ −→ A+ζ −→ C → 0

and then defineUAζ = Ker

[U(A+

ζ ) −→ UC = S1].

Stability implies thatKn(A) ∼= πn−1(UAζ).

and we see that the homotopy groups are periodic.If the Dixmier-Douady invariant is trivial then Aζ

∼= C(S3,K) and so UAζ∼=

F (S3, UK). Thus we see directly that

πn(UAζ) ∼= πn(F (S3, UK))

and an easy homotopy analysis shows this group to be Z for each n. This corre-sponds to K0(S3) = K1(S3) ∼= Z.

If the Dixmier-Douady class is s times the generator of H3(S3;Z) then Rosen-berg [5, 6] gives a spectral sequence argument to conclude that K0(Aζ) = 0 andK1(Aζ) ∼= Z/s. In particular, if the Dixmier-Douady class is a generator then

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6 KLEIN, SCHOCHET, AND SMITH

K1(Aζ) = 0 and so Aζ is K-contractible. Thus if the Dixmier-Douady class of ζis a generator then π∗(UAζ) = 0, and for any non-trivial Dixmier-Douady class wehave π∗(Aζ)⊗Q = 0. We hope to study Rosenberg’s result from the perspective ofrational homotopy in subsequent work.

5. If One Grading is Good, then Two Gradings...

We note that Theorem A gives a natural bigrading to π∗(UAζ) ⊗ Q. In theX = S3 example considered above, the classes 1 ⊗ s2j−1 have bidegree (0, 2j − 1)and total degree 2j − 1, and the classes x3 ⊗ s2j−1 have bidegree (−3, 2j − 1) andtotal degree −3 + (2j − 1) = 2j − 4. The bigrading has quite a bit of naturalityassociated with it. In the simplest case, with A = C(X) ⊗Mn it is keeping trackboth of the cohomological degree and of the size of the matrix!

This bidegree is of course completely lost when passing to Z/2-gradedK-theory.For an elementary example, consider the case X = CP 2. The rational (indeed, inte-gral in this case) cohomology ring is a truncated polynomial algebra on a generatorc ∈ H2(CP 2) with c3 = 0. Take A = F (CP 2,M3). Then the classes c ⊗ s3 andc2 ⊗ s5 have different bidegrees, and hence are distinguished, but they have thesame total degree and hence have the same degree when one passes to K-theory,even when K-theory is Z+-graded!

Now, take A = C(X) ⊗ Mn. One might reasonably ask for a calculation of[A,A], the homotopy classes of unital ∗-homomorphisms [A,A]. One source of suchmaps are the induced maps from functions f : X → X and so one might hope todetermine [A,A] as some functor of [X,X]. (Determining [X,X] itself is extremelydifficult even in fairly simple cases.) Another source of maps are the induced mapsfrom ∗-homomorphisms Mn → Mn. These are known, of course. Non-trivial mapsmust be isomorphisms, every isomorphism is inner, and hence every such map isgiven by conjugation by a unitary, so we are back to PUn. The real problem is thatthere are other maps besides these two types that intertwine the two.

There is a natural commuting diagram

[A,A]φK−−−−→ EndZ/2(K

∗(X))φ

End∗∗

(π∗(UA)⊗Q

)−−−−→ EndZ/2(K

∗(X)⊗Q).

It might seem at first glance that one would be better off using φK as an invariantrather than φ. This is illusory. The problem is that the map φK is only definedif we understand EndZ/2(K

∗(X)) to mean endomorphisms of Z/2-graded abeliangroups since if A is non-commutative then there is no ring structure on K∗(A).

We can say something about the map φ. If the map h : A → A arises froma map f : X → X then of course φ(h) = f∗ ⊗ 1. If h arises from conjugation bya unitary then φ(h) is just the identity, since Un is path-connected. We hope tocompute φ(h) in some intertwined cases and expect it to be a helpful invariant.This is work in progress.

References

[1] J. F. Adams, Vector fields on spheres, Annals of Math. 75 (1962), 603 - 632.[2] M. G. Barratt and J. Milnor, An example of anomolous singular homology, Proc. Amer.

Math. Soc. 13 (1962), 293–297.

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FROM RATIONAL HOMOTOPY TO K-THEORY 7

[3] G. Lupton, N. C. Phillips, C. L. Schochet, S. B. Smith, Banach algebras and rational homo-topy theory, Trans. Amer. Math. Soc. 361 (2009), 267–295.

[4] J. R. Klein, C. L. Schochet, and S. B. Smith, Continuous trace C∗-algebras, gauge groupsand rationalization, J. Topology and Analysis 1 (2009), 261-288.

[5] J. Rosenberg, Homological invariants of extensions of C∗-algebras, Operator algebras andapplications, Part 1 (Kingston, Ont., 1980), pp. 35–75, Proc. Sympos. Pure Math., 38, Amer.Math. Soc., Providence, RI, 1982.

[6] J. Rosenberg, Continuous - trace algebras from the bundle - theoretic point of view , J. Austral.Math. Soc. (Series A) 47 (1989), 368-381.

[7] C. Schochet, Topological methods for C∗-algebras. II. Geometric resolutions and the Kunnethformula, Pacific J. Math. 98 (1982), no. 2, 443–458.

Department of Mathematics, Wayne State University, Detroit MI 48202

E-mail address: klein@math.wayne.edu

Department of Mathematics, Wayne State University, Detroit MI 48202

E-mail address: claude@math.wayne.edu

Department of Mathematics, Saint Joseph’s University, Philadelphia PA 19131

E-mail address: smith@sju.edu

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Proceedings of Symposia in Pure Mathematics

Distances between Matrix Algebrasthat Converge to Coadjoint Orbits

Marc A. Rieffel

Abstract. For any sequence of matrix algebras that converges to a coadjoint

orbit we give explicit formulas that show that the distances between the matrixalgebras (viewed as quantum metric spaces) converges to 0. In the process wedevelop a general point of view that is likely to be useful in other relatedsettings.

Introduction

In earlier papers [6, 7, 9] I provided ways to give a precise meaning to state-ments in the literature of high-energy physics and string theory of the kind “Ma-trix algebras converge to the sphere”. I did this by equipping the matrix algebraswith suitable “Lipschitz seminorms” that make the matrix algebras into “compactquantum metric spaces”, and then by defining convergence by means of a suitable“quantum Gromov-Hausdorff distance” between quantum metric spaces. By nowa number of variations on this approach have been studied [1, 2, 3, 4, 5, 10].

When I then began to examine what consequences the convergence of quantummetric spaces had for the convergence of “vector bundles” (i.e. projective modules)over them [8], I found that it is very important that the Lipschitz seminormssatisfy a suitable Leibniz property. In [9] I showed that a very convenient sourcefor seminorms that satisfy this Leibniz property consisted of normed bimodules,and in [9] I also constructed explicit normed bimodules that worked well for matrixalgebras converging to coadjoint orbits.

However, for our approach to work well, it should be the case that for a conver-gent sequence of matrix algebras the quantum Gromov-Hausdorff distances betweenthe matrix algebras go to 0; but when I required that all of the seminorms satisfythe Leibniz property I did not see at first how to show this convergence directly.The purpose of the present paper is to give explicit normed bimodules and corre-sponding Leibniz Lipschitz seminorms that demonstrate this convergence to 0. In

2000 Mathematics Subject Classification. Primary 46L87; Secondary 53C23, 58B34, 81R15,

81R30.Key words and phrases. quantum metric space, Gromov–Hausdorff distance, Leibniz semi-

norm, coadjoint orbits, matrix algebras, coherent states, Berezin symbols.The research reported here was supported in part by National Science Foundation grant

DMS-0753228.

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2 MARC A. RIEFFEL

the process we develop a general point of view that is likely to be useful in otherrelated situations. This point of view is motivated by the “nuclear distance” in-troduced and studied by Hanfeng Li [2, 4, 5], in which all of the bimodules arerequired to be C∗-algebras. I have so far not seen how to apply Hanfeng Li’s ap-proach directly to obtain explicit normed bimodules for the matrix-algebra case.This is because Li’s nuclear distance requires implicitly that it always be the iden-tity element of the C∗-algebra that is used to define the needed inner derivation,and I have not seen how to successfully arrange this for the situation discussed inthis paper. But by trying just to arrange at least that all of the normed bimodulesthat I used be C∗-algebras, I was led to see the path to the explicit bimodules thatI sought.

The first section of this paper recalls the setting for matrix algebras converg-ing to coadjoint orbits, reformulates the bimodules from [9] so that they are C∗-algebras, and then uses these reformulated bimodules to construct candidates forC∗-bimodules between matrix algebras whose Leibniz Lipschitz seminorms mightshow that the distances go to 0. In Section 2 we place matters in a general frame-work, and obtain a basic theorem in this general framework. In Section 3 we provethat the candidate bimodules and corresponding Lipschitz seminorms of Section 1do indeed show that the distances between the converging matrix algebras go to 0.An important step in the proof comes from the general theorem in Section 2. Thefull statement of the main theorem is given at the end of Section 3.

1. The Bimodules

We recall the setting from [7, 9]. We let G be a compact connected semisimpleLie group, and we let g denotes the complexification of the Lie algebra of G. Wechoose a maximal torus in G, with corresponding Cartan subalgebra of g, its setof roots, and a choice of positive roots. We fix a specific irreducible unitary repre-sentation, (U,H), of G, and we choose a highest-weight vector, ξ, for (U,H) with‖ξ‖ = 1. For any n ∈ Z≥1 we set ξn = ξ⊗n in H⊗n, and we let (Un,Hn) be therestriction of U⊗n to the U⊗n-invariant subspace, Hn, of H⊗n that is generated byξn. Then (Un,Hn) is an irreducible representation of G with highest-weight vectorξn, and its highest weight is just n times the highest weight of (U,H). We denotethe dimension of Hn by dn.

We let Bn = L(Hn). The action of G on Bn by conjugation by Un will bedenoted simply by α. We assume that a continuous length function, , has beenchosen for G, and we denote the corresponding C∗-metric on Bn by LB

n . It isdefined by

LBn (T ) = sup‖αx(T )− T‖/(x) : x /∈ eG

for T ∈ Bn. (The term “C∗-metric” is defined in definition 4.1 of [9].) We let Pn

denote the rank-one projection along ξn. Then the α-stability subgroup, H, forP = P 1 will also be the α-stability subgroup for each Pn. We let A = C(G/H),and we let LA be the C∗-metric on A for and the left-translation action of G onG/H, defined much as is LB

n .Roughly speaking, our goal is to obtain estimates on the distance between

(Bm, LBm) and (Bn, LB

n ) that show that the distance goes to 0 as m and n go to∞. We want to do this in the setting of [9], where we insist that the Lipschitzseminorms involved satisfy a strong Leibniz property. We require this because of

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DISTANCES BETWEEN MATRIX ALGEBRAS 3

its importance for treating vector bundles (and projective modules), as shown in[8].

But in contrast to [9], our presentation here is influenced by Hanfeng Li’sdefinition of the “nuclear distance” between quantum metric spaces, although Ihave not seen how to use his nuclear distance directly. The effect of this influenceis that we try to arrange that all of the bimodules that we consider are actuallyC∗-algebras.

To motivate the construction of our bimodules, we first reformulate the corre-sponding constructions from [9] in terms of C∗-algebras. For any given n we formthe C∗-algebra A ⊗ Bn = C(G/H,Bn). There are canonical injections of A andBn into A⊗Bn, and by means of these we view A⊗Bn as an A-Bn-bimodule. Letωn ∈ C(G/H,Bn) be defined by

ωn(x) = αx(Pn).

We use the distinguished element ωn and the bimodule structure to define a semi-norm, Nn, on A⊕Bn by

Nn(f, T ) = ‖fωn − ωnT‖.This seminorm is easily seen to be the same as the seminorm Nσ described by othermeans in proposition 7.2 of [9]. It is also easy to see that Nn satisfies the strongLeibniz property defined in definition 1.1 of [9], for the reasons discussed in example2.3 of [9] if A⊗Bn is viewed as an (A⊗Bn)-bimodule in the evident way.

For a suitable choice of the constant γ, as discussed in propositions 8.1 and 8.2of [9], γ−1Nn is a bridge, as defined in definition 5.1 of [6]. This implies that the∗-seminorm Ln on A⊕Bn defined by

Ln(f, T ) = LA(f) ∨ LBn (T ) ∨ γ−1(Nn(f, T ) ∨Nn(f , T

∗))

is a C∗-metric on A ⊕ Bn (where ∨ means “maximum of”) that has the furtherproperty that its quotients on A and Bn agree with LA and LB

n on self-adjointelements. (See notation 5.5 and definition 6.1 of [9].) This quotient condition onseminorms is exactly what we required in [6, 7, 9] in order to define distancesbetween C∗-algebras such as A and Bn. Specifically, for our situation, let S(A)denote the state space of A, and similarly for Bn and A ⊕ Bn. Then S(A) andS(Bn) are naturally viewed as subsets of S(A ⊕ Bn). Now Ln defines a metric,ρLn

, on S(A⊕Bn) by

ρLn(μ, ν) = sup|μ(f, T )− ν(f, T )| : Ln(f, T ) ≤ 1.

(By definition this supremum should be taken over just self-adjoint f and T , butby the comments made just before definition 2.1 of [6] it can equivalently be takenover all f and T because Ln is a ∗-seminorm. This fact is also used later for other∗-seminorms.) The corresponding ordinary Hausdorff distance

distρLn

H (S(A), S(Bn))

gives, by definition, an upper bound for distq(A,Bn) as defined in definition 4.2of [6] when we don’t require the strong Leibniz condition, and for prox(A,Bn) asdefined in definition 5.6 of [9] when we do require the strong Leibniz condition. Itis shown in theorem 4.3 of [6] that distq satisfies the triangle inequality. But proxprobably does not satisfy the triangle inequality, basically because the quotient of aseminorm that satisfies the Leibniz condition need not satisfy the Leibniz condition.

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4 MARC A. RIEFFEL

We always have distq(A,B) ≤ prox(A,B), so if we can show that prox(A,B) is“small” than it follows that distq(A,B) is “small” too.

Since for our specific situation prox(A,Bn) converges to 0 as n goes to ∞, asseen in theorem 9.1 of [9], (and similarly for its matricial version, proxs, by theorem14.1 of [9]), it is natural to expect that prox(Bm, Bn) converges to 0 as m and ngo to ∞. But because we can not invoke the triangle inequality, we need to givea direct proof of this fact. In the process of doing this we will construct a specificseminorm that gives quantitative estimates.

Towards our goal we seek to construct a suitable Bm-Bn-bimodule. We can,of course, view the C∗-algebra A⊗Bm as being the Bm-A-bimodule Bm ⊗A, andthen it is natural to form an “amalgamation” over A of these two C∗-algebras, toobtain the Bm-Bn-bimodule

(Bm ⊗A)⊗A (A⊗Bn) = Bm ⊗A⊗Bn,

which we can view as C(G/H,Bm ⊗ Bn). Notice that this is again a C∗-algebra,and that we have natural injections of Bm and Bn into it. Inside this bimodule wechoose a distinguished element, namely ωmn = ωm ⊗ ωn, viewed as defined by

ωmn(x) = αx(Pm)⊗ αx(P

n) = αx(Pm ⊗ Pn).

In terms of ωmn we define a seminorm, Nmn, on Bm ⊕Bn by

Nmn(S, T ) = ‖Sωmn − ωmnT‖,where the norm is that of the C∗-algebra C(G/H,Bm ⊗ Bn). We can now hopeto find constants γ such that γ−1Nmn is a bridge between Bm and Bn. In thenext section we describe a more general setting within which to choose such bridgeconstants.

2. The Bridge Constants

In this section we consider the following more general setting. We are giventhree compact C∗-metric spaces, (A,LA), (B,LB) and (C,LC). We are also givenunital C∗-algebras D and E together with injective unital homomorphisms of Aand B into D, and of B and C into E. (Actually, we do not need the unitalhomomorphisms to be injective, but then we should provide notation for them,and that would clutter our calculations.) Thus we can consider D to be an A-B-bimodule and E to be a B-C-bimodule. We assume further that we are givendistinguished elements d0 and e0 of D and E respectively. For convenience weassume that ‖d0‖ = 1 = ‖e0‖. We then define seminorms ND and NE on A ⊕ Band B ⊕ C by

ND(a, b) = ‖ad0 − d0b‖Dand similarly for NE . We assume that there are constants γD and γE such thatγ−1D ND and γ−1

E NE are bridges for (LA, LB) and (LB, LC) respectively. This meansthat when we form the ∗-seminorm

LAB(a, b) = LA(a) ∨ LB(b) ∨ γ−1D (ND(a, b) ∨ND(a∗, b∗),

its quotients on A and B agree with LA and LB on self-adjoint elements, andsimilarly for LBC . Note that LAB and LBC are C∗-metrics by theorem 6.2 of [9].

Motivated by Hanfeng Li’s treatment of his nuclear distance [5], we considerany amalgamation, F , of D and E over B. This means that there are unitalinjections of D and E into F whose compositions with the injections of B into D

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DISTANCES BETWEEN MATRIX ALGEBRAS 5

and E coincide. We denote the images of d0 and e0 in F again by d0 and e0, andwe set f0 = d0e0. Unfortunately in this generality it could happen that f0 = 0. (InHanfeng Li’s definition of his nuclear distance this problem does not occur since hisdistinguished elements are, implicitly, the identity elements.)

Theorem 2.1. Let notation be as above, and assume that f0 = 0. View F asan A-C-bimodule in the evident way, and define a seminorm, NF , on A⊕ C by

NF (a, c) = ‖af0 − f0c‖F .Then for any γ ≥ γD + γE the seminorm γ−1NF is a bridge for (LA, LC).

Proof. It is clear that γ−1NF (1A, 0C) = 0 since f0 = 0, and that γ−1NF isnorm-continuous. Thus the first two conditions of definition 5.1 of [6] are satisfied.We must verify the third, final, condition. To simplify notation, we identify A, B,C, D and E with their images in F . For any a ∈ A, b ∈ B and c ∈ C we have

NF (a, c) = ‖af0 − f0c‖F ≤ ‖ad0e0 − d0be0‖F + ‖d0be0 − d0e0c‖F≤ ‖ad0 − d0b‖D‖e0‖E + ‖d0‖D‖be0 − e0c‖E= ND(a, b) +NE(b, c).

Now let a ∈ A with a = a∗ be given, and let ε > 0 be given. Since γ−1D ND is a

bridge for (LA, LB), there is by definition a b ∈ B with b∗ = b such that

LB(b) ∨ γ−1D ND(a, b) ≤ LA(a) + ε.

Then since γ−1E NE is a bridge for (LB, LC), there is a c ∈ C with c∗ = c such that

LC(c) ∨ γ−1D ND(b, c) ≤ LB(b) + ε.

Consequently

LC(c) ≤ LB(b) + ε ≤ LA(a) + 2ε,

and, from the earlier calculation,

NF (a, c) ≤ ND(a, b) +NE(b, c) ≤ γD(LA(a) + ε) + γE(LB(b) + ε)

≤ (γD + γE)LA(a) + ε(γD + 2γE).

The situation is basically symmetric between A and C, so one can make a similarcalculation but starting with a c ∈ C to obtain a b ∈ B and then an a ∈ A satisfyingthe corresponding inequalities. This shows that (γD + γE)

−1NF is indeed a bridge.Then also γ−1NF will be a bridge for any γ ≥ γD + γE .

However, I have so far not seen any good general conditions that yield estimatesshowing that if the corresponding seminorm

LAB = LA ∨ LB ∨ γ−1(ND ∨N∗D)

brings (A,LA) and (B,LB) close together, and similarly for LBC , then LAC us-ing (γD + γE)

−1NF brings (A,LA) and (C,LC) close together, in the sense that

distρLAC

H (S(A), S(C)) is small. In Hanfeng Li’s nuclear distance, in which the dis-tinguished elements are all, implicitly, the identity elements, this aspect works muchbetter. And since the nuclear distance satisfies the triangle inequality, it is clearthat distnu(B

m, Bn) converges to 0 as m and n go to ∞. But so far I find thenuclear distance to be more elusive, as I discuss briefly in section 6 of [9], though itis certainly attractive. I do not yet see how to obtain for the nuclear distance thekind of quantitative estimates that we will obtain here for prox.

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6 MARC A. RIEFFEL

3. The Proof and Statement of the Main Theorem

For the context of Section 1 the role of F of Section 2 is played by C(G/H,Bm⊗Bn), while the roles of d0 and e0 are played by ωm and ωn, with f0 being ωmn. LetγAm be defined as in proposition 8.1 of [9] but for P = Pm, and let γB

m be defined asin proposition 8.2 of [9] but for P = Pm. Let γm = maxγA

m, γBm. All that we need

to know here about γm is that propositions 8.1 and 8.2 of [9] tell us that, for Nm asdefined in Section 1 above, γ−1

m Nm is a bridge for (LA, LBm), and that propositions

10.1 and 12.1 of [9] tell us that γm converges to 0 as m goes to ∞. From Theorem2.1 above and from the identifications made above, it follows immediately that forany γ with γ ≥ γm + γn the seminorm γ−1Nmn is a bridge for (LB

m, LBn ).

We now investigate how close S(Bm) and S(Bn) are in the metric from thecorresponding seminorm Lmn on Bm⊕Bn. Given μ ∈ S(Bm), we want a systematicway to find a ν ∈ S(Bn) that is “relatively close” to μ. For this purpose we usethe Berezin symbols σn and σn that we used in [7, 9]. We recall that σn is thecompletely positive unital map from Bn to A defined by σn

T (x) = tr(αx(Pn)T ),

while σn is the completely positive unital map from A to Bn defined by

σnf = dn

∫G/H

f(x)αx(Pn)dx,

where we recall that dn is the dimension of Hn, and the G-invariant measure onG/H gives G/H measure 1. Then σm σn will be a completely positive unital mapfrom Bn to Bm, whose transpose will map S(Bm) into S(Bn), for any m and n.For any T ∈ Bn we have

σm(σnT ) = dm

∫G/H

αx(Pm) tr(αx(P

n)T )dx.

Let Nmn be the seminorm on Bm ⊕Bn determined by ωmn, so that

Nmn(S, T ) = ‖Sωmn − ωmnT‖= sup‖(S ⊗ In)αx(P

m ⊗ Pn)− αx(Pm ⊗ Pn)(Im ⊗ T )‖ : x ∈ G/H.

Then Lmn is defined on Bm ⊕Bn by

Lmn(S, T ) = LBm(S) ∨ LB

n (T ) ∨ γ−1(Nmn(S, T ) ∨Nmn(S∗, T ∗))

for some γ ≥ γm+γn. Let μ ∈ S(Bm) be given, and as state ν ∈ S(Bn) potentiallyclose to μ we choose ν to be defined by ν(T ) = μ(σm(σn

T )). We then want an upperbound on ρLmn

(μ, ν). Now

ρLmn(μ, ν) = sup|μ(S)− ν(T )| : Lmn(S, T ) ≤ 1,

and|μ(S)− ν(T )| = |μ(S)− μ(σm(σn

T ))| ≤ ||S − σm(σnT )||.

So we need to understand what the condition Lmn(S, T ) ≤ 1 implies for ||S −σm(σn

T )||. This seems difficult to do directly, so we use a little gambit that we haveused before, e.g. shortly before notation 8.4 of [9], namely

||S − σm(σnT )|| ≤ ||S − σm(σm

S )||+ ||σm(σmS )− σm(σn

T )||≤ δBmLB

m(S) + ‖σmS − σn

T ‖∞,

where for the last inequality we have used theorem 11.5 of [9], which includes thedefinition of δBm. (We remark that theorem 11.5 of [9] is the same as theorem 6.1of [7], but [9] gives a simpler proof of this theorem.) Note that Lmn(S, T ) ≤ 1

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implies that LBm(S) ≤ 1. Thus we see that it is ‖σm

S − σnT ‖∞ that we need to

control. In preparation for this we establish some additional notation in order toput the situation into a comfortable setting. Notice that Bm⊗Bn = L(Hm⊗Hn).Furthermore ξm ⊗ ξn is a highest-weight vector in (Um ⊗ Un,Hm ⊗ Hn), and itsweight is just the sum of the highest weights of (Um ⊗Hm) and (Un ⊗Hn), whichis just the highest weight of (U,H) multiplied by m+ n. Thus ξm ⊗ ξn is just thehighest-weight vector for a copy of (Um+n,Hm+n) inside Hm ⊗ Hn. To simplifynotation we now just set ξm+n = ξm⊗ξn, and view Hm+n as being the G-invariantsubspace of Hm ⊗ Hn generated by ξm+n. Then the rank-1 projection Pm+n onξm+n is exactly Pm ⊗ Pn. We let Πmn denote the projection from Hm ⊗ Hn

onto Hm+n. Our notation will not distinguish between viewing the domain ofPm+n as being Hm ⊗ Hn or as being Hm+n, and we will use below the fact thatαx(P

m+n) = αx(Pm+n)Πmn for any x ∈ G.

Lemma 3.1. For any S ∈ Bm and T ∈ Bn we have

σmS − σn

T = σm+nR

where R = Πmn(S ⊗ In − Im ⊗ T )Πmn, viewed as an element of Bm+n.

Proof. For any x ∈ G we have

σmS (x)− σn

T (x) = trm(αx(Pm)S)− trn(Tαx(P

n))

= (trm ⊗ trn)(αx(Pm ⊗ Pn)(S ⊗ In − Im ⊗ T )αx(P

m ⊗ Pn))

= trm+n(αx(Pm+n)Πmn(S ⊗ In − Im ⊗ T )Πmn)

= σm+nR (x).

Notice now that for R defined as just above, because the rank of Pm+n is 1,we have for any x ∈ G

|σm+nR (x)| = | trm+n(αx(P

m+n)Πmn(S ⊗ In − Im ⊗ T )Πmn)|= ‖αx(P

m+n)(S ⊗ In − Im ⊗ T )αx(Pm+n)‖

≤ ‖αx(Pm+n)(S ⊗ In)− (Im ⊗ T )αx(P

m+n)‖,and consequently

‖σm+nR ‖ ≤ Nmn(S

∗, T ∗).

But if Lmn(S, T ) ≤ 1, then Nmn(S∗, T ∗) ≤ γm + γn if we have taken γ = γm + γn.

Thus we find that

|μ(S)− ν(T )| ≤ δBm + γm + γn.

Since the situation is symmetric in m and n, we conclude that

distρLmn

H (S(Bm), S(Bn)) ≤ maxδBm, δBn +maxγAm, γB

m+maxγAn , γ

Bn .

As mentioned in part above, it is shown in proposition 10.1, theorem 11.5, andproposition 12.1 of [9] that, respectively, γA

m, δBm, and γBm all converge to 0 as m

goes to ∞. We thus obtain the main theorem of this paper:

Theorem 3.2. With notation as above, for all m and n we have

prox(Bm, Bn) ≤ maxδBm, δBn +maxγAm, γB

m+maxγAn , γ

Bn ,

and in particular, prox(Bm, Bn) converges to 0 as m and n go to ∞.

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8 MARC A. RIEFFEL

One can also obtain matricial versions of this theorem along the lines discussedin section 14 of [9].

References

[1] David Kerr, Matricial quantum Gromov-Hausdorff distance, J. Funct. Anal. 205 (2003),no. 1, 132–167. arXiv:math.OA/0207282. MR 2020211(2004m:46153)

[2] David Kerr and Hanfeng Li, On Gromov-Hausdorff convergence for operator metric spaces,J. Operator Theory 62 (2009), no. 1, 83–109, arXiv:math.OA/0411157.

[3] Hanfeng Li, Order-unit quantum Gromov-Hausdorff distance, J. Funct. Anal. 231 (2006),no. 2, 312–360, arXiv:math.OA/0312001. MR 2195335 (2006k:46119)

[4] , C*-algebraic quantum Gromov-Hausdorff distance, arXiv:math.OA/0312003.[5] , Metric aspects of noncommutative homogeneous spaces, J. Funct. Anal. 257 (2009),

no. 7, 2325–2350, arXiv:0810.4694.[6] Marc A. Rieffel, Gromov-Hausdorff distance for quantum metric spaces, Mem. Amer. Math.

Soc. 168 (2004), no. 796, 1–65, arXiv:math.OA/0011063. MR 2055927[7] , Matrix algebras converge to the sphere for quantum Gromov-Hausdorff distance,

Mem. Amer. Math. Soc. 168 (2004), no. 796, 67–91, arXiv:math.OA/0108005. MR 2055928[8] , Vector bundles and Gromov-Hausdorff distance, J. K-Theory, published online 2009,

arXiv:math.MG/0608266.

[9] , Leibniz seminorms for “Matrix algebras converge to the sphere”, arXiv:0707.3229.[10] Wei Wu, Quantized Gromov-Hausdorff distance, J. Funct. Anal. 238 (2006), no. 1, 58–98,

arXiv:math.OA/0503344. MR 2234123

Department of Mathematics, University of California, Berkeley, CA 94720-3840

E-mail address: rieffel@math.berkeley.edu

180

Geometric and Topological Structures Related to M-branes

Hisham Sati

Abstract. We consider the topological and geometric structures associated

with cohomological and homological objects in M-theory. For the latter, wehave M2-branes and M5-branes, the analysis of which requires the underlyingspacetime to admit a String structure and a Fivebrane structure, respectively.For the former, we study how the fields in M-theory are associated with the

above structures, with homotopy algebras, with twisted cohomology, and with

generalized cohomology. We also explain how the corresponding charges shouldtake values in Topological Modular forms. We survey background material andrelated results in the process.

Contents

1. Introduction and Setting2. The M-theory C-Field and String Structures3. The M-theory Dual C-Field and Fivebrane Structures4. The Gauge Algebra of Supergravity in 6k − 1 Dimensions5. Duality-Symmetric Twists6. M-brane Charges and Twisted Topological Modular FormsReferences

1. Introduction and Setting

String theory is concerned with a worldsheet Σ2, usually a Riemann surface, atarget spacetime M , usually of dimension ten, and the space of maps φ : Σ2 → Mbetween them. The study of the field theory on Σ2 is the subject of two-dimensionalconformal field theory (CFT). The study of the maps is the sigma (σ-) model andthe study of the target space is the target theory where low energy limits, i.e. fieldand supergravity theories, can be taken.

The target space theory involves fields called the Ramond-Ramond (RR) or theNeveu-Schwarz (NS) fields. These are differential forms or cohomology classes which

2010 Mathematics Subject Classification. Primary 53C08, 55R65; Secondary 81T50, 55N20,

11F23.Key words and phrases. String structures, Fivebrane structures, n-bundles, differential co-

homology, K-theory, topological modular forms, generalized cohomology theories, anomalies,dualities.

1

Proceedings of Symposia in Pure MathematicsVolume 81, 2010

c©2010 American Mathematical Society

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2 HISHAM SATI

can be paired with homology cycles, the branes, namely D-branes and NS-branesrespectively. These extended objects carry charges, generalizing those carried byelectrons. The RR fields are classified by K-theory [MM] [W6] and are twisted bythe NS fields leading to a twisted K-theory classification [W6] (see also [BM]), inthe sense of [Ro] [BCMMS]. Such a description has been refined to more general-ized cohomology theories, most notably elliptic cohomology. This was approachedfrom cancellation of anomalies in type IIA string theory [KS1], studying the com-patibility of generalized cohomology twists with S-duality in type IIB string theory[KS2], and the study of modularity in the actions of type IIB string theory andF-theory [KS3] [S4].

The study of sigma models involves loop spaces as follows. The definition ofspinors require lifting the classifying map of the tangent bundle from the specialorthogonal group SO(n) to the spinor group Spin(n), which corresponds to killingthe first homotopy group π1(SO(n)) of SO(n). In string theory a further step isneeded, namely lifting the spinor group to the String group String(n) by killing1 π3(SO(n)), giving rise to String structures. Existence of such structures is acondition for the vanishing condition of the anomaly of a string in the context ofthe index theory of Dirac operators on loop space [W3] [Ki]. The String structureis regarded as a lift of an LSpin(n)-bundle over the free loop space LX through

the Kac-Moody central extension LSpin(n)-bundle [Ki] [CP] [PW] [Mc]. Thislift can also be interpreted as a lift of the original Spin(n)-bundle down on targetspace X to a principal bundle for the topological group String(n) [ST1]. This is, infact, the realization of the nerve of a smooth categorified group, the String 2-group[BCSS] [H]. The above classification of String-bundles coincided with that of 2-bundles with structure 2-group the String 2-group [BS] [BBK] [BSt]. In additionto the above infinite-dimensional models, now there is a finite-dimensional modelfor the String 2-group [S-P]. The elliptic genus is a loop space generalization

of the A-genus as the index of the Dirac-Ramond operator [SW] [PSW] [W2][AKMW]. The Green-Schwarz [GS] anomaly can be computed as essentially theelliptic genus [LNSW]. Mathematically, the connection between elliptic generaand loop spaces has been studied, notably in [A] and [Liu]. The String structure ,required by modularity, provides an orientation [AHS] [AHR] for TMF, the theoryof Topological Modular Forms [Ho] [Go].

At the level of conformal field theory, which is the quantum field theory onthe worldsheet of the string, one has fields that are pulled back from the spacetimetheory via the sigma model map, in addition to other fields. In two-dimensionalsupersymmetric quantum field theory the partition function, which is an integralmodular function, is argued in [ST1] [ST2] to be an element in TMF. Anothergeometric description of elliptic cohomology via CFT is given in [HK2], whichbuilds on Segal’s definition of CFT and on vertex operator algebras. A variantthat has features of both [ST1] and [HK2] is proposed in [GHK] using a newly-introduced notion of infinite loop spaces.

We emphasize two main points, central to the theme of this paper, concerningthe above mathematical structures:

1This might perhaps more correctly be called “cokilling” since it corresponds to the White-head tower rather than to the Postnikov tower.

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GEOMETRIC AND TOPOLOGICAL STRUCTURES RELATED TO M-BRANES 3

(1) First, generally, that various structures appearing in this part of theoret-ical physics are much deeper (and thereby richer and more interesting)than the sketchy physics literature about them indicates.

(2) Second, more specifically, that the above mathematical structures appear-ing in string theory are beginning to appear, even in a perhaps richer form,within the study of another theory, namely M-theory.

M-theory (cf. [W5] [To3]) is a conjectured theory in eleven dimensions that unifiesall five ten-dimensional superstring theories. The theory is best understood throughthese string theories and also via its classical low energy limit, eleven-dimensionalsupergravity theory [CJS]. Thus one strategy in studying the theory is to takeeleven-dimensional supergravity and perform semi-classical quantization. Due toquantum effects the process is only selectively reliable. Among the reliable termsare the topological terms, i.e. the terms that are not sensitive to the metric. Metric-dependent quantities might undergo drastic changes due to quantum gravitationaleffects. One way of keeping the metric requires taking some large volume limit,making sure that the scale is larger than the critical scale at which Riemanniangeometry can no longer provide a good description. As the theory is supersymmetrc,it will at least have a fermion (in this case, a section of the tensor product of thetangent bundle and the spin bundle), and since it involves gravity, it will also containa metric, or graviton. There is also an a priori metric-independent field, called theC-field. This is a higher-degree analog of a connection whose field strength – theanalog of a curvature – is denoted by G4.

The fields (aside from the metric and fermions) in string theory and M-theoryare differential forms at the rational level, i.e. at the level of description of thecorresponding supergravity theories. Gauge invariance leads to a description of thefields in terms of de Rham cohomology. Quantum mechanically, these genericallybecome integral-valued and hence one needs to go beyond de Rham cohomology tointegral cohomology. Dually, these fields can be described via homology cycles thatadmit extra geometric structures, such as Spin structures and vector bundles. Thisdual homological picture is captured by the notion of branes, namely D-branes instring theory and M-branes in M-theory. The fields and the branes are of specifieddimensions, determined by the corresponding theory. In particular, D-branes haveodd (even) spacetime dimensions, and hence even (odd) spatial dimensions, fortype IIA (type IIB) string theory. For M-branes, spatial dimensions two for theM2-brane [BST] and five for the M5-brane [Gu] occur.

The five string theories in ten dimensions are related through a web of dualities(see e.g. [Sch2] for a survey). The first kind of duality is called T-duality ( “T” forTarget space), which relates two different theories on torus bundles, where the firsttheory with fiber a torus is related to a second theory with a fiber the dual torus.An example of this is T-duality between type IIA and type IIB string theories. Thesecond kind of dualities is S-duality, or strong-weak coupling duality, which relatesa theory at a high value of some parameter (strong coupling) to another theoryat a low value of the same parameter (weak coupling). This generalizes the usualelectromagnetic duality between electric and magnetic fields in four dimensions.Whenever we discuss dualities in this paper we will focus mostly on S-duality. Aduality within the same theory is called self-duality. An example of this is the self-duality among the RR fields in type IIA/B string theory. At the rational level this is

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4 HISHAM SATI

simply a manifestation of Hodge duality. A delicate discussion of such matters canbe found in [Fr2] [FMS]. Another important example of this is S-(self)duality intype IIB string theory. Some subtleties on the relation of this duality to generalizedcohomology are discussed in [BEJMS] [KS2].

The above dualities can be most directly seen at the level of fields. By Poincareduality, such dualities also manifest themselves at the level of branes. Branes of even(odd) spatial dimensions in type IIA (type IIB) are dual to odd (even) branes in typeIIB (IIA) string theory. Furthermore, within the same type II theory, D-p-branesare dual to D-(6− p)-branes, p even or odd for type IIA or type IIB, respectively.This is a homological manifestation of the self-duality of the RR fields. Thereis a similar duality in eleven dimensions which relates the C-field C3 to its dualC6, which, at the rational level, is Hodge duality between the corresponding fieldsstrengths G4 = ∗G7. This duality on the fields also has a homological interpretationas a duality between the M2-brane and the M5-brane.

String theories in ten dimensions can be obtained from eleven-dimensional M-theory via dimensional reduction and/or duality transformations. M-theory on thetotal space of a circle bundle gives rise to type IIA string theory on the base space.By pulling back the fields along the section of the circle bundle π, assumed trivial,one gets a D2-brane from an M2-brane [To2] and a NS5-brane from an M5-brane.On the other hand, upon integration over the fiber of π, the M2-branes give riseto strings [DHIS] and the M5-brane give rise to D4-branes [To2]. The branesof type IIB string theory can also be obtained from those of M-theory on a torusbundle [Sch1]. Similar relations hold at the level of fields: Integrating G4 in elevendimensions over the circle gives H3 = π∗(G4), the field strength of the NS B-field.On the other hand, pulling back G4 along a section s, again assuming the circlebundle is trivial, gives a degree four field F4 = s∗G4, which is one component ofthe total RR field. An invariant description of this is given in [FS] and [MSa].

Branes carry charges– a notion that can be made mathematically precise– thatcan be viewed either as classes of bundles in generalized cohomology or as theirimages in rational or integral cohomology under a (normalized) version of the Cherncharacter map. A working mathematical definition of D-branes and their chargescan be found in [BMRS]: A D-brane in ten-dimensional spacetime X is a triple(W,E, ι), where ι : W → X is a closed, embedded submanifold and E ∈ Vect(W ) isa complex vector bundle overW . The submanifold W is called the worldvolume andE the Chan-Paton bundle of the D-brane. The charges of the D-branes [Po] can beclassified, in the absence of the NS fields, by K-theory of spacetime [MM], namelyby K0(X) for type IIB [W6] and by K1(X) for type IIA [Ho]. The fields are alsoclassified by K-theory of spacetime but with the roles of K0 and K1 interchanged[MW] [FH]. In the presence of the NS B-field, or its field strength H3, the relevantK-theory is twisted K-theory, as was shown in [W6] [FW] [Ka] by analysis ofworldsheet anomalies for the case the NS field [H3] ∈ H3(X,Z) is a torsion class,and in [BM] for the nontorsion case. Twisted K-theory has been studied for sometime [DK] [Ro]. More geometric flavors were given in [BCMMS]. Recently, thetheory was fully developed by Atiyah and Segal [AS1] [AS2]. The identification oftwisted D-brane charges with elements in twisted K-theory requires a push-forwardmap and a Thom isomorphism in the latter, both of which are established in [CW].

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GEOMETRIC AND TOPOLOGICAL STRUCTURES RELATED TO M-BRANES 5

The study of the M-brane charges and their relation to (generalized) cohomol-ogy is one of the main goals of this paper. Given the relation between generalizedcohomology and string theory on one hand, and between string theory and M-theoryon the other, it is natural to ask whether elliptic cohomology or TMF have to dowith M-theory directly. Suggestions along these lines have been given in [KS1][S1] [S2] [S3]. In particular in [KS1] it was proposed that the M-branes should bedescribed by TMF in the sense that the elliptic refinement of the partition functionoriginates from interactions of M2-branes and M5-branes. Furthermore, in [S3] itwas observed that the M-theory field strength G4, rationally, can be viewed as partof a twist of the de Rham complex and suggested that the lift to generalized coho-mology would be related to a twisted version TMF . A twisted differential is one ofthe form d+α∧ acting on differential forms, where d is the de Rham differential andα is a differential form. Twisted rational cohomology is then the kernel modulo theimage of d+ α∧. When α is a 3-form then one gets the image of twisted K-theoryunder the twisted Chern character (cf. [AS2] [BCMMS] [MSt]).

The field strength G4 on an eleven-manifold Y 11 satisfies the shifted quantiza-tion condition [W7]

(1.1) [G4] +1

2λ(Y 11) = a ∈ H4(Y 11;Z) ,

with λ(Y 11) = 12p1(Y

11), where p1(Y11) is the first Pontrjagin class of the tangent

bundle TY 11 of Y 11 and a is the degree four class that characterizes an E8 bundlein M-theory [DMW]. There is a one-to-one correspondence between H4(M,Z)and isomorphism classes of principal E8 bundles on M , when the dimension of Mis less than or equal to 15, which is the case for Y 11 in M-theory. This followsfrom homotopy type of E8 being of the form (3, 15, · · · ). Up to the 14th-skeletonE8 is homotopy equivalent to the Eilenberg-MacLane space K(Z, 3) so that upto the 15th-skeleton the classifying space BE8 is ∼ K(Z, 4). For the homotopyclasses of maps [M,E8] = [M,K(Z, 3)] = H3(M,Z) if dimM ≤ 14, and similarlyEquivalence classes of E8 bundles on M = [M,BE8] = [M,K(Z, 4)] = H4(M,Z)if dimM ≤ 15. Therefore, corresponding to an element a ∈ H4(M,Z) we have anE8 principal bundle P (a) → M with p1(P (a)) = a.

In [Wa] the notion of a twisted String structure was defined, where the twistis given by a degree four cocycle. This degree four generalization of the twistedSpinc structure in degree three was anticipated there to be related to the fluxquantization condition (1.1). This was made explicit and precise in [SSS3], wherealso the Green-Schwarz anomaly was shown to be more precisely the obstructionto having a refined twisted String structure.

The study the dual NS- and M-theory fields of degree eight leads to even higherstructures. The first appears in a dual form of the Green-Schwarz anomaly cancel-lation. The second satisfies a condition analogous to (1.1) in degree eight, as shownin [DFM]. In [SSS1] Fivebrane structures were introduced and the correspondingdifferential geometric structures, i.e. the higher bundles, are constructed. They aresystematically studied in [SSS2] and shown to emerge within the description of thedual of the Green-Schwarz anomaly, involving the Hodge dual of H3 in ten dimen-sions, as well as in the description of the dual of the M-theory C-field in eleven

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6 HISHAM SATI

dimensions. Fivebrane structures can be twisted in the same way that String struc-tures can. Twisted Fivebrane structures are defined and studied in [SSS3], andtheir obstructions can be matched with the dual of the Green-Schwarz anomaly.In fact, both twisted String and Fivebrane structures are refined in [SSS3] to thedifferential case, using a generalization of the discussion in [HS]. The notion of(twisted) String and, to some extent, Fivebrane structures can in fact be describedin various ways, via:1. Principal and associated bundles.2. Gerbes and differential characters.3. Cech cohomology and Deligne cohomology.4. Loop bundles.5. 2-bundles and 6-bundles and their 2-algebras and 6-algebras, respectively.

Another purpose of this paper is to provide the generalized cohomology aspect.The appearance of λ in (1.1) and the subsequent interpretation in terms of twistedString structures suggests a relation to a theory that admits that structure as anorientation. A Spin manifold M has a characteristic class λ such that 2λ = p1(M).The paper [AHS] shows thatM admits a TMF orientation if λ = 0. More precisely,a String structure on a Spin manifold M is a choice of trivialization of λ, and in[AHS] it is shown that a String structure determines an orientation of M in TMF -cohomology. In this paper we argue that the shifted quantization condition (1.1)provides a twist for TMF and hence that (1.1) defines a twist for TMF . Moreprecisely, the cohomology class of G4 is 1

2λ − a, which we view as a twist of theTMF orientation by the degree four class a of the E8 bundle. Since E8 ∼ K(Z, 4)up to dimension 14 then a a priori can be any class in K(Z, 4). However, fixing anE8 bundle completely fixes a. Conjectures for using a twisted form of TMF andelliptic cohomology – and hence that such structures should exist– to describe thefields in M-theory go back to [S2] [S3].

Given the interpretation of G4 and its dual in terms of twisted String andFivebrane structures and the proposed connection to twisted TMF , it is naturalto consider the corresponding homological objects. The charge of the M5-brane is,rationally, the value of the integral of G4 over the unit sphere in the normal bundlein the eleven-dimensional manifold. We interpret the charge of the M5-brane infull and not just rationally, in a Riemann-Roch setting, as the direct image ofan element in twisted TMF on the M-brane. The interpretation of the chargesas elements of twisted TMF uses some recent work [ABG] on push-forward andThom isomorphism in TMF . This is a higher degree generalization of the case ofD-branes, where the H-field defines a twist for the Spinc structure and the chargeis defined using the push-forward and Thom isomorphism in twisted K-theory.

Given the topological structures defined by G4 and its dual, it is natural toask for geometric models for the corresponding potentials, i.e. the C-field C3 andits dual. There is the E8 model of the C-field [DMW] [DFM], mentioned above,which is essentially a Chern-Simons form, or more precisely a shifted differentialcharacter, where the shift on the E8 bundle class a is given by the factor 1

2λ. Inan alternative model in terms of 2-gerbes [AJ], picking a connection A on the E8

principal bundle P (a) gives a Deligne class, the Chern-Simons 2-gerbe CS(a), witha its characteristic class. The interpretation we advocate in terms of twisted String

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GEOMETRIC AND TOPOLOGICAL STRUCTURES RELATED TO M-BRANES 7

structures allows for other interpretations of the C-field, where the term 12λ is now

the ‘main’ term, since it is responsible for defining the String structure, and theterm a coming from the E8 bundle is merely a twist for that structure. This givesthe tangent bundle-related term a more prominent role. In fact, this model canbe described at the differential twisted cohomology level [SSS3]. In this paper weprovide an identification of the C-field as the String class, i.e. as the class thatprovides the trivialization of the String structure. We also do the same for the dualof the C-field which we identify with the Fivebrane class, the class that provides thetrivialization for the Fivebrane structure. We also give an alternative descriptionusing differential characters.

The appearance of higher chromatic phenomena in [KS1] [KS2] [KS3] in rela-tion to string theory, and the appearance of Fivebrane structures in degree eight instring theory and M-theory naturally leads to the question of whether higher degreetwists exist in this context. Indeed, in [S6] it was shown that a degree seven twistoccurs in heterotic string theory, manifested via a differential of the form d+H7∧.The lift of this twisted rational cohomology to generalized cohomology suggests theappearance of the second generator v2 at the prime 2 in theories descending fromcomplex cobordism MU . This generalizes the situation in degree three, where theBott generator u = v1 in K-theory appears via d+v1H3∧. In this paper we considerthe fields in M-theory as part of a twist in de Rham cohomology, extending andrefining the discussion in [S1] [S2] [S3]. In addition we consider duality-symmetrictwists, i.e. twisted differentials whose twists are uniform degree combinations ofthe H-field and its dual as well as G4 and its dual. The second case leads to aninteresting appearance of the M-theory gauge algebra, which in turn leads to thesuper-tranlation algebra. We also provide an L∞-algebra description of this gaugealgebra.

String Theory M-Theory

sigma model φ : Σ2 → X10 sigma model Φ : M3 → Y 11

ψ ∈ Γ(SΣ2 ⊗ φ∗TX10), B2 1-gerbe ψ ∈ Γ(SM3 ⊗N (M3 → Y 11), C3 2-gerbe

D-brane ⊃ ∂Σ2 M5-brane ⊃ ∂M3

Freed-Witten condition W3 + [H3] = 0 Witten Flux quantization 12λ+ [G4] = a

Table 1. Extended objects in string theory and in M-theory.

This paper is written in an expository style, even though it is mainly aboutoriginal research. In fact, there are three types of material:

• Survey of known results, with some new perspectives a well as providingsome generalization.

• New research established here.• New research announced and outlined here and to be more fully developedin the future. This material is mostly based on discussions with MatthewAndo, Chris Douglas, Corbett Redden, Jim Stasheff, and Urs Schreiber.

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8 HISHAM SATI

We hope that the expository style makes it more self-contained and accessible tomathematicians interested in this area of interaction between physics on one handand geometry and topology on the other.

2. The M-theory C-Field and String Structures

Consider an eleven-dimensional Spin manifold Y 11 with metric g. Correspond-ing to the tangent bundle TY 11 with structure group SO(11) there is the Spinbundle SY 11 with structure group the Spin group Spin(11), the double cover ofSO(11). Let ω be the Spin connection on SY 11 with curvature R. Using themetric we can identify the cotangent bundle T ∗Y 11 with the tangent bundle. Thefield content of the theory is the metric g, a spinor one-form ψ , i.e. a section∈ Γ(SY 11 ⊗ T ∗Y 11), and a degree three form C3.

2.1. The Quantization Condition and the E8 model for the C-field.The quantization condition on G4 on Y 11 is given in equation (1.1). This can beobtained either from the partition function of the membrane or from the reductionto the heterotic E8 × E8 theory on the boundary [W7]. We will reproduce thisresult using the membrane partition function in a fashion that is essentially thesame as appears in [W7] but with details retained. Since the Spin cobordism

groups ΩSpin4k+3 are zero, we can extend both the membrane worldvolume M3 and

the target spacetime Y 11 as Spin manifolds to bounding manifolds X4 and Z12,respectively. In fact we can also extend E8 bundles on M3 and Y 11 to E8 bundleson X4 and Z12, respectively, since MSpini (K(Z, 4)) = 0 for i = 3, 11 [Stg] whereBE8 has the homotopy of K(Z, 4) in our range of dimensions. The effective actioninvolves two factors:

(1) The ‘topological term’ exp i∫M3 C3,

(2) The fermion term exp i∫M3 ψDψ, where the integrand in the exponential

is the pairing in spinor space of the spinor ψ with the spinor Dψ, whereD is the Dirac operator.

The first factor will simply give exp i∫X4 G4. Now consider the second factor.

Corresponding to the map φ : X4 → Z12 we have the index of the Dirac operatorfor spinors that are sections of S(X4)⊗φ∗TZ12 given via the index theorem by thedegree two expression

IndexD =

[∫X4

A(X4) ∧ ch(φ∗TZ12)

]

(2)

=

∫X4

[1− 1

24p1(TX

4)

] [rank(TZ12) + ch2(φ

∗TZ12)]

=

∫X4

1

2p1(φ

∗TZ12)− 1

2p1(X

4) .(2.1)

On the other hand we have a split of the restriction of the tangent bundle TZ12 toX4 as TZ12 = TX4 ⊕ NX4 with NX4 the normal bundle of X4 in Z12. Takingthe characteristic class λ of both sides we get that the index is equal to λ(NX4).The effective action involves the square root of the index so that the contributionfrom the second factor in the effective action is exp 2πi 12λ(NX4). This gives the

result in [W7] provided we assume that N (M3 → Y 11) ∼= N (X4 → Z12).

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GEOMETRIC AND TOPOLOGICAL STRUCTURES RELATED TO M-BRANES 9

Proposition 2.1. The M-theory field G4 satisfies the quantization condition(1.1).

Remarks 2.2. The condition (1.1) has the following consequences [W7]:

(1) When λ is divisible by two then G4 cannot be set to zero.(2) When λ

2 is integral then the tadpole anomaly of [SVW] vanishes.

(3) The relation is parity-invariant, i.e. requiring G4 − λ2 to be integral is

equivalent to G4 +λ2 being integral.

The E8 model of C-field. The C-field at the level of supergravity will be just areal-valued three-form C3 ∈ Ω3(Y 11;R). The field strength is G4 = dC3 ∈ Ω4(Y 11).This is invariant under gauge transformations C3 → C3 + dφ2, where φ2 is a two-form. Factoring out by the gauge transformations amounts to declaring the fieldsto be in cohomology. Upon quantization, several features become important: inte-grality via holonomy, the presence of torsion, and possible appearance of anomalies.Taking these into account, a model for the C-field was obtained in [W7] and furtherdeveloped in [DMW] [DFM] [Mo].

Let P be a principal E8 bundle over Y 11 with the characteristic class a pulledback from H4(BE8;Z). Let A be a connection on P with curvature two-formF . The C-field in this model is given by C = CS3(A) − 1

2CS(ω) + c, whereCS3(A) is the Chern-Simons invariant for the connection A, CS(ω) is the Chern-Simons invariant of the connection ω on the Spin bundle, and c is the harmonicrepresentative of the C-field which dominates at long distance approximation, i.e.in the supergravity regime. The Chern-Simons forms and the Pontrjagin forms arerelated as dCS3(A) = TrF ∧ F , dCS3(ω) = TrR ∧ R, so that the field strength ofthe C-field is given by

(2.2) G4 = TrF ∧ F − 1

2TrR ∧R+ dc .

The cohomology classes are [TrF ∧ F ]DR = aR, [TrR ∧R]DR = 12

(p1(TY

11))R.

Globally, the C-field can be described as the pair [DFM]

(2.3) (A, c) ∈ EP (Y 11) := A(P (a))× Ω3(Y 11),

where A(P (a)) is the space of smooth connections on the bundle P with class a.

2.2. Twisted String structures.

Definition 2.3. An n-dimensional manifold X admits a String structure ifthe classifying map X → BO(n) of the tangent bundle TX lifts to the classifyingspace BString := BO〈8〉.

(2.4)

BO(n)〈8〉

M

f

f

BO(n) .

Remarks 2.4. (1) The obstruction to lifting a Spin structure on X to aString structure on X is the fractional first Pontrjagin class 1

2p1(TX).(2) The set of lifts, i.e. the set of String structures for a fixed Spin structure is

a torsor for a quotient of the third integral cohomology group H3(X;Z).

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10 HISHAM SATI

Definition 2.5 ([Wa] [SSS3]). An α-twisted String structure on a brane ι :M → X with Spin structure classifying map f : M → BO(n)〈4〉 is a cocycleα : X → K(Z, 4) and a map c : BO(n)〈4〉 → K(Z, 4) such that there is a homotopyη between BO(n)〈4〉 and X,

Mf

ι

BO(n)〈4〉

c

X α

K(Z, 4)

η

.

The homotopy η is the coboundary that relates the cocycle cf to the cocycle αι.Hence on cohomology classes it says that the fractional Pontrjagin class of M doesnot necessarily vanish, but is equal to the class ι∗[α].

The twisted String case was originally considered in [Wa]. The definition isrefined to structures dubbed F〈m〉 that account for obstructions that are fractionsof the ones for the String structures. For example, the fractional class 1

4p1 showsup in (1.1). As an application, which was originally the motivation:

Theorem 2.6 ([SSS3]). (1) The Green-Schwarz anomaly cancellationcondition defines a twisted String structure pulled back from BO(10)〈4〉 =BSpin(10). The twist α in this case is given by (minus) the degree fourclass of the E8 × E8 bundle.

(2) The anomaly cancellation condition in heterotic M-theory and the fluxquantization condition in M-theory each define a twisted String structurepulled back from F〈4〉 = BO〈 14p1〉. The twist α in this case is given by[G4] minus the class of the E8 bundle.

The division of λ by 2 require some refinement of the structure [SSS3] asmentioned in Remarks 6.11. The relation to orientation in generalized cohomol-ogy is discussed in the study of the membrane partition function in section 2.4.Note that when we consider M-theory with a boundary ∂Y 11, where essentiallythe heterotic string theory is defined, [G4] would be zero when restricted to ∂Y 11.In this case the flux quantization condition defines a a-twisted String structure12p1(∂Y

11) = a|∂Y 11 = 0 on that boundary. This is discussed further in section 6.3.

2.3. The C-field as a String class. Let 〈·, ·〉 be a suitably normalized Ad-invariant metric on the Lie algebra spin(11) of Spin(11). Then the 4-dimensionalChern-Weil form 〈R ∧ R〉 ∈ Ω4(Y 11) on S(Y 11) is one-half the first Pontrjaginclass (restoring normalization) 1

2p1(S, ω) = − 116π2Tr(R ∧R). We will later assume

the following condition on the Pontrjagin class 12p1(Y

11) = 0 in H4(Y 11;R). This

means that the bundle S(Y 11) admits a String structure. A choice of String struc-ture is given by a particular cohomology class S ∈ H3

(S(Y 11);Z

). This element

restricts to the fiber as the standard generator of H3(Spin(11);Z) ∼= Z.

In terms of the curvature R of the connection ω on S(Y 11), the condition12p1(S(Y

11)) = 0 ∈ H4(Y 11;R) means that [TrR ∧R] = 0 ∈ H4(Y 11;R). The

Chern-Simons 3-form of the connection ω on the principal bundle S(Y 11) is theright-invariant form CS3(ω) := 〈ω∧R〉− 1

6 〈ω∧ [ω, ω]〉 ∈ Ω3(S), whose pull-back to

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GEOMETRIC AND TOPOLOGICAL STRUCTURES RELATED TO M-BRANES 11

the fiber via the inclusion map i : Spin(11) → S is the real cohomology class − 16 〈ω∧

[ω, ω]〉 ∈ H3(Spin(11);R) associated to the real image of the standard generator ofH3(Spin(11);Z) ∼= Z. The Chern-Simons form CS3(ω) provides a trivialization forthe zero cohomology class above: dCS3(ω) = TrR ∧R ∈ Ω4(S(Y 11)).

Now we consider the geometry on the total space of the Spin bundle. Cor-responding to a choice of Riemannian metric g on Y 11 and a connection ω onS(Y 11) gives rise to a metric gS on the total space S(Y 11). Under the decomposi-tion TS(Y 11) ∼= π∗(TY 11 ⊕ spin(11)) of the tangent space into orthogonal verticaland horizontal subspaces, the metric decomposes as gS := π∗(g ⊕ g

Spin(11)), where

gSpin(11)

is the metric on the fiber.

In relating the fields on the base to classes on the total space, one is forced touse the adiabatic limit, introduced in [W1], of the metric on the total space. Inparticular, as is constructed in [R1], the String structure S on S(Y 11) is relatedto a form on the base in this fashion. This way there is a one-parameter family ofmetrics gδ = π∗ ( 1

δ2 g ⊕ gSpin(11)

)on the bundle S(Y 11) with parameter δ. This is

reminiscent of a Kaluza-Klein ansatz frequently used in supergravity. Now considerthe adiabatic limit δ → 0 of gδ. Note that metrics in the adiabatic limit have beenused in this form e.g. in [MSa] in the reduction from M-theory in eleven dimensionsto type IIA string theory in ten dimensions.

Relative Chern-Simons form on Y 11. Before considering the C-field we willneed the following. Consider an E8 bundle P over Y 11 with connection A ∈Ω1(P ; e8) and curvature π∗Ω = dA+ 1

2 [A,A] ∈ Ω2(P ; e8), where Ω ∈ Ω2(Y 11; adP ),a two-form with values in adP . The space AP of connections on P is an affine spacemodeled on Ω1(Y 11; adP ). Using [Fr1] we can define a relative Chern-Simons in-variant on the base. Given two connections A0 and A1 in AP , the straight linepath At = (1 − t)A0 + tA1, 0 ≤ t ≤ 1, determines a connection A on the bundle

[0, 1]× P [0, 1]× Y 11 . The relative Chern-Simons form is then

(2.5) CS3(A1, A0) := −∫[0,1]

TrF 2(A) ∈ Ω3(Y 11).

Using Stokes’ theorem

dCS3(A1, A0) = −d

∫[0,1]

TrF 2(A)

= −∫[0,1]

dTrF 2(A) + (−1)(11−4)

∫∂[0,1]

TrF 2(A)

= 0 + TrF 2(A1)− TrF 2(A0) .(2.6)

Now we are ready to consider the C-field.

Invariance of the C-field. The C-field is invariant under the following trans-formations [DFM]: A′ = A + α and C ′

3 = C3 − CS3(A,A + α) + Λ3, whereα ∈ Ω1

(adS(Y 11)

)and Λ3 is a closed 3-form on Y 11. The Chern-Simons invariant

takes values in R/Z so that it is not defined as a differential form unless exponen-tiated. The relative Chern-Simons invariant is defined as in (2.5).

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12 HISHAM SATI

If we include the Spin bundle S(Y 11) then we also have an invariance of theconnection ω and a corresponding shift in the relative Chern-Simons form of ω.Thus we have

Proposition 2.7. The C-field is invariant under the following transformations

(1) ω′ = ω + β,(2) A′ = A+ α,(3) C ′

3 = C3 − CS3(A,A+ α) + 12CS3(ω, ω + β) + Λ3,

where α ∈ Ω1(adP ), β ∈ Ω1(adS), and Λ3 ∈ Ω3(Y 11) is a closed differential formon Y 11.

Note that in terms of differential characters, Λ3 will be integral as in [DFM].

Harmonic part of the C-field. The C-field has a classical harmonic part, whichwe now characterize. The Bianchi identity and equation of motion for the C-fieldin M-theory are

dG4 = 0(2.7)

1

3pd ∗G4 =

1

2G4 ∧G4 − I8,(2.8)

where I8 is the one-loop term [VW] [DLM] I8 = 148

(p2 − ( 12p1)

2), a polynomial

in the Pontrjagin classes pi of Y 11, ∗ is the Hodge duality operation in elevendimensions, and p is the scale in the theory called the Planck constant. Theclassical (or low energy) limit given by eleven-dimensional supergravity, is obtainedby taking p → 0 and is dominated by the metric-dependent term. The other limitis the high energy limit probing M-theory and is dominated by the topological, i.e.metric-independent terms.

Let Δ3g :(Ω3(Y 11), g

)−→

(Ω3(Y 11), g

)be the Hodge Laplacian on 3-forms on

the base Y 11 with respect to the metric g given by Δ3g = d d

∗+ d

∗d, where d

∗is

the adjoint operator to the de Rham differential operator d. Assuming [G4] = 0 inH4(Y 11;R) so that G4 = dC3, then applying the Hodge operator on (2.8) gives

Proposition 2.8. In the Lorentz gauge, d∗C3 = 0, we have

(1) Δ3gC3 = ∗je, where je is the electric current associated with the membrane

given by

(2.9) je = 3p

(1

2G4 ∧G4 − I8

).

(2) C3 is harmonic if p → 0 and/or there are no membranes.

The space of harmonic 3-forms on Y 11 is H3g(Y

11) := kerΔ3g ⊂ Ω3(Y 11).

We would like to consider harmonic 3-forms on the Spin bundle S(Y 11). LetΔ3

gδ:(Ω3(S(Y 11)), gδ

)→(Ω3(S(Y 11)), gδ

)be the Hodge Laplacian for 3-forms

on S(Y 11) with respect to the metric gδ. The harmonic forms, which are in kerΔ3gδ,

on the Spin bundle can be calculated in the adiabatic limit δ → 0. The expressionfor kerΔ3

0 := limδ→0 kerΔ3gδ

was calculated in [R1], using the spectral sequence of[MaM], further developed in [D] [Fo]. Applying the results of [R1] to our casegives

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GEOMETRIC AND TOPOLOGICAL STRUCTURES RELATED TO M-BRANES 13

Proposition 2.9. (1) When 12p1(Y

11) = 0 ∈ H4(Y 11;R), i.e. [TrR ∧R] = 0 ∈ H4(Y 11;R) then kerΔ3

0 = π∗H3g(Y

11) ⊂ Ω3(S(Y 11)).

(2) When 12p1(Y

11) = 0 then

(2.10) kerΔ30 = R [CS3(ω)− π∗h]⊕ π∗H3

g(Y11) ⊂ Ω3(S(Y 11)) ,

where h ∈ Ω3(Y 11) is the unique form such that dh = TrR ∧ R, h ∈d

∗Ω4(Y 11) .

We see that for the C-field in M-theory we have

Proposition 2.10. (1) When Y 11 is a Spin manifold such that12p1(Y

11) = 0, the little c-field is a harmonic form both on Y 11 and on

S(Y 11).(2) When Y 11 is a String manifold, i.e. with 1

2p1(Y11) = 0 in cohomology,

so that [G4] = a, then there is a gauge in which the 3-form part of theC-field is that defining a String class, as in the above discussion.

Remark 2.11. The combination of forms appearing in proposition 2.9 areexactly the ones also appearing in heterotic string theory. Indeed, the Chapline-Manton coupling is a statement about the String class.

The String class from the String condition on the target Y 11. From (2.2),we see that when G4 = TrF ∧ F then 1

2TrR ∧ R = dc. At the level of cohomology

this means that 12p1(S, ω) = 0, i.e. that our space admits a String structure S.

Let us form the combination CS3(ω) − π∗c ∈ Ω3(S). Consider a choice of Stringstructure S ∈ H3(S(Y 11);Z). From (2.10), using the results in [R1], the adiabaticlimit of the harmonic representative of S is given by

(2.11) [S]0 := limδ→0[S]gδ = CS3(ω)− π∗c3 ∈ Ω3(S),

where the form c3 ∈ Ω3(Y 11) has the properties:

dc3 =1

2p1(S, ω),(2.12)

d∗c3 = 0 (Lorentz condition) .(2.13)

Observation 2.12. Under these assumptions, the C-field can be identified witha String class.

Remark 2.13. Consider a change of the String structure S. If the Stringstructure is changed by ξ ∈ H3(Y 11;Z) then the cohomology class of the Stringstructure changes, in the adiabatic limit, as

(2.14) [S + π∗ξ]0 = [S]0 + π∗[ξ]g .

Since ξ is a degree three cohomology class, the field strength G4 does not see thechange in String structure. However, at the level of the exponentiated C-field, i.e.at the level of holonomy or partition function of the membrane, there will be aneffect. See the discussion leading to observation 2.16. This will be formalized andconsidered in more detail in a future work.

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14 HISHAM SATI

2.4. The String class from the membrane. Consider the embedding ofthe membrane Σ3 → Y 11 with normal bundle N . The fields on the membraneworldvolume include a metric h, the pullback of the C-field and a spinor ψ ∈Γ(S(Y 11)|Σ3). Restricting S(Y 11) to Σ3 gives the splitting

(2.15) S(Y 11)|Σ3 = S(Σ3)⊗ S−(N )⊕ S(Σ3)⊗ S+(N ),

where kappa symmetry– a spinorial gauge symmetry– requires the fermions to besections of the first factor [BST]. Taking the membrane as an elementary object,the exponentiated action will contain a factor exp

[i∫Σ3(C3 + i−3

p vol(h)]. This is

one part of the partition function, with another being the spinor part given by thePfaffian of the Dirac operator. Neither of the factors in

(2.16) ZM2 = Pfaff(DS(Σ3)⊗S−(N )) exp

[i

∫Σ3

(C3 + i−3p vol(h)

]

are separately well-defined, but the product is [W7]. Taking Σ3 to be the boundaryof a 4-manifold B4 we get

∫B4 G4 in place of the first factor in the exponent in (2.16).

The partition function is independent of the choice of bounding manifold B4.

The quantization condition (1.1) for the C-field in M-theory was derived in[W7] by studying the partition function of the membrane of worldvolume M3

embedded in spacetime Y 11. There, the manifold M3 was assumed to be Spin.Here we notice that M3 already admits a String structure because 1

2p1(M3) = 0,

by dimension reasons. Since this is automatic, one might wonder what is gained byassuming this extra structure. We will proceed with justifying this.

The idea is that while M3 always admits a String structure, we can have morethan one String structure. We have the following diagram

(2.17) K(Z, 3) BString

∗

M3

ψ

σ BSpin

λ K(Z, 4) .

Choosing a String structure ψ is equivalent to trivializing λ σ. If we fix oneString structure ψ then any other is classified by maps from M3 to K(Z, 3), whichis H3(M3;Z). If we take K(Z, 3) = BK(Z, 2) then we can say that the set ofString structures on M3 is a torsor over the group of equivalence classes of gerbeson M3. From one given (equivalence class of) String structure we obtain for each(equivalence class of a) gerbe another (equivalence class of a) String structure.Notice that in the non-decomposible part of the C-field c3+h3, h3 is the curvatureof the gerbe. It is closed as a differential form. We can see that there is a gerbe onthe membrane worldvolume by taking the membrane to be of open topology andhaving a boundary on the fivebrane. There is then a gerbe connection C3 − db2,where b2 is the chiral 2-form on the M5-brane worldvolume (see e.g. [AJ]). In thiscase we have the exponentiated action

(2.18) exp

[i

(∫∂Σ3

b2 + i

∫Σ3

−3p vol(h)

)].

(See also Remark 2.13 and the discussion around equation (2.25)).

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GEOMETRIC AND TOPOLOGICAL STRUCTURES RELATED TO M-BRANES 15

Now we consider the bounding 4-dimensional space B4, ∂B4 = M3. Startingwith a Spin M3, there are two cases to consider, according to whether B4 is Spinor String. We will make use of an approach due to David Lipsky and to CorbettRedden. Let us start with the Spin case. Including B4 in diagram (2.17) we get

(2.19) K(Z, 3) BString

∗

M3

ψ

σ

BSpinλ K(Z, 4)

B4

τ

.

Let CM3 be the cone on M3. The fact that

(2.20) M3ψ BString

∗

BSpin K(Z, 4)

commutes up to homotopy means precisely that there is a strictly commuting dia-gram

(2.21) M3

i1

∗

M3

ψ

i0 M3 × I

BString

BSpin K(Z, 4)

.

Moreover, the cone is precisely the pushout

(2.22) M3

∗

M3 × I CM3

,

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16 HISHAM SATI

through which, hence, diagram (2.21) factors

(2.23) M3

i1

∗

M3

ψ

i0 M3 × I

CM3

Cψ

BString

BSpin K(Z, 4)

.

Here Cψ is the extension of ψ from M3 to the cone on M3. Therefore, we find thatthe following diagram

(2.24) M3

λσ

CM3Cψ

ρ

∗

B4

λτ

B4⋃

M3CM3 λ(τ,ψ) K(Z, 4)

commutes. Note that the fact that ψ extends from M3 to the cone of M3, asindicated, is crucially another incarnation of the fact that ψ is homotopic to themap through the point. The map ρ is equivalent to a String structure, and the mapλ(τ, ψ) is the relative String class. Let [B4,M3] be the relative fundamental classand 〈 , 〉 the pairing between cohomology and homology. For this pairing we willstudy integrality and (in)dependence on the choice of B4 or structures on B4. Thelong exact sequence for relative cohomology is

· · · → H3(M3) −→ H4(B4,M3) −→ H4(B4) −→ H4(M3) · · ·αω3 → λ(τ, ψ) + ∂(αω3) −→ λ(τ ) ,(2.25)

where ω3 is the volume form on M3 and α is a real number. The sequence thenis explained as follows. Given a choice of initial String structure on M3, any otherchoice will be given by the difference with multiples of the volume form ω3. Notethat the only parameter which is varying is α ∈ R.

Then, let B′4 be another bounding manifold with corresponding Spin structureτ ′. Then, by the index theorem,

(2.26)

∫B4

λ(τ, ψ)

24−∫B′4

λ(τ ′, ψ)

24= x ∈ Z ,

so that

(2.27)

∫B4

λ(τ, ψ)−∫B′4

λ(τ ′, ψ) = 24x, x ∈ Z .

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GEOMETRIC AND TOPOLOGICAL STRUCTURES RELATED TO M-BRANES 17

Now taking e2πi of both sides gives that the expression is an integer. So thistogether with what we explained above also gives that integral does not depend onthe choice of B4 or structures on B4, and we have the following

Proposition 2.14. In the case when M3 is String and B4 is Spin, the relativepairing 〈λ(τ, ψ) , [B4,M3]〉 =

∫B4 λ(τ, ψ) is well-defined mod 24.

Now consider the case when B4 also admits a String structure, so that∫B4 λ =

0. In this case, the question simply reduces to a statement in cobordism of String

3-manifolds Ω〈8〉3

∼=−→Z/24 defined by (M3, ψ) −→∫B4 λ(τ, ψ). Therefore

Proposition 2.15. In the case when both M3 and B4 are String manifolds,the relative pairing 〈λ(τ, ψ) , [B4,M3]〉 =

∫B4 λ(τ, ψ) is well-defined mod 24.

Dependence of the membrane partition function on the String structure.Taking (2.16) into account and the fact mentioned above (in the proof of proposition2.14 that changing the String structure of the membrane amounts to changing itsvolume, we have for membrane worldvolumes with String structure

Observation 2.16. The membrane partition function depends on the choice ofString structure on the membrane worldvolume.

Remarks 2.17. (1) There are nonperturbative effects, namely instan-tons, resulting from membranes wrapping 3-cycles in spacetime. See forexample [HM].

(2) In the case of string theory, the partition function depends crucially onthe Spin structure of the string worldsheet. Modular invariance requiressumming over all such structures [SW]. The observations we made abovethen suggest that the membrane theory would require a careful consider-ation of dependence on the String structure, and possibly summing oversuch structures. We hope to address this important issue elsewhere.

The framing in Chern-Simons theory. We can look at the dependence of themembrane partition function on the String structure through the dependence ofChern-Simons theory on the choice of framing. The partition function of Chern-Simons theory on M3 depends on [W4]: M3, the structure group G, the Chern-Simons coupling k, and a choice of framing f of the manifold. In particular, thesemiclassical partition function, while independent of the metric, it does dependon the choice of framing, and different framings generally give different values forthe partition function. However, there are transformations that map the valuecorresponding to one framing to the value corresponding to another. A framing ofM3 is a homotopy class of a trivialization of the tangent bundle TM3. Given aframing f : M3 → TM3 of M3 the gravitational Chern-Simons term can be definedas

(2.28) IM3(g, f) =1

4π

∫M3

f∗CS(ω) ,

where g is the metric on M3, ω is the Levi-Civita connection on M3, and theintegrand is the pullback via f of the Chern-Simons form on TM3.

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18 HISHAM SATI

In dimension three, there is an isomorphism between the String cobordism

group ΩString3 and the framed cobordism group Ωfr

3 . Thus, the study of Chern-Simons theory with a String structure is then equivalent to the study of Chern-Simons theory with a framing. Therefore, it is natural to consider a String structurein Chern-Simons theory, and hence on the membrane worldvolume, since the latteris essentially described by Chern-Simons theory.

We argue that not only is a String structure allowable, but is in fact desirable.This is because such a structure explains the framing anomaly in a very natural way.Under the transformation IM3 → IM3 + 2πs, where s is the change in framing, thepartition function transforms as [W4] ZM3 → ZM3 ·exp

(2πis · d

24

), for d a constant

related to the level of the theory. This factor of 24, making the partition functionessentially a 24th root of unity, is reflection of the fact that both the String- and theframed cobordism group are isomorphic in dimension three to Z/24. Any Lie groupG has two canonical String structures defined by the left invariant framing fL andthe right invariant framing fR of the tangent bundle TG. For example takingM3 tobe G = SU(2), there are three framings: a left framing and a right framing (relatedby orientation reversal) and a trivial framing given by taking S3 = ∂D4 to be theboundary of the 4-disk. The invariants associated to these framings are the images

of ΩString3 under the σ-map (the String orientation [AHS]) in tmf−3 ∼= Z/24. This

map depends on the String structure in an analogous way that its more classical

cousin, the Atiyah α-invariant, refining the A-genus from Z to KO, depends on theSpin structure. From the isomorphisms π3S

0 → π3MString → π3tmf [Ho] thesevalues are as follows

ΩString3 −→ tmf−3

[SU(2), fL] −→ − 1

24

[SU(2), fR] −→ 1

24[SU(2), ∂D4

]−→ 0 .(2.29)

The transformation of the partition function can then be soon more transparentlyusing String cobordism. Thus, we get more confirmation to observation 2.16 and,in fact, we can also add

Observation 2.18. The dependence on framing of Chern-Simons theory (andhence also for the membrane partition function) can be seen more naturally withinString cobordism. We thus conjecture that the membrane partition function takesvalues in (twisted) tmf .

Note that within Spin cobordism there would be no nontrivial expressions in

dimension three. This is because ΩSpin3 = 0 and also the target for the Atiyah α-

invariant, KO3(pt), is also zero. Furthermore any generalization of α, for exampleto the Ochanine genus with target KO3(pt)[[q]] would also be trivial.

Further relation to generalized cohomology. There is further a connectionof the membrane partition function to generalized cohomology from another angleas follows. The Stiefel-Whitney class w4 is the mod 2 reduction of λ = 1

2p1. Thisimplies that λ is even if and only if w4 = 0. The latter is in fact an orientation con-dition in real Morava E-theory EO(2) [KS1] (worked out there for a different but

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GEOMETRIC AND TOPOLOGICAL STRUCTURES RELATED TO M-BRANES 19

related purpose). Therefore, in order to remove the ambiguity in the quantization,we require EO(2)-orientation.

Proposition 2.19. The membrane partition function is well-defined in EO(2)-theory. In particular, G4 is an integral class when the underling spacetime is EO(2)-oriented.

Remarks 2.20. (1) This means that eleven-dimensional spacetime back-grounds in M-theory with no fluxes should be EO(2)-oriented.

(2) Note that EO(2) is closely related to the theory EO2 of Hopkins andMiller, which in turn is closely related to TMF .

(3) The M-theoretic partition function via E8 gauge theory of [DMW] isconsidered for the String case in [S7].

The case when the Spin bundle is trivial. Consider the case when S(Y 11) istrivial as a principal bundle. This then means that there is a global section. Sucha section s : Y 11 → S(Y 11) gives an isomorphism of principal bundles

(2.30) S(Y 11)s∼= Y 11 × Spin(11) .

Using the fact that Hi (Spin(11);Z) = 0 for i = 1, 2, the Kunneth formula gives aninduced isomorphism on integral cohomology

H3(S;Z) ∼= H3(Y 11;Z)⊕H3(Spin(11);Z)

S ←→ (0 , 1Spin) .(2.31)

The String structure is then determined by s and is an element S ∈ H3(S;Z) whichcorresponds to the pullback of the generator 1Spin ∈ H3(Spin(11);Z).

Note that the differential d acting on p∗CS3(ω) is 12p1(ω), which is the same

as dc. This means that [p∗CS3(ω) − c] ∈ H3(Y 11;R). Consider the membraneworldvolume M3, taken as a 3-cycle X ∈ Map(M3, Y 11). Then p(X) ⊂ Y 11 ×pt ⊂ S under the isomorphism (2.30) induced by the global section s. Thetriviality of the bundle implies that any class ξ ∈ H3(S;Z) is zero when evaluatedon 3-cycles Σ3 in Y 11 ⊂ S, 〈ξ, [s(Σ3)]〉, where [s(Σ3)] is the fundamental class ofthe 3-cycle s(Σ3) in S(Y 11). Then, in real cohomology

(2.32) 0 =

∫s(Σ3)

[ξ]0 =

∫s(Σ3)

(CS3(ω)− π∗c) =

∫Σ3

(s∗CS3(ω)− c) .

This holds for an arbitrary 3-cycle Σ3 so that [s∗CS3(ω) − c] = 0 ∈ H3(Y 11;R).Now if d

∗(s∗CS3(ω)) = 0, the since d

∗c = 0, then s∗CS3(ω) − c is harmonic,

and hence zero, so s∗CS3(ω) = c. Therefore, in this case, following the generalconstruction [R1] [R2], s∗CS3(ω) and c are equal to the coexact and harmoniccomponents, respectively.

The C-field as a Chern-Simons 2-gerbe. In [CJMSW] a Chern-Simons bun-dle 2-gerbe is constructed, realizing differential geometrically the Cheeger-Simonsinvariant [CS]. This is done by introducing a lifting to the level of bundle gerbesof the transgression map from H4(BG;Z) to H3(G;Z). A similar construction isgiven in [AJ]. Both groups of interest, Spin(n) and E8 are simply-connected, a factthat removes some subtleties from the discussion.

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20 HISHAM SATI

For any integral cohomology class in H3(Spin(n);Z), there is a unique stableequivalence class of bundle gerbes [Mu, MuS] whose Dixmier-Douady class is thegiven degree three integral cohomology class. Geometrically H4(BSpin(n);Z) canbe regarded as stable equivalence classes of bundle 2-gerbes over BSpin(n), whoseinduced bundle gerbe over Spin(n) has a certain multiplicative structure. Moreprecisely, given a bundle gerbe G over Spin(n), G is multiplicative if and only if itsDixmier-Douady class is transgressive, i.e., in the image of the transgression mapτ : H4(BSpin(n);Z) → H3(Spin(n);Z) [CJMSW].

Consider a principal Spin(n)-bundle P with connection A on a manifoldM . Forthe Chern-Simons gauge theory canonically defined by a class in H4(BSpin(n);Z),there is a Chern-Simons bundle 2-gerbe Q(P,A) associated with the P which isdefined to be the pullback of the universal Chern-Simons bundle 2-gerbe by theclassifying map f of (P,A) [CJMSW].

With the canonical isomorphism between the Deligne cohomology and Cheeger-Simons cohomology, the Chern-Simons bundle 2-gerbe Q(P,A) is equivalent[CJMSW] in Deligne cohomology to the Cheeger-Simons invariant S(P,A) ∈H3(M,U(1)), which is the differential character that can be associated with eachprincipal G-bundle P with connection A [CS]. A bundle gerbe version of the dis-cussion of the invariance of the C-field in section 2.3 is provided by the following

Proposition 2.21 ([AJ] (also [CJMSW])). A Chern-Simons 2-gerbe is con-tained in the data of the C-field.

The metric torsion part of C-field. Connections with torsion come in variousclasses. Especially interesting is the case when the torsion is totally antisymmetric.In this case the new connection is metric and geodesic-preserving, and the Killingvector fields coincide with the Riemannian Killing vector fields. Connections withtorsion arise in in eleven-dimensional supergravity [E] and in heterotic string theoryand type I [Str]. In the first case an ansatz is taken such that the little c-field isproportional to the torsion tensor T ∈ Ω3(Y 11), most prominently when Y 11 is anS7 bundle over 4-dimensional anti-de Sitter space AdS4, in which case the torsionis parallelizing – see [DNP]. In the second case, the H-field in heterotic stringtheory acts as torsion, which is important for compactification to lower dimensions[Str].

3. The M-theory Dual C-Field and Fivebrane Structures

3.1. The E8 model for the dual of the C-field. In [DFM] the electriccharge associated with the C field is studied. From the nonlinear equation of motion(2.8) of the C-field, the induced electric charge that is given by the cohomology class

(3.1) [1

2G2 − I8]DR ∈ H8(Y,R) .

In [DFM] the integral lift of (3.1) is studied and denoted ΘY (C) (and also ΘY (a)),where C = (A, c).

A tubular neighbourhood of the M5-brane worldvolume V in Y is diffeomorphicto the total space of the normal bundle N → V . Let X = Sr(N) be the 10-

dimensional sphere bundle of radius r, so that the fibers of Xπ→V are 4-spheres.

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GEOMETRIC AND TOPOLOGICAL STRUCTURES RELATED TO M-BRANES 21

An 11-dimensional manifold Yr with boundary X is then constructed by removingfrom Y the disc bundle Dr(N) of radius r. Yr = Y − Dr(N), the bulk manifold,is the complement of the tubular neighbourhood Dr(N)). There are two pathintegrals or wavefunctions:

(1) The bulk C field path integral Ψbulk(CX) ∼∫exp[G∧∗G]Φ(CYr

) where the

integral is over all equivalence classes of CYrfields that on the boundary

assume the fixed value CX . This wavefunction is a section of a line bundleL on the space of CX fields.

(2) The M5-brane partition function ΨM5(CV ), which depends on the C fieldon an infinitesimally small (r → 0) tubular neighbourhood of the M5-brane.

In general, ΨbulkΨM5 is not gauge invariant and therefore it is a section of a linebundle. However, one can consider a new partition function Ψ′

M5 that is obtainedfrom multiple M5-branes stacked on top of one another instead of just a singleM5-brane. This stack gives rise to a twisted gerbe on V as follows [AJ]. In orderfor ΨbulkΨ

′M5 to be well defined, the twisted gerbe has to satisfy

(3.2) [CS(π∗(ΘX))]− [ϑijkl, 0, 0, 0] = [DH ] + [1, 0, 0, CV ] ,

where, CS(π∗(ΘX)) is the Chern-Simons 2-gerbe associated with π∗(ΘX) and achoice of connection on the E8 bundle with first Pontryagin class π∗(ΘX) (all other2-gerbes differ by a global 3-form), while [ϑijkl, 0, 0, 0] is the 2-gerbe class associatedwith the torsion class θ on V , β(ϑ) = θ, and [1, 0, 0, CV ] is the trivial Deligne classassociated with the global 3-form CV . In particular (3.2) implies

(3.3) π∗(ΘX)− θ = ξDH,

where the RHS is the characteristic class of the lifting 2-gerbe.

3.2. Twisted Fivebrane structures. Fivebrane structures are obtained bylifting String structures as follows.

Definition 3.1 ([SSS1][SSS2]). An n-dimensional manifold X has a Five-brane structure if the classifying map X → BO(n) of the tangent bundle TX liftsto the classifying space BFivebrane := BO〈9〉.

(3.4)

BO(n)〈9〉

M

f

f

BO(n) .

Theorem 3.2 ([SSS2]). (1) The obstruction to lifting a String structureon X to a Fivebrane structure on X is the fractional second Pontrjaginclass 1

6p2(TX).

(2) The set of lifts, i.e. the set of Fivebrane structures for a fixed Stringstructure in the real case, or the set of BU〈9〉 structures for a fixed BU〈7〉structure in the complex case, is a torsor for a quotient of the seventhintegral cohomology group H7(X;Z).

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22 HISHAM SATI

Twisted Fivebrane structures. Twisted cohomology refers to a cohomologywhich is defined in terms of a twisted differential on ordinary differenial forms. Thetwisted de Rham complex Ω•(X, dH2i+1

) means the usual de Rham complex butwith the differential d replaced by dH2i+1

:= d+H2i+1∧, which squares to zero byvirtue of the Bianchi identity for H2i+1, provided that H2i+1 is closed. The formon which this twisted differential acts would involve components that are 2i form-degrees apart, e.g. of the form F =

∑mn=0 Fk+2in. The main case considered in

[S6], and which corresponds to heterotic string theory, corresponds to i = 3, k = 2,and m = 1. There it was observed that, at the rational level, the (abelianized) fieldequation and Bianchi identity in heterotic string theory can be combined into anequation given by a degree seven twisted cohomology. This is discussed further insection 5. In [S6] it was also proposed that such a differential should correspond toa twist of what was there called a higher String structure and later in [SSS2] wasdefined, studied in detail and given the name Fivebrane structure.

Definition 3.3 ( [SSS3]). A β-twisted Fivebrane structure on a brane ι :M → X with String structure classifying map f : M → BO(n)〈8〉 is a cocycleβ : X → K(Z, 8) and a map c : BO(n)〈8〉 → K(Z, 8) such that there is a homotopyη between BO(n)〈8〉 and X,

Mf

ι

BO(n)〈8〉

c

X

β K(Z, 8)

η

.

The fractional class 148p2 show up in physics. As an application, which was

originally the motivation:

Theorem 3.4 ([SSS3]). (1) The dual formula for the Green-Schwarzanomaly cancellation condition on a String 10-manifold M is the ob-struction to defining a twisted Fivebrane structure, with the twist givenby ch4(E), where E is the gauge bundle with structure group E8 × E8.

(2) The integral class in M-theory dual to G4 defines an obstruction to twistedFivebrane structure, which is the obstruction to having a well-defined par-tition function for the M-fivebrane.

The precise description actually requires some refinement of the Fivebranestructure to account for appearance of fractional classes such as 1

48p2 rather than16p2. This is called F〈9〉-structure in [SSS3]. Also, a corresponding fivebrane Lie6-algebra is defined in [SSS3], where the description of the twist in terms of L∞-algebras is also given.

Higher bundles: The following applications are of interest (see [SSS1]).1. Chern-Simons 3-forms arise as local connection data on 3-bundles with connec-tion which arise as the obstruction to lifts of ordinary bundles to the correspondingString 2-bundles and are shown to be governed by the String Lie 2-algebra. For gan ordinary semisimple Lie algebra and μ its canonical 3-cocycle, the obstructionto lifting a g-bundle to a String 2-bundle is a Chern-Simons 3-bundle with charac-teristic class the Pontrjagin class of the original bundle.

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GEOMETRIC AND TOPOLOGICAL STRUCTURES RELATED TO M-BRANES 23

2. The formalism immediately allows the generalization of this situation to higherdegrees. Indeed, certain 7-dimensional generalizations of Chern-Simons 3-bundlesobstruct the lift of ordinary bundles to certain 6-bundles governed by the FivebraneLie 6-algebra. The latter correspond what was defined in [SSS1] as the Fivebranestructure, for which the degree seven NS field H7 plays the role that the degree threedual NS fieldH3 plays for the n = 2 case. Using the 7-cocycle on so(n), lifts throughextensions by a Lie 6-algebra, defined as the Fivebrane Lie 6-algebra, is obtained.Accordingly, Fivebrane structures on String structures are indeed obstructed bythe second Pontrjagin class.

3.3. Harmonic part of the dual of the C-field. Consider the equation ofmotion (2.8). Set C7 := ∗G4, so that we get

(3.5) dC7 = je,

where je is the electric current given in (2.9). Let

(3.6) Δ7g :(Ω7(Y 11), g

)−→

(Ω7(Y 11), g

)

be the Hodge Laplacian on 7-forms on Y 11 with respect to the metric g. Taking d∗

of both sides of equation (3.5) we get

Proposition 3.5. In the Lorentz gauge, d∗C7 = 0, we have

(1) Δ7gC7 = ∗jm, where jm is the fivebrane magnetic current, related to the

electric current je (2.9) by

(3.7) jm = d(∗je) .(2) C7 is harmonic if p → 0 and/or there are no fivebranes.

This is the degree seven analog of proposition 2.8. The geometry of the de Rhamrepresentatives of the Fivebrane class for the dual C-field is considered in [RS].

3.4. The integral lift. The quantization law (1.1) on the C-field leads to anintegral lift of the electric charge, the left hand side of the equation of motion (2.8),as mandated by Dirac quantization. The lift is [DFM]

[G8] =1

2

(a− 1

2λ

)(a− 1

2λ

)+ I8

=1

2a(a− λ) + 30A8 .(3.8)

Proposition 3.6. Properties of [G8].

(1) [G8] and [G4] obey the multiplicative structure on KSpin.(2) G8 defines an obstruction to having a (twisted) Fivebrane structure.

The first part is proved in [S7] and the second part in [SSS3]. Let us expanda bit on the first point. The quadratic refinement defined in [DFM] is encodedin the multiplicative structure in the K-theory for Spin bundles. Starting with areal unoriented bundle ξ, the condition w1(ξ) = 0 turns ξ into an oriented bundle,and the condition w2(ξ) = 0 further makes ξ a Spin bundle. Obviously then, areal O-bundle becomes a Spin bundle when W = 0, and so the kernel of W is the

reduced group KSpin(X). Thus W fits into the exact sequence [LD]

(3.9) 0 −→ KSpin(X) = kerW −→ KO(X)W−→H1(X;Z2)×H2(X;Z2).

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24 HISHAM SATI

Among the properties of this class proved in [DFM] is that it is a quadraticrefinement of the cup product of two degree four classes a1 and a2

(3.10) Θ(a1 + a2) + Θ(0) = Θ(a1) + Θ(a2) + a1 ∪ a2.

We would like to look at this from the point of view of the structure on the productof the cohomology groups H4( · ;Z)×H8( · ;Z). For this we consider the two classesa and Θ(a) as a pair (a,Θ(a)) in H4( · ;Z)×H8( · ;Z). Then the linearity of theaddition of the degree four classes a and the quadratic refinement property (3.10)of Θ(a) can both be written in one expression in the product H4( · ;Z)×H8( · ;Z),which makes use of the ring structure, namely

(3.11) (a1,Θ(a1)) + (a2,Θ(a2)) = (a1 + a2,Θ(a1) + Θ(a2) + a1 ∪ a2) .

The second entry on the RHS is just Θ(a1 + a2) − Θ(0), and so it encodes theproperty (3.10).

We can define the shifted class Θ0(a) as the difference Θ(a) − Θ(0), so that(3.11) is replaced by

(3.12)(a1,Θ

0(a1))+(a2,Θ

0(a2))=(a1 + a2,Θ

0(a1 + a2)),

corresponding to the special case

(3.13) Θ0(a1 + a2) = Θ0(a1) + Θ0(a2) + a1 ∪ a2.

This is then just a realization of the multiplication law on H4( · ;Z) ×H8( · ;Z)which, for (a, b) in the product group, is

(3.14) (a1, b1) + (a2, b2) = (a1 + a2, b1 + b2 + a1 ∪ a2).

Note that in order to get this law we had to use the modified eight-class Θ0(a), or

alternatively discard Θ(0) = 30A8.

We now make the connection to Spin K-theory. Similarly to the case of otherkinds of bundles, e.g. complex or real, one can get a Grothendieck group of iso-morphism classes of Spin bundles up to equivalence. The reduced KSpin group

of a topological space can be defined as KSpin(X) = [X,BSpin]. For the case ofBSpin, we will be interested in relating Spin K-theory to cohomology of degrees 4and 8. Such a homomorphism of abelian groups

(3.15) QX : KSpin(X) → H4(X;Z)×H8(X;Z)

is defined by [LD] QX (Q1(ξ), Q2(ξ)) for ξ ∈ KSpin(X), where Q1 and Q2 are the

Spin characteristic classes of [Th]. For two bundles ξ and γ in KSpin(X), and fork ≤ 3,

(3.16) Qk(ξ ⊕ γ) =∑

i+j=k

Qi(ξ) ∪Qj(γ).

Remark 3.7. The above multiplicative structure is a Z-analog (or 4k-analog)of the Z2-structure in the case of KO-theory. Given a topological space X, let

KO(X) be the reduced KO group for X and let

(3.17) W : KO(X) −→ H1(X;Z2)×H2(X;Z2)

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GEOMETRIC AND TOPOLOGICAL STRUCTURES RELATED TO M-BRANES 25

be the map W (ξ) = (w1(ξ), w2(ξ)), where wi(ξ) denotes the i-th Stiefel-Whitney

class of ξ ∈ KO(X). There is a group structure on H1(X;Z2)×H2(X;Z2) makingW a homomorphism, i.e. a map that preserves the group structure.

3.5. Invariance of the dual C-field. From (3.8) we see that we can writethe dual C-field at the level of differential forms as

(3.18) G8 =1

2G4 ∧G4 − I8 + dc7 .

3.5.1. Case I: Trivial cohomology, no one-loop term. In this case we have 12p1 =

0 = 16p2. From (3.18) we have dC7 = 1

2G4 ∧G4 + dc7. When C3 = CS3(A) we get

C7 = 12CS3(A) ∧G4.

3.5.2. Case II: 12p1 = 0. The invariance of G8 will include the invariance of the

three terms in (3.18), hence invariance of G4, of the Pontrjagin classes of S(Y 11),and of the differential form C7 ∈ Ω7(Y 11). Therefore, we have

Proposition 3.8. The dual field G8 is invariant under the following transfor-mations:

(1) The invariances of the C-field from proposition 2.7,(2) C7 → C7 + Λ7,(3) CS7(ω) → CS7(ω) + λ7,

where Λ7 and λ7 are closed differential forms in Ω7(Y 11).

As in the case for the C-field, the expressions above when recast in terms ofdifferential characters will result in requiring Λ7 and λ7 to be closed integral forms,i.e. to be in Ω7

Z(Y 11).

The Fivebrane class.

Remark 3.9. The Fivebrane class can be discussed in a manner that is verysimilar to that of the String class. The change of Fivebrane structure F can beseen from the M-theory fivebrane, similarly to the way the String structure S isseen from the membrane.

Here we assume that spacetime is ten-dimensional. We relate the dual H-field H7 to the degree seven Chern-Simons form CS7 above via a class C7 that weintroduce. Consider the principal bundle

(3.19) String(n) → STRING(M) → M,

where STRING(M) is the total space of the String(n) bundle on our ten-dimensionalspacetime M corresponding to heterotic string theory. Recall that this is one of thetwo bundles in that theory, namely the one obtained from the lift of the tangentbundle (non-gauge one). We can build a class C7 out of the Chern-Simons 7-formCS7(A) on STRING(X) for the given connection 1-form A on the gauge bundle as

(3.20) C7 = CS7(A)− π∗H7,

with dCS7 = π∗ (p2(M)) and such that∫Σ7

C7 = 1 where Σ7 is a fundamental

7-cycle in the fiber. Similarly to the Spin case [ASi], we have

Proposition 3.10. If X is 6-connected, C7 represents a generator ofH7(STRING(M);Z).

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26 HISHAM SATI

Proof. C7 is closed since

dC7 = dCS7(A)− dπ∗H7

= π∗ (p2(M))− π∗dH7

= π∗ (p2(M)− dH7)

= 0.(3.21)

Consider the homotopy exact sequence(3.22)

· · · → πi(String(n)) → πi(STRING(M)) → πi(M) → πi−1(String(n)) → · · ·corresponding to the bundle (3.19). Assuming πi(M) = 0 for i ≤ 6, the sequencegives that π7(STRING(M)) ∼= π7(String(n)) = Z. Therefore, C7 represents a gen-erator of H7 (STRING(M), Z).

Remark 3.11. If M is ten-dimensional and is 6-connected then it is topo-logically the ten-dimensional sphere. This follows from Poincare duality and thePoincare conjecture in ten dimensions. The first gives that the cohomology groupsin degrees 8 and 9 are the same as those in degrees 2 and 1, respectively, and henceare zero by 6-connectedness. M , having cohomology groups Z in degrees 0 and 10,is a homology ten-sphere. However, by the Poincare conjecture, which is a theoremin ten dimensions, M must be the sphere S10 itself.

3.6. Higher differential characters. We consider the space Map(Z,M),with Z of dimension six. The space Map(M, STRING(M)) is a bundle overMap(Z,M) with structure ‘group’ Map(M, String(10)). Let

(3.23) ev : Z ×Map(Z, STRING(M)) → STRING(M)

be the evaluation map.

Proposition 3.12. There exists a Cheeger-Simons differential 6-character B6

with dB6 = C7 and such that ev∗B6 exponentiates to a differential 0-character onMap(Z, STRING(M)) with values in U(1).

Proof. This is analogous to [ASi] where the degree two case is established.Our case corresponds to replacing B2 with B6 and the spin condition with the

String condition. For a trivial map Φ0 : Z → P0 ∈ STRING(M), let γ be a path

from Φ0 to Φ ∈ Map(Z, STRING(M)) so that γ maps the interval [0, 1] times Z to

STRING(M) with γ(1) = Φ. This path exists since πi(STRING(M)) = 0 for i ≤ 6.The function ev∗B6 will be given by

(3.24) exp

(2πi

∫γ([0,1]×Z)

C7

)= exp

(2πi

∫γ([0,1])

α1

)

with α1 is the one-form∫Zev∗C7. The function is independent of the path γ: if γ1

is another path then γ−11 γ is a map of S1 × Z → P0 and

(3.25)

∫(γ−1

1 γ)(S1×Z)

C7 =

∫γ(S1×Z)

C7 −∫γ1(S1×Z)

C7 ∈ Z,

i.e. α1 represents an integral 1-cocycle. Since C7 represents an element ofH7(STRING(M);Z), there exists a Cheeger-Simons differential 6-character B6

with dB6 = C7. Then∫Zev∗B6 exponentiates to a differential 0-character on

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GEOMETRIC AND TOPOLOGICAL STRUCTURES RELATED TO M-BRANES 27

Map(Z, STRING(M)) with values in S1.

Proposition 3.13. The curvature of the 0-character ev∗B6 is the one-formα1, which can be interpreted as a flat connection on a circle bundle overMap(Z, STRING(M)).

Remarks 3.14. 1. We consider the effect of gauge transformation on H7.Changing H7 to H7 + dβ6 leads to a shift in B6 as B6 + β6.

2. The topological term

(3.26) I =

∫Z

dx1 ∧ · · · ∧ dx6Bμ1···μ6∂1X

μ1 · · · ∂6Xμ6

can be interpreted as log∫Zev∗B6, in analogy with the case of the string.

3. Similar results can be extrapolated to eleven dimensions.

3.7. Mapping Space Description. As we recalled in the introduction, Stringstructures on a space M is related to Spin structure on the loop space LM . Is therea corresponding statement about Fivebrane structures? We do not fully answer thisquestion here but we do give possible scenarios.

In the case of String structure, the string class on the loop space LM is obtainedby pull-back of the second Chern character, ch2(E), of a bundle E on M to give abundle LE on LM via the evaluation map

(3.27) ev : S1 × LM → M,

so that the String class is∫S1 ev

∗ch2(E), where∫S1 : H∗(S1×LX;C) → H∗−1(LM ;C)

is the integration along S1.

Now the idea is to generalize this to the case of the Fivebrane structure. Wesee two directions for doing so:

(1) Replace the second Chern character ch2 by a higher degree Chern char-acter chq, q > 2, and keeping the same evaluation map (3.27). This willyield higher degree analogs of the String class, but still on the loop spaceLM .

(2) In addition, replace the circle S1 by a higher-dimensional space Y so thatthe loop space LM = [S1,M ] is replaced by a higher degree generalizationMap[Y,M ], the space of maps from Y to M .

We start with the first. Here, for a vector bundle E over M , one has the higherdegree analogs of the String class as

(3.28) Cq(LE) = −(2πi)q+1q!

∫S1

ev∗chq+1(E),

which gives a class of degree 2(q + 1) − 1 = 2q − 1 on LM . Such a generalizationof the usual String structure has been defined in [As]. Kuribayashi in [Ku] findsfairly special conditions under which it is still true that Cp(LE) = 0 if and only ifchp+1(E) = 0, namely when H∗(M ;R) is a tensor product of truncated polynomialand exterior algebras. This generalizes McLaughlin’s result [Mc] in the case p = 1for the usual String structure, where 1

2p1(P ) = 0 implies a String structure on abundle P on M only when π2(M) = 0.

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28 HISHAM SATI

An example of the higher classes (3.28) is the first term in the anomaly poly-nomial

(3.29) dH7 = 2π

[ch2(A)− 1

48p1(ω)ch2(A) +

1

64p1(ω)

2 − 1

48p2(ω)

],

where A and ω are the connections on the gauge bundle V and the tangent bundleTM , respectively. Note that we can use p = 3 in (3.28) to get a degree sevenclass upon integration over the circle

∫S1 ev

∗ch4(E), which gives 1(2π)3

14!C

3(LV ) +

decomposables + non− gauge factors.

In this case the fivebrane class can be described as follows. We have, for n ≥ 5,the isomorphism

(3.30)

∫S1

ev∗ : H8 (BString(n);Z) → H7 (LBString(n);Z) .

Since H7(BString(n);Z) = 0 and H7(String(n);Z) = Z, we have that the image in(3.30) is Z. In terms of the space itself, the evaluation map and integration overthe circle give

(3.31)

∫S1

ev∗ : H8(M ;Z) → H7(LM ;Z).

Next we consider the second case. In addition, replace the circle S1 by a higher-dimensional space Y so that the loop space LM = [S1,M ] is replaced by a higherdegree generalization Map[Y,M ]. Then what replaces the evaluation map (3.27)is ev : Y ×Map[Y,M ] → M and (3.28) would then become

∫Yev∗chp+1(E). The

result will be a class on Map[Y,M ] of degree 2p − dimY . Obviously, when p = 1and dimY = 1, we get back the String case. We will discuss further aspects of thegeneral case in section 3.8.

There are two special cases of interest, the first when Y is a torus and thesecond when Y is a sphere. Let the dimension of Y be m and that of X be n. Thenthe two cases are

• Y = Tm: This gives Map[Tm,M ] = LmM , the higher iterated loop spacesof M , i.e. LmM = LL · · ·LXn (m times). This is the iterated loop spaceof M which is obtained by looping on X m times. The bundle replacingthe loop bundle of the String case will be a bundle with structure groupthe toroidal group Map[Tm, G] = LmG.

• Y m = Sm: In this case the space to consider is Map[Sm;M ], correspond-ing to the homotopy groups πm(M) of M . Such spaces, at least for lowm, have been studied in connection to gauge theory in physics in [Mi].

In the physical situation under consideration, Y can be taken to be the spa-tial part W 5 of the fivebrane worldvolume in spacetime of dimension ten for theheterotic fivebrane and dimension eleven for the M5-brane. Then, for p = 3, theintegration over the worldvolume yields a degree three class on the fifth loop spaceL5X10 (likewise for eleven dimensions). The homotopy groups of the two groups,G and LmG are related by πn(L

mG) = πn+m(G) ⊕ πn(G), so that, in particular,π3(L

5G) = π8(G)⊕ π3(G). For example, 7-connectedness of G would be the sameas 2-connectedness of L5G.

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GEOMETRIC AND TOPOLOGICAL STRUCTURES RELATED TO M-BRANES 29

3.8. Determinantal 6-gerbes. We can look at the relation to higher gerbesby looking at the structure on the worldvolume itself. Consider the case whenthe worldvolume has five compact dimensions, i.e. time is considered to be R.The Fivebrane structure can be described in terms of a 5-gerbe in general. In thespecial case of an index 5-gerbe, this can be interpreted as the degree five part ofthe Family’s index theorem for Dirac operators [Lo] [BKTV].

The construction of the index 1-gerbe is given in [Lo] where also general featuresof the higher gerbes are given. The construction for those is given in [BKTV]which we follow below. The degree two component of the families index theoremis given by the first Chern class, or the curvature, of a determinant line bundle.There is an obstruction to realizing the component of the families index theorem ofdegree higher than two as a curvature of some geometric object. This, however, isautomatic if we choose our spacetime X to be 5-connected. so X is homeomorphicto S10, and the explicit construction is given in [Lo] in this case. The correspondingDeligne classes form a countable sets corresponding to different trivializations ofthe index bundle on the five-skeleton of the triangulation of X and are labeled by⊕2

j=1H5−2j(X;Z) = H1(X;Z)⊕H3(X;Z).

Assume Y m to be a compact oriented C∞-manifold of dimension m over whichwe have a smooth complex vector bundle E. Then the space of sections ΓE of thebundle is expected to give rise to a determinental 5-gerbe DetΓ(E), generalizing thedeterminant line bundle [BKTV]. Thus, in the case of the fivebrane, if we take thespatial part then the currently perceived wisdom leads us to a determinantal 5-gerbeand if we take the even-dimensional spacetime then we are led to a determinantal6-gerbe.

We now consider a family of fivebranes by considering the map to spacetimeq : W 5 → X10 of relative dimension 5 and a C∞ bundle E on W 5. Then thecharacteristic class of the 5-gerbe would be a class in H6(X10, C∞∗

X10) = H7(X10,Z).The class of the determinantal 6-gerbe in complex cohomology should be thought ofas a 6-fold delooping of the usual first Chern (determinantal) class. Still following[BKTV], the Real Riemann-Roch formula gives

(3.32) C1(q∗E) =

∫W 5

[ch(E) ∧ Td(TW 5)

]12

∈ H7(X10,C),

where we are integrating the degree twelve part of the index formula over thefive-dimensional spatial part of the fivebrane worldvolume to get a degree sevenclass. This involves the Dolbeault operator ∂ over a complex envelope of W 5. In

the smooth category, we just replace the Todd class Td in (3.32) with the A, theroof-genus of TW 5.

4. The Gauge Algebra of Supergravity in 6k − 1 Dimensions

Five-dimensional SO(2) supergravity on a five-dimensional Spin manifold X isthe theory obtained by coupling pure supergravity in five dimensions to an SO(2)vector multiplet [CN][C]. The former is made of a metric g on X and a Rarita-Schwinger field ψ, which is a section of the spin bundle SX coupled to the tangentbundle TX, ψ ∈ Γ(SX⊗TX). The latter contains an SO(2)-valued, hence abelian,one-form C1 with curvature two-form G2 = dC1. The Lagrangian L(5) will have

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30 HISHAM SATI

• a bosonic part L(5),bos for the bosonic fields (g,G2),• a fermionic part L(5),ferm for the fermionic field Ψ,• and an interaction part L(5),int for the terms that are mixed in the bosonic

and fermionic fields.

In L(5) = L(5),bos + L(5),ferm + L(5),int we will consider only the bosonic part,given by the five-form

(4.1) L(5),bos = R ∗1l− 1

2G2 ∧ ∗G2 −

1

6G2 ∧G2 ∧ C1,

where ∗ is the Hodge duality operator on differential forms in five dimensions, andR is the scalar curvature of the metric g of X.

Eleven-dimensional supergravity [CJS] has some common features with five-dimensional supergravity [C] [CN], described above. The bosonic field content isthe same, except that the potential C3, replacing C1, is now of degree three sothat the corresponding field strength G4 is of degree four. The Hodge dual ineleven dimensions to G4 is G7. The bosonic part of the Lagrangian is given by theeleven-form

(4.2) L(11),bos = R ∗1l− 1

2G4 ∧ ∗G4 −

1

6G4 ∧G4 ∧ C3 .

From here on we treat both theories at the same time. We thus take X to be a(6k− 1)-dimensional Spin manifold on which we define a supergravity with Chern-Simons term built out of the potential C2k−1, with corresponding field strengthG2k. The value k = 1 corresponds to the five-dimensional case and the value k = 2to the eleven-dimensional case.

The equations of motion are obtained from the Lagrangian via the variational

principle. The variationδL(6k−1),bos

δC2k−1= 0 for C2k−1 gives the corresponding equation

of motion

(4.3) d ∗G2k +1

2G2k ∧G2k = 0 .

We also have the Bianchi identity

(4.4) dG2k = 0 .

The second order equation (4.3) can be written in a first order form, by firstwriting d

(∗G2k +

12C2k−1 ∧G2k

)= 0 so that

(4.5) ∗G2k = G4k−1 := dC4k−2 −1

2C2k−1 ∧G2k ,

where C4k−2 is the potential of G4k−1, the Hodge dual field strength to G2k in6k − 1 dimensions.

The action Sbos =∫Xdvol(X)Lbos, and hence the equations of motion, are

invariant under the abelian gauge transformation δC2k−1 = dλ2k−2, where λ2k−2 isa (2k−2)-form. We can alternatively write the gauge parameter as Λ2k−1 = dλ2k−2.In fact, the first order equation (4.5) is invariant under the infinitesimal gaugetransformations

δC2k−1 = Λ2k−1, δC4k−2 = Λ4k−2 −1

2Λ2k−1 ∧ C2k−1,(4.6)

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GEOMETRIC AND TOPOLOGICAL STRUCTURES RELATED TO M-BRANES 31

where Λ4k−2 is the (4k−2)-form gauge parameter satisfying dΛ4k−2 = 0. Applyingtwo successive gauge transformations with different parameters Λi and Λ′

i, i =2k − 1, 4k − 2, for the first and second one, respectively, and forming the thecommutators gives [

δΛ2k−1, δΛ′

2k−1

]= δΛ′′

4k−2,

[δΛ2k−1

, δΛ4k−2

]= 0 ,[

δΛ4k−2, δΛ′

4k−2

]= 0 ,(4.7)

with the new parameter Λ′′4k−2 = Λ2k−1 ∧ Λ′

2k−1. Note that the transformationsare nonlinear, and this can be tracked back to the presence of the Chern-Simonsform in the Lagrangian (4.1).

We now introduce generators v2k−1 and v4k−2 for the Λ2k−1 and Λ4k−2 gaugetransformations, respectively. On the generators, from the commutation relations(4.7), we get the graded Lie algebra

v2k−1, v2k−1 = −v4k−2 ,

[v2k−1, v4k−2] = 0 ,

[v4k−2, v4k−2] = 0 ,(4.8)

Note that we can use a graded commutator, which unifies a commutator and ananticommutator, so that the above algebra (4.8) becomes

[v2k−1, v2k−1] = −v4k−2 ,

[v2k−1, v4k−2] = 0 ,

[v4k−2, v4k−2] = 0 ,(4.9)

where it is now understood that we are using graded commutators. The generatorssatisfy the following properties

(1) The generators v2k−1 and v4k−2 are constant: dv2k−1 = 0 = dv4k−2.(2) The grading on the generators v2k−1 and v4k−2 follow that of the potentials

A2k−1 and A4k−2, respectively. Hence, v2k−1 is odd and v4k−2 is even.Thus, d(v2k−1α) = −v2k−1dα and d(v4k−2α) = v4k−2dα, for any α.

We will think of these “generators” vi as elements of a graded Lie algebra, where wewill write C2k−1⊗v2k−1, etc. instead of just C2k−1 for the fields (see the discussionaround equation (4.26)).

The field strengths can be combined into a total uniform degree field strengthG by writing

(4.10) V = eC2k−1⊗ v2k−1eC4k−2⊗ v4k−2 ,

so that

G = dC2k−1 ⊗ v2k−1 + (dC4k−2 −1

2C2k−1 ∧ dC2k−1)⊗ v4k−2

= G2k ⊗ v2k−1 +G4k−1 ⊗ v4k−2

= G2k ⊗ v2k−1 + ∗G2k ⊗ v4k−2 .(4.11)

Note the analogy with usual (i.e. not higher-graded) nonabelian gauge theory. Vis the analog of g and G = dVV−1 is the analog of dgg−1.

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32 HISHAM SATI

The equation of motion for C2k−1 (= Bianchi identity for C4k−2) and theBianchi identity for C2k−1 are obtained together from

(4.12) dG − G ∧ G = −dV ∧ dV−1 − dVV−1 ∧ dVV−1 = 0 .

Indeed, using the commutators (4.9), we have

G ∧ G = (G2k ⊗ v2k−1) ∧ (G2k ⊗ v2k−1) + (G2k ⊗ v2k−1) ∧ (∗G2k ⊗ v4k−2)

+(∗G2k ⊗ v4k−2) ∧ (G2k ⊗ v4k−2)

=1

2[G2k ⊗ v2k−1, G2k ⊗ v2k−1] + [G2k ⊗ v2k−1, ∗G2k ⊗ v4k−2]

=1

2G2k ∧G2k ⊗ [v2k−1, v2k−1]−G2k ∧ ∗G2k ⊗ [v2k−1, v4k−2]

= −1

2G2k ∧G2k ⊗ v4k−2 .(4.13)

Hence, (4.12) follows from the equation of motion (4.3) and the Bianchi identity(4.4), which indeed correspond, respectively, to the coefficient of v2k−1 and v4k−2

in the expression for dG. The case k = 2, corresponding to eleven-dimensionalsupergravity, was derived in [CJLP].

4.1. Models for the M-Theory Gauge Algebra. In the previous sectionwe have seen that G4 and its dual can be written in terms of the total uniformdegree field strength G, the generators in which satisfy an algebra. It is naturalto ask about the nature of the generators and the graded structure in which theyresult. In this section we provide a description in terms of homotopy (or higher-categorical) Lie algebras: L∞-algebras based on the constructions in [SSS1], andsuperalgebras corresponding to (1|1) supertranslations.

4.2. The gauge algebra as an L∞-algebra. One connection to L∞-algebrasis the appearance of higher form abelian Chern-Simons theory. Recall that for g

any semisimple Lie algebra and μ = 〈·, [·, ·]〉 the canonical 3-cocycle on it, we callgμ the corresponding (skeletal version of the) stringμ(g) Lie 2-algebra. Similarly,for g any semisimple Lie algebra and μ7 the canonical 7-cocycle on it, we call gμthe corresponding (skeletal version of the) Fivebrane Lie 6-algebra [SSS1] [SSS3].

Reminder on L∞-algebra valued diferential forms. Recall from [SSS1] thatfor g any L∞-algebra with CE(g) its Chevalley-Eilenberg differential graded com-mutative algebra (DGCA), the space of GCA-morphisms CE(g) → Ω(X) is iso-morphic to the degree zero elements in the graded vector space Ω•(X)⊗ g, whereg is in negative degree. Flat L∞-algebra valued forms can be realized as gradedtensor products A ∈ Ω•(Y )⊗g of forms with L∞-algebra elements with the specialproperty that

• A is of total degree 0 ,• A satisfies a flatness constraint of the form

(4.14) dA+ [A ∧A] + [A ∧A ∧A] + · · · = 0 ,

where d and ∧ are the operations in the deRham complex and where[·, · · · , ·] are the n-ary brackets in the L∞-algebra.

It is usually more convenient to shift g by one into non-positive degree (hence withthe usual Lie 1-algebra part in degree 0) and accordingly take A to be of totaldegree 1.

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GEOMETRIC AND TOPOLOGICAL STRUCTURES RELATED TO M-BRANES 33

Recall one of the central constructions of [SSS1] involving the Weil algebraW(g). For g any L∞-algebra with degree (n+ 1)-cocycle μ that is in transgressionwith an invariant polynomial P

(4.15) 0 P P

μdCE(g)

csdW(g)

CE(g) W(g) W(g)basic

we can form the String-like extension Lie n-algebra gμ and the corresponding Chern-Simons Lie (n+1)-algebra csP (g) with the property that W(gμ) CE(csP (g)). In[SSS1] this construction was of interest for the case that μ was a nontrivial cocycleon a semisimple Lie 1-algebra. Another interesting case in which the constructionworks is when an invariant polynomial P suspends to 0, i.e. if it is in transgressionwith the 0-cocycle μ = 0.

Notice that in particular all decomposable invariant polynomials P = P1 ∧P2, for P1 and P2 nontrivial and with transgression elements csi, dW(g)csi = Pi ,suspend to 0, since for them we can choose the Chern-Simons element cs = cs1∧P2,which vanishes in CE(g) because P2 does, by definition:

(4.16) 0 P1 ∧ P2 P1 ∧ P2

0dCE(g)

cs1 ∧ P2

dW(g)

CE(g) W(g) W (g)basic

.

Higher abelian Chern-Simons forms. A very simple but useful example arethe decomposable invariant polynomials on shifted u(1) in an even number of shifts:b2k−2u(1), for k any positive integer. In this case

(4.17) CE(b2k−2u(1)) = (

•∧( 〈c〉︸︷︷︸2k−1

), d = 0)

and

(4.18) W(b2k−2u(1)) = (•∧( 〈c〉︸︷︷︸2k−1

⊕ 〈g〉︸︷︷︸2k

), dc = g, dg = 0) .

The invariant polynomials are all the wedge powers g, , g ∧ g, g ∧ g ∧ g of the singleindecomposable one P := g

(4.19) inv(b2k−2u(1)) = (•∧( 〈P 〉︸︷︷︸

2k

), dP = 0) .

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34 HISHAM SATI

Notice that in this case the CE-algebra of the “String-like extension” b2k−2u(1)μ=0

is that of b2k−2u(1)⊕ b4k−1u(1):

(4.20) CE(b2k−2u(1)μ=0) = CE(b2k−2u(1)⊕ b4k−3u(1)) .

Abelian Chern-Simons L∞-algebras. Let k ∈ N be a positive integer. Then theLie (2k − 2)-algebra b2k−2u(1) has a decomposable degree 4k invariant polynomialP4k which is the product of two copies of the standard degree 2k-polynomial. Thecorresponding Chern-Simons Lie (4k− 1)-algebra csP4k

(b2(k−1)u(1)) is given by theChevalley-Eilenberg algebra of the form

(4.21) CE(csP4k(b2k−2)u(1))) =

( •∧(〈c2k−1, g2k, c4k−2, g4k−1〉), d

)

where

dc2k−1 = g2k ,

dc4k−2 = c2k−1 ∧ g2k + g4k−1 ,

dg2k = 0 ,

dg4k−1 = g2k ∧ g2k .(4.22)

This has a canonical morphism onto

(4.23) CE(b2k−2u(1)⊕ b4k−3u(1)) =

( •∧(〈c2k−1, c4k−2〉), d = 0

)

with respect to which we can form the invariant or basic polynomials

(4.24) CE(b2k−2u(1)⊕ b4k−3u(1)) CE(csP4k(b2k−2)u(1)))

i∗ basic(i∗) .

This is the DGCA

(4.25) basic(i∗) =

⎛⎜⎝

•∧(〈g2k〉︸ ︷︷ ︸

2k

⊕〈g4k−1〉︸ ︷︷ ︸4k−1

), (dg2k = 0 , dg4k−1 = g2k ∧ g2k)

⎞⎟⎠ .

We consider the L∞-algebra sa that admits the above as its Chevalley-Eilenbergalgebra. This sa is a graded Lie algebra with generators v3 and v6 in degree 3 and6, respectively, and with the graded Lie brackets being

[v3, v3] = v6

[v3, v6] = 0

[v6, v6] = 0 .(4.26)

Flat differential form data with values in this L∞-algebra is given by a degree1-element A = G4 ⊗ v3 +G7 ⊗ v6 ∈ Ω•(X)⊗ g, where

• G2k a closed 2k-form;• G4k−1 is a 4k − 1-form satisfying dG4k−1 = G2k ∧G2k.

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GEOMETRIC AND TOPOLOGICAL STRUCTURES RELATED TO M-BRANES 35

In total we have that a Cartan-Ehresmann connection with respect to i∗ is givenby differential form data as follows:(4.27)

Ω•vert(Y ) CE(b2k−2u(1)⊕ b4k−3u(1))

Avert

Ω•(Y )

CE(csP4k(b2k−2)u(1)))

i∗

(A,FA) G2k = dC2k−1

G4k−1 = dC4k−2 + C2k−1 ∧G2k

Ω•(X)

basic(i∗)

dG2k = 0dG4k−1 = G2k ∧G2k

.

This can be regarded as a certain Cartan-Ehresmann connection for the product ofa line (2k − 1)-bundle and a line 4(k − 2)-bundle

The situation for 11-dimensional supergravity. The local gauge connectiondata of 11-dimensional supergravity is given by a 3-form C3 with curvature 4-formG4 = dC3 which can be captured in a duality-symmetric manner by regardingC3 as the data giving a flat connection with values in the abelian Chern-SimonsLie 6-algebra obtained by setting k = 2, subject to a self-duality constraint: A =G4 ⊗ v3 + (∗G4)⊗ v6. The flatness condition satisfied by this is then equivalent tothe equations of motion for G4

(4.28) (dA+ [A ∧A] = 0) ⇔

dG4 = 0d ∗G4 = − 1

2G4 ∧G4

Therefore,

Theorem 4.1. The C-field and its dual in M-theory define an L∞-algebra astheir gauge algebra.

4.3. The gauge algebra as a Superalgebra. We next interpret the M-theory gauge algebra in another novel way. The commutator of v3 with v6 andthat of v3 and v6 are zero. Furthermore, the commutator of two v3’s gives v6, itis natural to suspect that each one of the two generators belongs to a differentsubspace in some grading. Indeed, these are the even and odd gradings, and wehave

Proposition 4.2. The generators v3 and v6 form a Lie superalgebra of trans-lations in (1|1) dimensions.

Remark 4.3. This is analogous to the generator ∂∂θ + θ ∂

∂x , where x is an evencoordinate and θ is an odd coordinate. Thus we see an analog of the supersymmetricquantum mechanical relation Q,Q = H, where Q is the supercharge and H isthe Hamiltonian of the system.

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36 HISHAM SATI

5. Duality-Symmetric Twists

In the twisted cohomology setting one can form uniform degree expressions forboth the fields, e.g. the cohomology classes, and the twisted differential. In thissection we consider twists of degrees higher than the familiar three. Also, given thediscussion of uniform degree fields in the previous section, it is natural to use thesefor more exotic twists.

5.1. Degree seven twists. The NS H-field in type II string theory serves asthe twist in the K-theoretic classification of the RR fields. This involves unifyingthe fields of all degrees into one total RR field. We investigate, based on [S6], thecase of heterotic string theory where there are E8 ×E8 or Spin(32)/Z2 gauge fieldsin addition to the H-field.

a. Rationally: Considering the gauge field and its dual as a unified field, theequations of motion at the rational level contain a twisted differential with a noveldegree seven twist. Consider the case where the Yang-Mills group G, is brokendown to an abelian subgroup, thus making the curvature F2 be simply dA. ThemanifoldsM10 are chosen such that this breaking via Wilson lines (= line holonomy)is possible. The result of the variation of the action S =

∫H3∧∗H3+

∫F2∧∗F2 with

respect to A gives (d−H7∧)F = 0, where F = F2+∗F2 is defined as the combinedcurvature, and H7 is equal to ∗H3 at the rational level, in analogy with the RRfields. In this analysis, following [S6], we used the “Chapline-Manton coupling”H3 = CS3(A), where CS3(A) is the Chern-Simons three-form for the connectionA, whose curvature is F2. This gives a twisted differential dH7

= d−H7∧ which innilpotent, i.e. squares to zero, d2H7

= 0, since H7 is closed.

(i) Let vn denote the nth generator of the complex oriented cobordism ring. Con-sider the case n = 2 and let R = R[[v2, v

−12 ]] be a graded ring. The generator

vn has dimension 2pn − 2, so that at the prime p = 2, v2 has dimension 6. LetdH7

= d − v−12 H7 be the twisted de Rham differential of uniform degree one. De-

note by ΩidH7

(M10;R) the space of dH7-closed differential forms of total degree i on

M10. The total curvature F is an element of degree two, i = 2, in the above spaceof forms. The equation dH7

F = 0 defining the complex is just the Bianchi identityand the equation of motion of the separate fields. Another possibility is to use thecombination F = u−1

1 F2+u−12 F8, where u1 has degree two and u2 has degree eight.

(ii) The second step is to ask whether the argument at the level of rational co-homology generalizes to some rational generalized cohomology theory. A twistingof complex K-theory over M is a principal BU⊗-bundle over M . From BU⊗ ≡K(Z, 2) × BSU⊗, the twisting is a pair τ = (δ, χ) consisting of a determinantaltwisting δ, which is a K(Z, 2)-bundle over M and a higher twisting χ, which isa BSU⊗-torsor. Twistings are classified, up to isomorphism, by a pair of classes[δ] ∈ H3(M,Z) and [χ] in the generalized cohomology group H1(X,BSU⊗). Theformer is twisted K-theory, where the twist is given by the Dixmier-Douady (DD)class. The twistings of the rational K-theory of X are classified, up to isomorphism,by [Te] the group

∏n>1 H

2n+1(M ;Q). This shows that, in addition to the usual

H3Q-twisting, one can in principle have twistings from H5Q and H7Q etc. It isthus possible that a degree seven twist comes from complex K-theory. However, itis not obvious how to isolate just the degree seven part from the tower of all n > 3odd-dimensional twists.

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GEOMETRIC AND TOPOLOGICAL STRUCTURES RELATED TO M-BRANES 37

b. Integrally: The above generalizes the usual degree three twist that lifts totwisted K-theory and raises the natural question of whether at the integral levelthe abelianized gauge fields belong to a generalized cohomology theory.

(i) The appearance of the higher degree generator connects nicely with the dis-cussion in [KS1][KS2] [KS3] on generalized cohomology in type II (and to someextent type I) string theories. Further the appearance of the w4 = 0 conditionin [KS1], interpreted in [S4] in the context of F-theory, which is a condition inheterotic string theory, is another hint for the relevance of generalized cohomol-ogy in the heterotic theory. Given the appearance of elliptic cohomology throughW7 = 0 and the H7-twist, then the condition W7+[H7] = 0 is expected to make anappearance, which would give rise to some notion of twisted structure in a similarway that the analogous condition W3 + [H3] = 0 of [FW] amounts to a twistedSpinc structure. This structure that we seek would be related to a twisted Stringstructure, but is not quite the same but is implied by it, since the String orientationcondition implies W7 = 0 via the action of the operation Sq3. Further, we expectthe modified condition to correspond to a differential d7 in the AHSS of twistedgeneralized theories, possibly Morava K-theory and elliptic cohomology, since the‘untwisted’ differential is the first nontrivial differential there, in analogy to the‘twist’ for d3 generated by [H3] in the K-theory AHSS.

Conjecture 5.1. The cohomology class W7+[H7] corresponds to a differentialin twisted Morava K-theory and twisted Morava E-theory, where [H7] acts as thetwist.

This is a generalization of the statement that W3 + [H3] corresponds to thedifferential d = Sq3 + [H3]∪ in twisted K-theory. Of course the construction ofsuch twisted generalized theories is not yet established. Nevertheless we note thefollowing.

(ii) In general, twists of a cohomology theory E are classified by BGL1(E), i.e. thetwisted forms of E∗(X) correspond to homotopy classes of maps [X,BGL1(E)]. Anequivalent way of saying this, which the more familiar one in the context of K-theory,is that the twists are classified by BAut(E), where Aut(E) is the automorphismgroup of E. The homotopy groups of BGL1(E) are given as units of the ringE0(pt) in degree 1, and as Ek−1(pt) in degree k > 1. For K-theory, K0(pt) = Z

gives π0BGL1(K) = Z/2, K2(pt) = Z gives π3BGL1(K) = Z, which is detected bya map K(Z, 3) → BGL1(K) giving the standard degree twist. In addition, there isπ7BGL1(K) = Z.

(iii) Another possibility is the following. Twists of TMF are classified byBGL1(TMF ). This may have nontrivial homotopy in degree 7 coming from thehomotopy in degree 6 of the connective theory tmf . 2

5.2. Duality-symmetric twists in ten-dimensional string theory. Intype II string theory one encounters the Ramond-Ramond (RR) fields F =

∑i Fiu

−i,where u is the Bott generator and i is the degree of the RR field, which is even fortype IIA and odd for type IIB. This satisfies the equation dH3

F = 0, where dH3

2The homotopy groups of the spectrum tmf and those of its periodic version TMF arerelated as π∗(TMF ) = π∗(tmf)

[(Δ24)−1

], where Δ is the discriminant of elliptic curves.

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38 HISHAM SATI

is the twisted differential, whose uniform degree expression is dH3= d + u−1H3∧.

Here H3 is the Neveu-Schwarz (NS) 3-form. This is explained very well in [Fr2].

As we saw in section 5.1, in [S6] a degree seven twist was uncovered in heteroticand type I string theory, where the twist is given by the dual H7 of the usual NSH-field H3 in ten dimensions. The differential is of the form dH7

= d + v−1H7∧.In this theory, for instance for v = u3, one can form the uniform total degree onefield strength [Fr2]

(5.1) H = u−1H3 + u−3H7 ,

with corresponding potentials, or B-fields, of total degree zero B = u−1B2+u−3B6.

Given that the total field strengths are built of more than one component, wecan ask whether the corresponding differential of uniform degree might be built outof a twist that has more than one component. Consider a candidate twisted deRham differential with an expression of the form

(5.2) dH = d+ u−1H3 ∧+u−3H7 ∧ .

The square is d2H contains the terms that are zero because dH3and dH7

are differ-entials. In addition, there is the cross-term u−4(H3 ∧H7 +H7 ∧H3), which is zeroby antisymmetry of the wedge product. Of course, another way to immediately seethis is to write dH as dH3

+ u−3H7∧ or as dH7+ u−1H3∧. Thus one can build a

twisted graded de Rham complex out of such a differential.

Remarks. 1. In fact, one can build a differential by adding to dH all expressionsof the form u−iH2i+1∧, i.e.

(5.3) d′H = d+∞∑i=0

u−iH2i+1 ∧ .

2. As differential forms, the u are constant, i.e. du = 0. We can conceive of twomodifications of this: First consider the generators appearing in front of the H2i+1

to be independent. For example, in [S6], instead of (5.1), we used the expression 3

(5.4) H = v−1(1)H3 + v−1

(2)H7 ,

where v(1) is still the Bott generator and v(2) is the generalization of that generator,i.e. identified as coming from a complex-oriented generalized cohomology theoryat the prime p = 2. In this case it still holds that v(1) and v(2) are constants asdifferential forms.

From the above discussion the following immediately follows.

Proposition 5.2. There is a twisted graded de Rham complex with differentiald+∑∞

i=1 v−1(i)H2i+1∧ , provided the differential forms H2i+1 are closed. The coeffi-

cients v(i) are constant as differential forms and can be taken to be either dependentor independent.

3In section 4 we used vi to indicate a generator of degree i, so to make a distinction weare using the notation v(i) to indicate a generator of level i in complex-oriented generalizedcohomology. We understand that the first notation is more standard for the second notion, butsince this is the only occurrence of the higher Bott generators, then we hope it will not cause aconfusion.

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GEOMETRIC AND TOPOLOGICAL STRUCTURES RELATED TO M-BRANES 39

One can associate analytic torsion [MWu] to this type of twisted de Rham complexand a spectral sequence for the corresponding twisted cohomology [LLW].

5.3. Duality-symmetric twists in eleven-dimensional M-theory. In M-theory the situation is much more interesting. In this case we will see that we canhave twists of even degree and the generators are not independent in the sense thatthey satisfy relations. Some aspects of this discussion have been observed in [S1][S2] [S3].

In the low energy limit of M-theory, in addition to the metric and the gravitino,there is the C-field C3 with field strength G4. We can build a differential with G4

a twist as follows. The square of the expression dG4= d+ v3G4∧ is 4

(5.5) d2G4= d2 + d(v3G4∧) + v3G4 ∧ d+ v3G4 ∧ v3G4 ∧ .

On the right hand-side of (5.5), the first term is always zero since the bare d is thede Rham differential. For the second term we need to decide whether v3 is even orodd as a differential form. Since G4 is even we see that we have to choose v3 to beodd in order to cancel the third term. In addition, for the left-over from the secondterm to be zero, G4 has to be closed. The last term has no other term againstwhich to cancel, so it has to be zero by itself. We need v3 to be idempotent. Thiscan be achieved either by the fact that the form degree is odd or by the strongercondition that it squares to zero, i.e. that it is a Grassmann variable. The abovediscussion generalizes in an obvious way to the case when the coefficient has degree2i− 1 and the field has degree 2i. Therefore we have

Proposition 5.3. The de Rham complex can be twisted by a differential of theform d + v2i−1G2i∧ provided that G2i is closed and v2i−1 is Grassmann algebra-valued.

In M-theory one can consider the field dual to the C-field. This is a fieldstrength G7, which at the rational level is Hodge dual to G4. We can use G7

to twist the de Rham differential in the same way that H7 did. Furthermore, inthe same way as in (5.4) one could form a duality-symmetric uniform degree fieldstrength G = v−1

3 G4 + v−16 G7. This expression can now be used to twist the de

Rham differential, leading to

(5.6) dG = d+G∧ = d+ v−13 G4 ∧+v−1

6 G7 ∧ .

The conditions for (5.6) to be a differential are given in the following.

Proposition 5.4. The de Rham complex can be twisted by the differential dGprovided that either

(1) dG7 = 0 and v23 = 0, or(2) v3, v3 = v6 and dG7 = − 1

2G4 ∧G4.

Furthermore, this differential on G is equivalent to the equations of motion and theBianchi identity of the C-field.

4In this section we have suppressed the tensor product between generators and fields for easeof notation.

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40 HISHAM SATI

Remarks 5.5. 1. The first case would hold when there is no Chern-Simonsterm in the M-theory action.2. The second case arises in M-theory and realizes the equation of motion forthe C-field. While this is what appears in M-theory, mathematically we can havecombination of even and odd fields of any degrees

(5.7) d+ v−12m−1G2n + v−1

2mG2m+1 .

3. The generators in proposition 5.4 have appeared in [CJLP] in the context ofthe M-theory gauge algebra (which is generalized in section 4). What we have doneabove is relate them to twisted cohomology.

6. M-brane Charges and Twisted Topological Modular Forms

6.1. Evidence for TMF.

6.1.1. Construction of anomaly-free partition functions. The E-theoretic par-tition function in type IIA.

The K-theoretic partition function encounters an anomaly [DMW], given bythe seventh integral Stiefel-Whitney class W7, whose cancellation [KS1] is the ori-entation condition in elliptic cohomology for Spin manifolds, and in second integral

Morava K-theory at p = 2 K(2) (cf. [Mor]) for oriented manifolds. The aboveclass W7 is the result of applying the Steenrod square operation Sq3 on w4, thefourth Z/2 Stiefel-Whitney class or, equivalently, the result of applying the Bock-stein operation β = Sq1 on the degree six class Sq2w4, W7 = Sq3(w4) = βSq2(w4),by the Adem relation Sq3 = Sq1Sq2.

Theorem 6.1 ([KS1]). (1) A 10-manifold X is orientable with respect

to K(2) iff W7(X) = 0.

(2) The M-theory partition function is anomaly-free when constructed on K(2)-orientable spaces.

Similar results hold also for Morava E(2)-theory. In [KS1] an elliptic refinementof the mod 2 index j is obtained. Assuming that X is orientable with respect to areal version EO(2) of E(2)-theory, there is an EO(2)-orientation class [X]EO(2) ∈EO(2)10(X). Now for x ∈ E0(X), the class xx lifts canonically to EO(2)0(X), soj(x) = 〈xx, [X]EO(2)〉 ∈ EO(2)10, the right hand side being EO(2)10 = Z/2[v3(1)v

−1(2) ]

by [HK1]. The assumption on EO(2)-orientation is made precise:

Theorem 6.2 ([KS1]). (1) A spin manifold X is orientable with respectto EO(2) if and only if it satisfies w4(X) = 0, where w4 is the fourthStiefel-Whitney class.

(2) When w4 = 0, this uncovers another anomaly to the existence of an ellipticcohomology partition function.

Remark 6.3. The class w4 is the mod 2 reduction of the integral class λ = 12p1,

so that the vanishing of λ implies the vanishing of w4, which in turn implies thevanishing of W7. Therefore, the String orientation condition λ = 0 is a necessarycondition for the cancellation of the DMW anomaly.

There is a “character map” E → K[[q]][q−1] where q is a parameter of dimension 0,with K[[q]] a product of infinitely many copies of K, and the notation [q−1] signifiesthat q is inverted. The map is determined by what happens on coefficients [AHS].

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GEOMETRIC AND TOPOLOGICAL STRUCTURES RELATED TO M-BRANES 41

Theorem 6.4 ([KS1]). The refined partition function is a one-parameter fam-ily of theta functions. At the prime p = 2, v3(1)v

−1(2) is of dimension zero and serves

as the expansion parameter q.

The E-theoretic partition function in type IIB. The construction of thepartition function in the IIB case is analogous to that of type IIA. Instead ofK1(X), one starts with E1(X) where E is a complex-oriented elliptic cohomol-ogy. The construction proceeds precisely analogously as in the K1(X) case. How-ever, the discussion of the phase is delicate. First there is the pairing in E1(X):E1(X) ⊗ E1(X) → E2(X) → E−8 = E0 where the second map is capping withthe fundamental orientation class in E10(X). To construct a θ-function, a qua-dratic structure is needed, which amounts to considering real elliptic cohomology:A product of an x ∈ E1(X) with itself can be given a real structure, which givesrise to an element of ω(x) ∈ ER1+αX, which when capped with the fundamentalorientation class in ER10(X) gives an element in Eα−9. This is a Z/2-vector spacegenerated by the classes v3n−1

(1) v2−n(2) σ−4a2, n ≥ 1[HK1]. Therefore,

Theorem 6.5 ([KS2]). There is a quadratic structure depending on one freeparameter, which leads to a precise IIB analog of the θ-function constructed for IIAusing real elliptic cohomology in [KS1].

TMF and the type IIB fields. A particularly convenient combination of thetwo degree-three fields in type IIB string theory is G3 = F3 − τH3, where τ is theparameter on the upper half plane. This is a field with modular weight −1 since ittransforms as G′

3 = G3 · (cτ + d)−1 under τ ′ = (aτ + b)/(cτ + d). In tmf , a classof modular weight k appears in tmf2k(X10). Therefore,

Proposition 6.6 ([KS3]). The fields of type IIB string theory as elements in

tmf, satisfy G3 ∈ tmf−2X10.

This points to the 12-dimensional picture: suppose, in the simplest possiblephysical scenario [V] that V 12 = X10 × E where E is an elliptic curve, thenG3 × μ ∈ tmf0(V 12) where μ ∈ tmf2(E) is the generator given by orienta-tion. It is consistent that the class ends up in dimension 0 and no odd numbershows up. Modular classes of weight 0, however, must be in dimension 0. Themathematical interpretation of τ appears only when we apply the forgetful mapEk(X) → Kk(X)[[q1/24]][q−1/24] with q = exp(2πiτ ). In fact, it is necessary togeneralize to an elliptic cohomology theory E which is in general modular onlywith respect to some subgroup Γ ⊂ SL(2,Z). For forms with such modularity,fractional powers of q are needed: in the case of complex-oriented cohomology, oneencounters q1/24. The map E → K[[q1/24]][q−1/24] whose induced map on coeffi-cients (homotopy groups) makes the k-th homotopy group modular of weight k/2

is not the correct normalization to use because then E0(Sk) = E−k(pt) wouldhave modular weight −k/2 and not 0. The correct normalization is given bycomposition with Adams operations (or alternatively with Ando operations [A])ψη : K[[q1/24]][q−1/24] → K[[q1/24]][q−1/24], where η is the Dedekind function(Δ1/24 where Δ is the discriminant form), which is a unit in K[[q1/24]][q−1/24].For general k, multiplication by ηk is needed.

More on elliptic curves and F-theory. Roughly speaking, there is an ellipticcohomology theory for every elliptic curve. There is no universal elliptic curve over

221

42 HISHAM SATI

a commutative ring, so on what basis should one ‘favor’ one elliptic curve over theother, and hence one elliptic cohomology over the other? If one considers all ellipticcurves at once then the corresponding generalized cohomology theory is tmf , whichis then, in a sense, the universal ‘elliptic’ cohomology theory. However, the price topay is that tmf is not an elliptic cohomology theory and also not even-periodic. Formore on this see e.g. [Lu]. In our identification of the F-theory elliptic curve withthe elliptic curve in elliptic cohomology (after suitable reduction of coefficients),we hence, along the lines of [S4], expect elliptic curves with a fixed modulus in F-theory to correspond to elliptic cohomology while ones with a modulus parametervarying in the base of the elliptic fibration, i.e. families, to correspond to tmf . It isthen tempting to propose that tmf sees all possible compactifications of F-theoryon an elliptic curve, i.e. all admissible elliptic fibrations.

S-duality and twisted K-theory are not compatible. Type IIB string theoryhas a duality symmetry, S-duality, which is analogous to electric/magnetic dualityin gauge theory. In the presence of H3, the description of the RR fields of type IIBusing twisted K-theory is not immediately compatible with S-duality. The origin ofthis is that the RR fields are considered as elements of K-theory while the S-dualfield H3 is taken to be a cohomology class, leading to the breaking of the symmetry.Furthermore, type IIB string theory has a five-form in place of the four-form in typeIIA and in M-theory. In [DMW] the apparent puzzle about the incompatibilityof twisted K-theory and S-duality is raised. A definite statement is proved in[KS2]. The condition for anomaly cancellation for F3 is (Sq3 + H3) ∪ F3 = 0,which is not invariant under the full SL(2,Z) group. The direct SL(2,Z)-invariantextension of the above equation is [DMW] F3 ∪ H3 + βSq2(F3 + H3) = 0. Oneimmediate question is that of justification (and interpretation) of the nonlinear termβSq2H3 = H3 ∪H3. The point is to exhibit this as a differential, or obstruction forthe cohomological pair (H3, F3) to lift to the theory.

The usual requirement that twistedK-theory be a module over K-theory, whichforces the ‘structure group’ of the bundle of K-theories in question to be the mul-tiplicative infinite loop space GL1(K) of K-theory, violates the condition. One canthen ask for some further generalized twisting, where, for a particularH3, the choiceof allowable F3’s would not form a vector space, i.e. whether one could consider aform of twisted K-theory which is not a module cohomology theory over ordinaryK-theory. In [KS2] this is shown not to exist if the twisting space is K(Z, 3). Theclassifying space, i.e. a topological space B such that the affine-twisted K1-groupwould be classified by homotopy classes of maps X → B, cannot exist because the‘group cohomology’ H2(CP∞, BU) vanishes. The above equation cannot occur asfirst Postnikov invariant, so

Theorem 6.7 ([KS2]). K(Z, 3)-twisted K-theory is not compatible with S-duality in type IIB string theory.

4. Re-interpreting the twist: Due to the above, one has to seek a solution inthe realm of higher generalized cohomology theories. However, a question ariseswhether or not to leave the twist. The twist introduces an intrinsic non-commutativitywhich seems to prevent further delooping of the theory into a modular second co-homology group, giving an indication for a an untwisted generalized cohomologytheory.

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GEOMETRIC AND TOPOLOGICAL STRUCTURES RELATED TO M-BRANES 43

There is a map K(Z, 3) → TMF coming from the String orientation. A twistof K-theory of the form X → K(Z, 3) then gives rise to an element of TMF (X)by composition X → K(Z, 3) → TMF , in fact defining “elliptic line bundles” (cf.[D]). Let us explain this. The classifying space BGL1K for elementary twistingsof complex K-theory splits, as an infinite loop space, as a product of two factorsA× B. The first factor is a K(Z, 3) bundle over K(Z/2, 1) which splits as a spacebut has nontrivial infinite loop structure classified by Sq3 ∈ H3(H(Z/2);Z). Thereis a natural infinite loop map B → TMF from B to the representing space fortopological modular forms, and so by projecting through B a map BGL1K →TMF . In particular an elementary twisting of K-theory for X determines a TMF -class on X [D]. The geometric interpretation of these TMF classes is simplified ifrestricted to those classes coming from twistings involving only the K(Z, 3) factorof B. Such a twisting is determined by a map X → K(Z, 3) or equivalently by aBS1 bundle on X. This bundle can be thought of as a stack 5 locally isomorphic tothe sheaf of line bundles on X and hence as a 1-dimensional 2-vector bundle on X.In this sense the TMF classes coming from K-theory twistings can be viewed as1-dimensional elliptic elements and twisted K-theory as K-theory with coefficientsin this “elliptic line bundle” [D].

Therefore, what looks like twisting to the eyes of K-theory, untwists and be-comes merely a multiplication by a suitable element in TMF or any suitable formof elliptic cohomology. Thus,

Observation 6.8 ([KS2]). If both F3 and H3 are viewed as elements of ellipticcohomology, i.e. symmetrically, and the twisting is replaced by multiplication thenthe S-duality puzzle is solved.

6.2. Review of D-brane charges and twisted K-theory. The analogywith the more familiar case of the NS field H3 is as follows. The cohomology class[H3] appears in the definition of a twisted Spinc-structure

(6.1) W3 + [H3] = 0,

a condition for consistent wrapping of D-branes around cycles in ten-dimensionalspacetime [FW], where W3 is the third integral Stiefel-Whitney class of the normalbundle, the vanishing of which allows a Spinc-structure. In the presence of the NSB-field, or its field strength H3, the relevant K-theory is twisted K-theory, as wasshown in [W6] [FW] [Ka] by analysis of worldsheet anomalies for the case theNS field [H3] ∈ H3(X,Z) is a torsion class, and in [BM] for the nontorsion case.Twisted K-theory has been studied for some time [DK] [Ro]. More geometric fla-vors were given in [BCMMS]. Recently, the theory was fully developed by Atiyahand Segal [AS1] [AS2]. It is a further result that [H3] acts as a determinantalK(Z, 2)-twist for complex K-theory. The left hand side of the expression (6.1) is infact the first differential d3 in the Atiyah-Hirzebruch spectral sequence for twistedK-theory – see [BCMMS] [AS1] [AS2]. Then, a twisted D-brane in a B-field(X,H3) is a triple (W,E, ι), where ι : W → X is a closed, embedded orientedsubmanifold with ι∗H = W3(W), and E ∈ K0(W) (see [BMRS]).

5These are yet another incarnation of the line bundle gerbes mentioned earlier, for instancein the paragraphs just before proposition 2.21. This stack incarnation is precisely the “gerbe” inthe original sense of the word (a locally non-empty and transitive stack).

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44 HISHAM SATI

A D-brane wrapping a homology cycle is inconsistent if it suffers from an anom-aly, and is sometimes inconsistent if the homology cycle cannot be represented byany nonsingular submanifold. This is detected by the first Milnor primitive coho-mology operation Q1 = βP 1

3 where β is the Bockstein and P 13 is the Steenrod power

operation, both at the prime p = 3. In contrast, the twisted Spinc condition is atp = 2. In fact, we have

Observation 6.9 ([ES2]). The twisted Spinc condition is not sufficient.

D-brane charges are classified by the twisted K-group K∗H(X). A rigorous

formulation of such D-brane charges requires a Thom isomorphism and a push-forward map. Indeed the Thom isomorphism and push-forward in twisted K-theoryare established in [CW]: Corresponding to the map ι : W → X there is

(1) Push-forward map: ι! : K•ι∗σ+W3(ι)

(W) → K•σ(X).

(2) Thom isomorphism: K•(W) ∼= K•ι∗σ+W3(ι)

(W).

With the use of the Riemann-Roch formula and index theorem in twisted K-theoryit is now established that the RR charges in the presence of an H-field are indeedclassified by twisted K-theory [CW2].

Now applying the push-forward map for ι in twisted K-theory, one can associatea canonical element in KH(X), the desired D-brane charge of the underlying D-brane [CW]

(6.2) ι! : K(W) ∼= Kι∗H+W3(W)(W) −→ K∗H(X),

Hence,

Definition 6.10. For any D-brane wrapping W determined by an elementE ∈ K(W), the charge is

(6.3) ι!(E) ∈ K∗H(X).

6.3. The M-brane charges and twisted TMF. In the case of M-theory,the object carrying charges with respect to G4 is the M5-brane and we will studyconditions for consistent wrapping of such branes on cycles in eleven-dimensionalspacetime. Hence, the candidate object to carry charges with respect to TMF isthe M-theory fivebrane. In this section we will provide a point of view on theinterpretation of Witten’s quantization condition (1.1) for G4, which will give thecontext within which we describe M-brane charges in the following section.

In the case of the string, the target spacetime is assumed to be Spin, i.e.w2(X

10) = 0. Then this also implies that X10 is certainly Spinc. Then the re-quirement that the brane’s worldvolume be Spinc is equivalent to requiring thenormal bundle to the D-brane to be Spinc. On the other hand, for the M-theorycase, Witten’s flux quantization is obtained from the embedding of the membranein spacetime Y 11. Taking Y 11 be Spinc will not be enough this time. We will seebelow how the twisted String condition [Wa] [SSS3] in the three bundles: tangentbundle to the worldvolume, the normal bundle, and the tangent bundle of the targetwill be related.

Remarks 6.11. The Interpretation of Witten’s quantization.

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GEOMETRIC AND TOPOLOGICAL STRUCTURES RELATED TO M-BRANES 45

1. The equation (1.1) makes sense only if λ is divisible by two. It means that itis not [G4/2π] but [G4/π] that is well defined as an integral cohomology class andthat this class is congruent to λ modulo two [W7].2. A weaker condition than (1.1) can be obtained by multiplying by two, providedthere is no 2-torsion,

(6.4) 2[G4] + λ = 2a.

Condition (6.4) gives condition (1.1) if there is no 2-torsion and once λ is divisibleby two.3. We rewrite condition (6.4) in turn in a suggestive way as λ−2 ([G4]− a) = 0, sothat we identify α := 2([G4]− a) as the twist of the String structure [Wa] [SSS3].4. Alternatively, we can work not with twisted String structure but rather withwhat was called twisted F〈4〉-structure in [SSS3] to account for the factor of 2dividing λ. There is no canonical description of F〈4〉 yet except through BO〈8〉.

We can consider 3 different cases:

• the case when a = 0 so that the E8 bundle is trivial and the twist isprovided by [G4],

• the case [G4] = 0 so that the flux is G4 = dC3 and the twist is providedby the E8 class a,

• the general case, where the twist is provided by α.

To make a comparison, let us briefly recall the model of [DFM]. The fieldstrength in M-theory is geometrically described as a shifted differential character[DFM] in the sense of [HS]. A shifted differential character is the equivalenceclass of a differential cocycle which trivializes a specific differential 5-cocycle relatedto the integral Stiefel-Whitney class W5(Y ). The Stiefel-Whitney class w4(Y ) ∈H4(Y ;Z2) defines a differential cohomology class w4 via the inclusion H4(Y ;Z2) →H4(Y ;R/Z) → H4(Y ;Z). On a Spin manifold, W5(Y ) = 0 is satisfied since λis an integral lift of w4(Y ). In this case, the differential cohomology class w4

can be lifted to a differential cocycle by defining W5(Y ) =(0, 1

2λ, 0)∈ Z5(Y ) ⊂

C5(Y ;Z) × C4(Y ;Z) × Ω5(Y ), and the C-field can be defined as the differentialcochain C = (a, h, ω) ∈ C4(Y ;Z) × C3(Y ;Z) × Ω4(Y ) trivializing W5, δC = W5,i.e. in components [DFM]

(6.5) δa = 0, δh = ω − aR +1

2λ, dω = 0 .

It was proposed in in [DFM] that G4 lives in

(6.6) H412λ(Y 11),

the space of shifted characters on Y 11 with shift 12λ, and similarly on the fivebrane

worldvolume.

Remark 6.12. In defining G4 to live in (6.6), the authors of [DFM] are takingthe point of view that the class 1

2λ acts as a twist for the differential character. The

point of view we would like to take here is that 12λ is what is being twisted, and

hence plays a more central role. After all, natural structures, i.e. ones related to thetangent bundle, should be in a sense more fundamental for describing structures ona manifold that are extra or auxiliary structures such as bundles not related to the

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46 HISHAM SATI

tangent bundle. The E8 gauge theory can then be seen as responsible for the twistof 1

2λ. This is the point of view also adopted in [SSS3], building on the definitionin [Wa]. Therefore

(6.7)

1

2λ− shifted E8 structure

=⇒ E8 − twisted String structure .

6.4. The M5-brane charge. There are two definitions for the the M5-branepartition function [W8] [W9] [HS], and hence for the M5-brane charge [DFM].One is intrinsic and uses the theory on the worldvolume. The other is extrinsicand uses anomaly inflow and hence the embedding in eleven dimensions. We willuse the extrinsic definition for the M5-brane charge as this is the one that usesthe Thom isomorphism and the push-forward (in the appropriate theory). Someaspects of the intrinsic approach were used in [SSS3] to relate twisted Fivebranestructures to the worldvolume theory of the fivebrane.

Consider the embedding ι : W → Y of the fivebrane with six-dimensionalworldvolumeW into eleven-dimensional spacetime Y . Consider the ten-dimensionalunit sphere bundle π : V → W of W with fiber S4 associated to the normalbundle N → W of the embedding ι. There is a corresponding 11-manifold Yr withboundary V obtained by removing the disk bundle of radius r, Yr = Y − Dr(N)[DFM]. Corresponding to the sphere bundle V there is the Gysin sequence. Usingthat the normal bundle has vanishing Euler class e(N) = 0, one can deduce that[DFM]

H3(V ;Z) ∼= H3(W ;Z)

H4(V ;Z) ∼= H4(W ;Z)⊕ Z (noncanonically).(6.8)

so that H3(V, U(1)) ∼= H3(W , U(1)). This relates G4 on the worldvolume to thaton the normal bundle, i.e. the tangential components to the transverse components.

The M5-brane is magnetically charged under the C-field, i.e. the former actsas a source for the latter. The charge is then measured by the value of the integralof G4 over the linking sphere S4 of W in Y

(6.9) QM5 =

∫S4

G4 = k ∈ Z .

Since the total Pontrjagin class of S4 is 1, then QM5is equal to the instanton

number of an E8 instanton of the E8 gauge theory with four-class a:∫S4 a = k ∈ Z.

The description in terms of instantons is further given in [ES1]. Note that thecharge defined this way highlights the role of E8. If the E8 bundle is trivial thenthe charge is zero.

The anomaly inflow cancellation allows for a partition function of the M5-braneto be well-defined. In order for this to be made precise, the C-fields on W mustbe related to the C-field on V . The gauge equivalence classes of C-fields on W isthe shifted differential character [C] ∈ H4

12λ(W ), and similarly for W . This then

requires the existence of the map

(6.10) i : H412λW

(W ) → H412λX

(X).

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GEOMETRIC AND TOPOLOGICAL STRUCTURES RELATED TO M-BRANES 47

In [DFM], a C-field C0 on X was chosen such that via the map i the C-fields arerelated as

(6.11) i[C] = [C0] + π∗[C].

We now give our main proposal.

Proposal 6.13. (1) As we mentioned earlier, the shifted differential char-acter gives the minor role to 1

2λ. This obscures an interpretation in termsof generalized cohomology. We would like to replace (6.10) with

(6.12) ρ :

(1

2λ+ α

)(W) −→

(1

2λ+ α

)(V ) .

(2) In order to properly define the M5-brane charge, what we need is then aThom isomorphism and a push-forward map in the appropriate theory.Our proposal is that the desired theory is TMF and hence we apply theThom isomorphism and the push-forward in TMF to obtain the M5-branecharge. This way, the analog of (6.11) will be canonical.

If X is a space, then the twisted forms of K∗(X) correspond to homotopyclasses of maps [X,BGL1K]. The third homotopy group of the parametrizing spaceis π0BGL1K = Z since K2(pt) = Z. This is the determinantal twist in K-theory anexample of which being the NSH-field in string theory. Twists of TMF are classifiedby BGL1TMF and there is a corresponding map K(Z, 4) → BGL1TMF .

The proof of the following theorem is explained by Matthew Ando.

Theorem 6.14. A class α ∈ H4(X;Z) gives rise to a twist tmf∗α(X) of tmf(X).

Moreover, if V is a (virtual) spin vector bundle over X with half-Pontrjagin classλ, then tmf∗

λ(X) ∼= tmf∗(XV ) as modules over tmf∗(X).

Then, armed with a Thom isomorphism and a pushforward map (see Ando’scontribution to these proceedings [ABG]), the main application is

Definition/Theorem 6.15. Given an embedding ι : W → Y , the charge ofthe M5-brane is given by

(6.13) ι!(E) ∈ TMF ∗α(Y )

Remarks 6.16. We consider the Witten quantization condition for the world-volume, normal bundle, and target for both the M2-brane and the M5-brane.

(1) M2-brane: The condition 12λ(M3) + [G4]|M3 − aE8

|M3 = 0 is satisfied onthe M2-brane worldvolume, by dimension reasons. Therefore, the condi-tion 1

2λ(N)+ [G4]|N −aE8|N = 0 on the normal bundle N is equivalent to

the same condition on the target. This is indeed what enters in Witten’sderivation of the quantization condition (1.1).

(2) M5-brane: Assuming as above that the condition holds for the normalbundle to the fivebrane, then this implies that the condition on the world-volume is equivalent to that on the target. So we have

1

2λ(W) + [G4]|W − aE8

|W = 0 =⇒ 1

2λ(Y ) + [G4]|Y − aE8

|Y = 0 .

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48 HISHAM SATI

Discussion and further evidence for the interpretation of M5-brane charge.

1. Two-gerbes. The system of M2-branes ending on M5-branes is an M-theoretic realization of the system of strings ending on D-branes. In the latter,there is a gauge field, or connection one form A, on the string boundary ∂Σ ⊂ Q.In addition, there is the B-field on Σ, which is acts as a twist for the Chan-Patonbundle, whose connection is A and curvature is F . Now in the case of M-theory, themembrane boundary ∂M2 ⊂ M5 (written schematically) has a degree two potential,essentially a B-field, which represents a gerbe made nonanbelian by the presenceof the pullback of the C-field. This is equivalent to a 2-gerbe system.

2. Loop variables for the membrane. The system of multiple M5-branes,generalizing the system of n D-branes leading to U(n) nonabelian gauge symmetry,

can be described by twisted ΩG- gerbes, i.e. twisted gerbes for the universal centralextension of the based loop group ΩG, where G is any of the Lie groups Spin(n),n ≥ 7, E6, E7, E8, F4, and G2 [AJ]. Indeed, it has been shown explicitly in [G3]that classical membrane fields are loops.

3. Fivebrane in loop space. The generalization of the abelian system offields, called tensor multiplet, to the nonabelian case leading to the nonabelian ten-sor multiplet, which appears in the worldvolume theory of the M5-brane, requiresloop space variables and a formulation in loop space [G1] [G2]. Given the generalprinciple that degree n phenomena in a space are captured by degree n−1 phenom-ena on its loop space, the situation for n = 3 suggests a relation between (twisted)K-theroy in loop space to be related to (twisted) TMF -cohomology of the space.

4. Relating TMF in M-theory to twisted K-theory in string theory.As we have reviewed it is known that twisted K-theory classifies D-branes andtheir charges in the presence of the NS B-field on a ten-dimensional space X10. Wehave also seen how M-branes and their charges should take values in TMF on aneleven-dimensional space Y 11. Given the relation between M-theory and type IIAstring theory, the situation when Y 11 is a (possibly trivial) circle bundle over X10,there should be a relation between the TMF description and the twisted K-theorydescription, in the sense that (possibly S1-equivariant twisted )TMF of X10 × S1

should give rise to twisted K-theory of X10. Current discussions with MatthewAndo and with Christopher Douglas suggest schematically the following

Conjecture 6.17. There is a map(Ωtmf × S1

)/S1 → k/∗.

5. Capturing the fields of degree 4k in M-theory. Twisted K-theory, underthe twisted Chern character map that lands in rational cohomology, leads to dif-ferential forms of all even degree up to the dimension of the manifold. These formsare the components of the RR field in the classical supergravity approximation. Ineleven dimensions, then, one should ask about some form of classical limit of theTMF description. What replaces the Chern chracter map should be a version forTMF (cf. the Miller character) of the Pontrjagin character map in KO-theory

(6.14) Ph : KO∗(X) −→⊗C K∗(X) −→ch H∗∗(X;Q) .

The range is degree 4k cohomology, which indeed captures the field G4 and its dualG8 (or Θ). We think of this as the combination v G4+w G8, for suitably identifiedgenerators v and w, which make H∗∗(X;Q). Some aspects of this in connection toSpin K-theory have been discussed in [S7] (see also section 3.4).6. Infinite number of fields. We know that, physically, an element in differ-ential K-theory corresponds to a collection of physical fields: all the RR fields.

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GEOMETRIC AND TOPOLOGICAL STRUCTURES RELATED TO M-BRANES 49

Analogously, an element in (differential twisted) elliptic cohomology or (differentialtwisted) TMF should be given by a large collection of physical fields. At the levelof differential forms, the large number of fields should be seen in the same way thatthe K-theoretic RR fields are seen at the level of differential forms via supergravityfields and their Hodge duals. But eleven-dimensional supergravity as traditionallyknown features only a single candidate field: the 3-form C-field (and its Hodgedual). This certainly cannot model generic elements in TMF by itself. Therefore,our previous discussion suggests a considerably richer structure hidden within andbeyond eleven-dimensional supergravity. Recall that such a rich structure is alsosuggested by hidden symmetries:

First, at the level of differential forms:• In the dimensional reduction of eleven-dimensional supergravity (and hence

type IIA supergravity) on tori Tn with fluxes one gets the Cremmer-Julia [CJ]exceptional groups En(n), which are infinite-dimensional Kac-Moody groups forn ≥ 9 (cf. [J]). In the latter case, hence, there are an infinite number of fields atthe classical level. Passing to the quantum theory, one has the U -duality groupsEn(n)(Z), the Z-forms of the above non-compact groups, and so we still have aninfinite number of fields.

• Already in eleven dimensions, the classic works of de Wit, Nicolai ( reviewedin [dWN]), and conjecture of [Du], imply the existence of Cremmer-Julia groupswithout compactification. One recent striking proposal is that of [We] in which theLorentzian Kac-Moody algebra E11 is proposed as a symmetry in M-theory. Thisproposal has withstood many checks. The algebra e11 admits an infinite Z-gradingas e11 = · · · ⊕ e8 ⊕ · · · .

Second, integrally: The above fields should have refinements at the quantumlevel to whichever (generalized) cohomology theory ends up arising.7. The topological term via higher classes. The topological part of the M-theory action is written in a suggestive compact form when lifted to the boundingtwelve-manifold Z12 in [S1] [S2] [S3]. To do so, the total String class (in thenotation of those papers) λ = 1 + λ1 + λ2 + · · · is introduced, where λ1 = p1/2 isthe usual ‘String class’, and λ2 = p2/2 which is well-defined for Spin manifolds. Theinterpretation of the class and the characters is as degree 4k analogs of the Chernclass and the Chern character, mapping from the cohomology theory describingM-theory to degree 4k cohomology. These are essentially the Spin characteristicclasses defined in [Th] and were precursors to the discussion of Fivebrane structuresin [SSS1] [SSS2] [SSS3].

Theorem 6.18. [S1] [S2] [S3][S5]

(1) The M-theory fields are elements of a unified field strength. The quanti-zation formula on the total M-theory field reproduces the quantization onG4 and its dual G8.

(2) G4 can be viewed as an index 2-gerbe.

8. Twisted cohomology in M-theory. The characters can be extended toinclude the dual fields and to account for their dynamics. For the dual field, onecan pick either the straightforward degree seven Hodge dual or its differential, thedegree eight field Θ. The dual formulation favors the degree four/eight combinationwhereas the duality-symmetric dynamics favors the degree four/seven combination.The second combination hints at a role for the prime p = 3 in M-theory analogous

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50 HISHAM SATI

to the role played by the prime p = 2 in K-theory, and consequently in string theory.The combination leads to twisted (generalized) cohomology. However, unlike thecase in K-theory, the twist is given by a degree four class, suggesting relation toTMF, since a twist of the latter can be seen, at least heuristically, as a K-theory(degree three) twist on the loop space.

Theorem 6.19. [S2] [S3] There is a twisted (graded) cohomology on the M-theory fields with a twist given by a degree four class.

9. The q-expansions. The q-expansions from the Miller character TMF →K[[q]] → H∗Q[[q]] will come from the comparison to type IIA string theory. Wehave the following diagram

(6.15) S1 W3

π′

Y 11

π

S1

=

Σ2 X10

.

There are two principal circle bundles: π and π′. In the dimensional reductionfrom M-theory to type IIA string theory the two fibers are identified and from amembrane in eleven dimensions we get a string in ten dimensions. At the level ofpartition functions of the targets (the fiber bundle π in the diagram) it was observedin [KS1] that the resulting partition function, formulated in elliptic cohomology,is a q-expansion of the K-theoretic partition function. There q was built out of thegenerators v1 and v2 at the prime 2, q = v31v

−12 . This essentially came from the

fact that EO2(pt) = Z2[[q]].

The E-theoretic quantization. In the untwisted case, the total field strength

F (x) of type II string theory as described by K-theory is 2π times

√A(X)ch(x).

The refinement of the description of the fields to elliptic cohomology is For anyelliptic cohomology theory E, there is a canonical map E → K((q)) (where q is asabove), so compose with the Chern character to get a map chE : E → H∗((q)).

The term

√A(X) should be replaced by an analogous term related to the Witten

genus σ(X)1/2 where

Theorem 6.20 ([KS3]). The E-theoretic quantization condition for the RRfields is given by the formula for the elliptic field strength associated with x: F (x) =σ(X)1/2chE(X), where σ(X) is the characteristic class of X associated with thepower series

σ(z) = (ez/2 − e−z/2)∏

n≥1

(1− qnez)(1− qne−z)

(1− qn)2.

The σ-function, in the q → 0 limit, reduces to the characteristic function ofthe A-genus, thus reducing this field strength to the type II field strength in the10-dimensional limit.

The connection of this to charges is as follows. Considering the bundle π′, we getq-expansions on Σ2 upon taking Fourier modes of the circle. The boundary ∂Σ2 has

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GEOMETRIC AND TOPOLOGICAL STRUCTURES RELATED TO M-BRANES 51

Chan-Paton charges on it. Since the boundary of the string ends on a D-brane thencertainly the q-expansions will be seen by the D-brane as well. The result is that theChan-Paton bundle on the D-brane gets replaced by its appropriate q-expansionsE → S•E,Λ•E. The charges of D-branes have been previously calculated at q = 1,i.e. at Λ• = S• = 1, as they are essentially the Chern character

(6.16) K∗(q = 1) −→ H∗Q(q = 1) .

The q-refinement of the D-brane charge formula should then be

(6.17) Q = ΦW (X10)chell(E) ,

the Witten genus of X10 twisted by the appropriate exterior and symmetric powersof the Chan-Paton bundle.

The higher modes for the supermultipet. In an orthogonal discussion to[KS1], we could also view q as coming from Fourier modes on the circle, i.e. Kaluza-Klein modes and interpret q accordingly. Consider the supergravity fields (g, C3, ψ).The coupling to vector bundles V gives V ⊗ Lk). For example V is an E8 bundlewith characteristic class a and set c1(L). Consider the connection A on the vectorbundle V . Coupling it to Lk leads to the connection A⊗ e−ikθ. Then the C-field,which is essentially the Chern-Simons form of the E8 bundle, will also be also haveFourier components as C3 ⊗ e−ikθ. This is of course compatible with, and is infact the same as, just the dimensional reduction of the C-field directly from elevendimensions without use of the E8 bundle.

10. Concluding remark. We have reviewed and indicated further developmentthat a closer examination of the deep structures involved in string- and M-theoryindicates and shows that very rich cohomological phenomena are at work in thebackground. While it is now well-known that (differential) K-theory encodes muchof the interesting structure of Type II string theory, we have shown and argued thatby further reasoning along such lines one finds various generalized cohomologicalstructures that go beyond K-theory. This includes Morava K-theory and E-theory,elliptic cohomology, and TMF . In addition, we have considered higher structuresgeneralizing Spin structures, such as String and Fivebrane structures, and haveemphasized a delicate interplay between all these.

Acknowledgements

I would like to thank Matthew Ando, David Lipsky, Corbett Redden, Urs Schreiber,and Jim Stasheff for very useful discussions, and Dan Freed for useful remarks.I also thank the organizers of the CBMS conference on “Topology, C∗-algebras,and String Duality”, Robert Doran and Greg Friedman, for the kind invitationto the conference. Special thanks are due to the anonymous referee, who kindlygave many helpful remarks and suggestions for improving the presentation andstrengthening the points made in the article. I also thank Arthur Greenspoon foreditorial suggestions.

References

[AKMW] O. Alvarez, T. P. Killingback, M. L. Mangano, and P. Windey, String theory and loopspace index theorems, Commun. Math. Phys. 111 (1987) 1-10.

[ASi] O. Alvarez and I. M. Singer, Beyond the elliptic genus, Nuclear Phys. B 633 (2002)309-344, [arXiv:hep-th/0104199].

231

52 HISHAM SATI

[A] M. Ando, Power operations in elliptic cohomology and representations of loop groups,Trans. AMS 352 (2000) 5619–5666.

[ABG] M. Ando, A. Blumberg, and D. Gepner, Twists of K-theory and TMF, to appear inProc. Symp. Pure Math.

[AHR] M. Ando, M. Hopkins, and C. Rezk, Multiplicative orientations of KO-theory and ofthe spectrum of topological modular forms, preprint 2009,[http://www.math.uiuc.edu/ mando/papers/koandtmf.pdf].

[AHS] M. Ando, M. J. Hopkins, and N. P. Strickland, Elliptic spectra, the Witten genus andthe theorem of the cube, Invent. Math. 146 (2001), no. 3, 595–687.

[As] A. Asada, Four lectures on the geometry of loop group and non abelian de Rhamtheory, Chalmers University of Technology/The University of Goteborg, 1990.

[AJ] P. Aschieri and B. Jurco, Gerbes, M5-brane anomalies and E8 gauge theory J. HighEnergy Phys. 0410 (2004) 068, [arXiv:hep-th/0409200].

[AS1] M. Atiyah and G. Segal, Twisted K-theory, Ukr. Math. Bull. 1 (2004), no. 3, 291–334,[arXiv:math/0407054] [math.KT].

[AS2] M. Atiyah and G. Segal, Twisted K-theory and cohomology, Inspired by S. S.Chern, 5-43, Nankai Tracts Math., 11, World Sci. Publ., Hackensack, NJ, 2006,[arXiv:math/0510674] [math.KT].

[BBK] N. A. Baas, M. Bokstedt, and T. A. Kro, Two-categorical bundles and their classifyingspaces, [arXiv:math/0612549] [math.AT].

[BCSS] J. Baez, A. Crans, U. Schreiber, and D. Stevenson, From loop groups to 2-groups,Homology, Homotopy Appl. 9 (2007), no. 2, 101- 135, [arXiv:math/0504123] [math.QA].

[BS] J. Baez, and U. Schreiber, Higher gauge theory, Categories in Algebra, Geometry andMathematical Physics, 7–30, Contemp. Math., 431, Amer. Math. Soc., Providence,RI, 2007, [arXiv:math/0511710v2] [math.DG].

[BSt] J. Baez and D. Stevenson, The classifying space of a topological 2-group,[arXiv:0801.3843] [math.AT].

[BST] E. Bergshoeff, E. Sezgin, and P.K. Townsend, Supermembranes and eleven-dimensionalsupergravity, Phys. Lett. B189 (1987) 75-78.

[BCMMS] P. Bouwknegt, A. L. Carey, V. Mathai, M. K. Murray, and D. Stevenson, Twisted

K-theory and K-theory of bundle gerbes, Commun. Math. Phys. 228 (2002) 17-49,[arXiv:hep-th/0106194].

[BEJMS] P. Bouwknegt, J. Evslin, B. Jurco, V. Mathai, and H. Sati, Flux compactification onprojective spaces and the S-duality puzzle, Adv. Theor. Math. Phys. 10 (2006) 345-94,[arXiv:hep-th/0501110].

[BM] P. Bouwknegt and V. Mathai, D-branes, B-fields and twisted K-theory, J. High EnergyPhys. 0003 (2000) 007, [arXiv:hep-th/0002023].

[BKTV] P. Bressler, M. Kapranov, B. Tsygan, and E. Vasserot, Riemann-Roch for real varieties,[arXiv:math/0612410].

[BMRS] J. Brodzki, V. Mathai, J. Rosenberg, and R. J. Szabo, D-branes, RR-fields and dual-ity on noncommutative manifolds, Commun. Math. Phys. 277, no. 3 (2008) 643-706,[arXiv:hep-th/0607020v3].

[CJM] A. L. Carey, S. Johnson and M. K. Murray, Holonomy on D-branes, J. Geom. Phys.52 (2004) 186-216, [arXiv:hep-th/0204199].

[CJMSW] A. L. Carey, S. Johnson, M. K. Murray, D. Stevenson, and B.-L. Wang, Bundle gerbesfor Chern-Simons and Wess-Zumino-Witten theories, Commun. Math. Phys. 259(2005) 577-613, [arXiv:math/0410013] [math.DG].

[CW] A. L. Carey and B.-L. Wang, Thom isomorphism and push-forward map in twistedK-theory, [arXiv:math/0507414v4] [math.KT].

[CW2] A. L. Carey and B.-L. Wang, Riemann-Roch and index formulae in twisted K-theory,[arXiv:0909.4848] [math.KT].

[CN] A. H. Chamseddine and H. Nicolai, Coupling the SO(2) supergravity through dimen-sional reduction, Phys. Lett. B96 (1980) 89-93.

[CS] J. Cheeger and J. Simons, Differential characters and geometric invariants, In Geom-etry and Topology, 50-80, LNM 1167, Springer, 1985.

[CP] R. Coquereaux and K. Pilch, String structures on loop bundles, Commun. Math. Phys.120 (1989) 353-378.

232

GEOMETRIC AND TOPOLOGICAL STRUCTURES RELATED TO M-BRANES 53

[C] E. Cremmer, Supergravities in 5 dimensions, in Superspace and supergravity, eds. S.W.Hawking and M. Rocek, Cambridge Univ. Press, Cambridge 1981.

[CJ] E. Cremmer and B. Julia, The SO(8) supergravity, Nuclear Phys. B159 (1979) 141–212.

[CJLP] E. Cremmer, B. Julia, H. Lu, and C.N. Pope, Dualization of dualities II, Nucl. Phys.B535 (1998) 242–292, [arXiv:hep-th/9806106].

[CJS] E. Cremmer, B. Julia, and J. Scherk, Supergravity theory in eleven-dimensions, Phys.

Lett. B76 (1978) 409-412.[D] X. Dai, Adiabatic limits, nonmultiplicativity of signature, and Leray spectral sequence,

J. Amer. Math. Soc. 4 (1991), no. 2, 265–321.[dWN] B. de Wit and H. Nicolai, Hidden symmetries, central charges and all that, Class.

Quant. Grav. 18 (2001) 3095-3112, [arXiv: hep-th/0011239].[DFM] E. Diaconescu, D. S. Freed and G. Moore, The M-theory 3-form and E8 gauge theory,

Elliptic cohomology, 44–88, London Math. Soc. Lecture Note Ser., 342, CambridgeUniv. Press, Cambridge, 2007, [arXiv:hep-th/0312069].

[DMW] E. Diaconescu, G. Moore and E. Witten, E8 gauge theory, and a derivationof K-Theory from M-Theory, Adv. Theor. Math. Phys. 6 (2003) 1031–1134,[arXiv:hep-th/0005090].

[DK] P. Donovan and M. Karoubi, Graded Brauer groups and K-theory with local coeffi-

cients, Inst. Hautes Etudes Sci. Publ. Math. 38 (1970) 5–25.[D] C. L. Douglas, On the twisted K-homology of simple Lie groups, Topology 45 (2006),

no. 6, 955–988.[Du] M. J. Duff, E8 ×SO(16) symmetry of d = 11 supergravity?, in Quantum Field Theory

and Quantum Statistics, eds. I. A. Batalin, C. Isham and G. A. Vilkovisky, AdamHilger, Bristol, UK 1986.

[DHIS] M. J. Duff, P. S. Howe, T. Inami and K. S. Stelle, Superstrings in D = 10 fromsupermembranes in D = 11, Phys. Lett. 191B (1987) 70–74.

[DLM] M. J. Duff, J. T. Liu, and R. Minasian, Eleven-dimensional origin of string-stringduality: A One loop test, Nucl. Phys. B452 (1995) 261-282, [arXiv:hep-th/9506126].

[DNP] M. J. Duff, B. E. W. Nilsson, and C. N. Pope, Kaluza-Klein supergravity, Phys. Rept.130 (1986) 1-142.

[E] F. Englert, Spontaneous compactification of eleven-dimensional supergravity, Phys.Lett. B119 (1982) 339–342.

[ES1] J. Evslin and H. Sati, SUSY vs E8 gauge theory in 11 dimensions, J. High EnergyPhys. 0305 (2003) 048, [arXiv:hep-th/0210090].

[ES2] J. Evslin and H. Sati, Can D-Branes Wrap Nonrepresentable Cycles?, J. High EnergyPhys. 0610 (2006) 050, [arXiv:hep-th/0607045].

[FS] J. Figueroa-O’Farrill, J. Simon, Supersymmetric Kaluza-Klein reductions of M2 andM5-branes, Adv. Theor. Math. Phys. 6 (2003) 703-793, [arXiv:hep-th/0208107].

[Fr2] D. S. Freed, Dirac charge quantization and generalized differential cohomol-ogy, in Surv. Differ. Geom. VII, 129–194, Int. Press, Somerville, MA, 2000,[arXiv:hep-th/0011220].

[Fr1] D. S. Freed, Classical Chern-Simons theory II, Houston J. Math. 28 (2002), no. 2,293–310.

[FH] D. S. Freed and M. J. Hopkins, On Ramond-Ramond fields and K-theory, J. HighEnergy Phys. 05 (2000) 044, [arXiv:hep-th/0002027].

[FMS] D. S. Freed, G. W. Moore, and G. Segal, The uncertainty of fluxes, Commun. Math.Phys. 271 (2007) 247-274, [arXiv:hep-th/0605198].

[FW] D. S. Freed and E. Witten, Anomalies in string theory with D-branes, Asian J. Math.3 (1999), no. 4, 819–851, [arXiv:hep-th/9907189].

[Fo] R. Forman, Spectral sequences and adiabatic limits Comm. Math. Phys. 168 (1995),no. 1, 57–116.

[Go] P. G. Goerss, Topological modular forms (aftern Hopkins, Miller, and Lurie),[arXiv:0910.5130] [math.AT].

[GHK] J. M.Gomez, P. Hu and I. Kriz, Stringy bundles and infinite loop space theory, preprint2009.

[GS] M. B. Green and J. H. Schwarz, Anomaly cancellation in supersymmetric D = 10gauge theory and superstring theory, Phys. Lett. B 149 (1984) 117-122.

233

54 HISHAM SATI

[Gu] R. Gven, Black p-brane solutions of D = 11 supergravity theory, Phys. Lett. B276(1992) 49–55.

[G1] A. Gustavsson, The non-Abelian tensor multiplet in loop space, J. High Energy Phys.0601 (2006) 165, [arXiv:hep-th/0512341].

[G2] A. Gustavsson, Loop space, (2, 0) theory, and solitonic strings, J. High Energy Phys.0612 (2006) 066, [arXiv:hep-th/0608141].

[G3] A. Gustavsson, Selfdual strings and loop space Nahm equations, J. High Energy Phys.

0804 (2008) 083, [arXiv:0802.3456] [hep-th].[HM] J. A. Harvey and G. Moore, Superpotentials and membrane instantons,

[arXiv:hep-th/9907026].[H] A. Henriques, Integrating L∞-algebras, Compos. Math. 144 (2008), no. 4, 1017–1045,

[arXiv:math/0603563] [math.AT].[Ho] M. J. Hopkins, Algebraic topology and modular forms, Proceedings of the ICM, Beijing

2002, vol. 1, 283–309, [arXiv:math/0212397] [math/AT].[HS] M. J. Hopkins and I. M. Singer, Quadratic functions in geometry, topology, and

M-theory, J. Differential Geom. 70 (2005), no. 3, 329–452, [arXiv:math/0211216][math.AT].

[HK1] P. Hu and I. Kriz, Real-oriented homotopy theory and an analogue of the Adams-Novikov spectral sequence, Topology 40 (2001) 317–399.

[HK2] P. Hu and I. Kriz, Conformal field theory and elliptic cohomology, Adv. Math. 189(2004), no. 2, 325–412.

[J] B. Julia, in Lectures in Applied Mathematics, AMS-SIAM, vol. 21, (1985), p. 335.[Ka] A. Kapustin, D-branes in a topologically nontrivial B-field, Adv. Theor. Math. Phys.

4 (2000) 127-154, [arXiv:hep-th/9909089].[Ki] T. P. Killingback, Global anomalies, string theory and space-time topology, Class.

Quant. Grav. 5 (1988) 1169–1186.[KS1] I. Kriz and H. Sati, M Theory, type IIA superstrings, and elliptic cohomology, Adv.

Theor. Math. Phys. 8 (2004) 345–395, [arXiv:hep-th/0404013].[KS2] I. Kriz and H. Sati, Type IIB string theory, S-duality and generalized cohomology,

Nucl. Phys. B715 (2005) 639–664, [arXiv:hep-th/0410293].

[KS3] I. Kriz and H. Sati, Type II string theory and modularity, J. High Energy Phys. 08(2005) 038, [arXiv:hep-th/0501060].

[Ku] K. Kuribayashi, On the vanishing problem of string classes, J. Austral. Math. Soc.(series A) 61 (1996), 258-266.

[LNSW] W. Lerche, B. E. W. Nilsson, A. N. Schellekens and N. P. Warner, Anomaly cancellingterms from the elliptic genus, Nucl. Phys. B 299 (1988) 91–116.

[LD] B. Li and H. Duan, Spin characteristic classes and reduced KSpin group of a low-dimensional complex, Proc. Amer. Math. Soc. 113 (1991) 479–491.

[LLW] W. Li, X. Liu, and H. Wang, On a spectral sequence for twisted cohomologies,[arXiv:0911.1417] [math.AT].

[Liu] K. Liu, On Modular Invariance and Rigidity Theorems, Harvard thesis 1993.[Lo] J. Lott, Higher degree analogs of the determinant line bundles, Comm. Math. Phys.

230 (2002) 41–69, [arXiv:math/0106177][math.DG].[Lu] J. Lurie, A survey of elliptic cohomology, preprint, 2007,

[http://www.math.harvard.edu/ lurie/papers/survey.pdf].[MSa] V. Mathai and H. Sati, Some relations between twisted K-theory and E8 gauge theory,

J. High Energy Phys. 03 (2004) 016, [arXiv:hep-th/0312033].[MSt] V. Mathai and D. Stevenson, Chern character in twisted K-theory: equi-

variant and holomorphic cases, Commun. Math. Phys. 236 (2003) 161-186,[arXiv:hep-th/0201010].

[MWu] V. Mathai and S. Wu, Analytic torsion for twisted de Rham complexes,[arXiv:0810.4204] [math.DG].

[MaM] R. Mazzeo and R. Melrose, The adiabatic limit, Hodge cohomology and Leray’s spectral

sequence for a fibration, J. Differential Geom. 31 (1990), no. 1, 185–213.[Mc] D. A. McLaughlin, Orientation and string structures on loop space, Pacific J. Math.

155 (1992) 143-156.[Mi] J. Mickelsson, Current Algebras and Groups, Plenum Press, London and New York,

1989.

234

GEOMETRIC AND TOPOLOGICAL STRUCTURES RELATED TO M-BRANES 55

[MP] J. Mickelsson and R. Percacci, it Global aspects of p-branes, J. Geom. and Phys. 15(1995) 369-380, [arXiv:hep-th/9304054].

[MM] R. Minasian and G. Moore, K-theory and Ramond-Ramond charge, J. High EnergyPhys. 11 (1997) 002, [arXiv:hep-th/9710230].

[Mo] G. W. Moore, Anomalies, Gauss laws, and Page charges in M-theory, Comptes RendusPhysique 6 (2005) 251-259, [arXiv:hep-th/0409158].

[MW] G. Moore and E. Witten, Self duality, Ramond-Ramond fields, and K-theory, J. High

Energy Phys. 05 (2000) 032, [arXiv:hep-th/9912279].[Mor] J. Morava, Forms of K-theory, Math. Zeitschrift 201 (1989) 401–428.[Mu] M. K. Murray, Bundle gerbes, J. London Math. Soc. (2) 54 (1996), no. 2, 403416.[MuS] M. K. Murray and D. Stevenson, Bundle gerbes: Stable isomorphsim and local theory,

J. London Math. Soc. (2). 62 (2002), no. 3, 925-937, [arXiv:math/9908135] [math.DG].[PSW] K. Pilch, A. N. Schellekens, and N. P. Warner, Path integral calculation of string

anomalies, Nucl. Phys. B 287 (1987) 362–380.[PW] K. Pilch and N. P. Warner, String structures and the index of the Dirac-Ramond

operator on orbifolds, Commun. Math. Phys. 115 (1988) 191–212.[Po] J. Polchinski, Dirichlet-branes and Ramond-Ramond charges, Phys. Rev. Lett. 75

(1995) 4724–4727, [arXiv:hep-th/9510017].[R1] C. Redden, String structures and canonical three-forms, [arXiv:0912.2086] [math.DG].[R2] C. Redden, Harmonic forms on principal bundles, [arXiv:0810.4578] [math.DG].[RS] C. Redden and H. Sati, work in progress.[Ro] J. Rosenberg, Continuous trace C∗-algebras from the bundle theoretic point of view,

J. Aust. Math. Soc. A47 (1989) 368–381.[S1] H. Sati, M-theory and Characteristic Classes, J. High Energy Phys. 0508 (2005) 020,

[arXiv:hep-th/0501245].[S2] H. Sati, Flux quantization and the M-theoretic characters, Nucl. Phys. B727 (2005)

461–470, [arXiv:hep-th/0507106].[S3] H. Sati, Duality symmetry and the form-fields in M-theory, J. High Energy Phys. 0606

(2006) 062, [arXiv:hep-th/0509046].[S4] H. Sati, The Elliptic curves in gauge theory, string theory, and cohomology, J. High

Energy Phys. 0603 (2006) 096, [arXiv:hep-th/0511087].[S5] H. Sati, E8 gauge theory and gerbes in string theory, [arXiv:hep-th/0608190].[S6] H. Sati, A higher twist in string theory, J. Geom. Phys. 59 (2009) 369-373,

[arXiv:hep-th/0701232].[S7] H. Sati, An Approach to anomalies in M-theory via KSpin, J. Geom. Phys. 58 (2008)

387-401, [arXiv:0705.3484] [hep-th].[S7] H. Sati, On anomalies of E8 gauge theory on String manifolds, [arXiv:0807.4940]

[hep-th].[SSS1] H. Sati, U. Schreiber and J. Stasheff, L∞-connections and applications to String- and

Chern-Simons n-transport, in Recent Developments in QFT, eds. B. Fauser et al.,Birkhauser, Basel (2008), [arXiv:0801.3480] [math.DG].

[SSS2] H. Sati, U. Schreiber, and J. Stasheff, Fivebrane structures, to appear in Rev. Math.Phys. [arXiv:math/0805.0564] [math.AT].

[SSS3] H. Sati, U. Schreiber, and J. Stasheff, Twisted differential String- and Fivebrane struc-tures, [arXiv:0910.4001] [math.AT].

[SW] N. Seiberg and E. Witten, Spin structures in string theory, Nucl. Phys. B276 (1986)272-290.

[SW] A. N. Schellekens and N.P. Warner, Anomalies and modular invariance in string the-ory, Phys. Lett. B 177 (1986) 317–323.

[S-P] C. J. Schommer-Pries, A finite-dimensional String 2-group, preprint 2009.[Sch1] J. H. Schwarz, The power of M-theory, Phys. Lett. B367 (1996) 97-103,

[arXiv:hep-th/9510086].[Sch2] J. H. Schwarz, Lectures on superstring and M-theory dualities, Nucl. Phys. Proc. Suppl.

55B (1997) 1-32, [arXiv:hep-th/9607201].[SVW] S. Sethi, C. Vafa, and E. Witten, Constraints on low-dimensional string compactifica-

tions, Nucl. Phys. B480 (1996) 213-224, [arXiv:hep-th/9606122].[ST1] S. Stolz and P. Teichner, What is an elliptic object?, in Topology, geometry and quan-

tum field theory, 247–343, Cambridge Univ. Press, Cambridge, 2004.

235

56 HISHAM SATI

[ST2] S. Stolz and P. Teichner, Super symmetric field theories and integral modular functions,[http://math.berkeley.edu/ teichner/Papers/SusyQFT.pdf]

[Stg] R. E. Stong, Appendix: calculation of ΩSpin11 (K(Z, 4)), in Workshop on unified string

theories (Santa Barbara, Calif., 1985), 430–437, World Sci. Publishing, Singapore,1986.

[Str] A. Strominger, Superstrings with torsion, Nucl. Phys. B274 (1986) 253–284.[Te] C. Teleman, K-theory of the moduli space of bundles on a surface and deformations of

the Verlinde algebra, in Topology, Gometry and Quantum Field Theory, U. Tillmann

(ed.), Cambridge University Press, 2004.[Th] E. Thomas, On the cohomology groups of the classifying space for the stable spinor

groups, Bol. Soc. Mat. Mexicana (2) 7 (1962) 57–69.[To1] P. K. Townsend, String-Membrane Duality in Seven Dimensions, Phys. Lett. B354

(1995) 247-255, [arXiv:hep-th/9504095].[To2] P. K. Townsend, D-branes from M-branes, Phys. Lett. B373 (1996) 68-75,

[arXiv:hep-th/9512062].[To3] P. K. Townsend, Four lectures on M-theory, Summer School in High Energy Physics

and Cosmology Proceedings, E. Gava et. al (eds.), Singapore, World Scientific, 1997,[arXiv:hep-th/9612121].

[V] C. Vafa, Evidence for F-theory, Nucl. Phys. B469 (1996) 403–418,[arXiv:hep-th/9602022].

[VW] C. Vafa and E. Witten, A one-loop test of string duality, Nucl. Phys. B447 (1995)261-270, [arXiv:hep-th/9505053].

[Wa] B.-L. Wang, Geometric cycles, index theory and twisted K-homology, J . Noncommut.Geom. 2 (2008), 497–552, [arXiv:0710.1625] [math.KT].

[We] P. C. West, E11 and M-theory, Class. Quantum Grav. 18 (2001) 4443-4460.[W1] E. Witten, Global gravitational anomalies, Commun. Math. Phys. 100 (1985) 197–229.[W2] E. Witten, Elliptic genera and quantum field theory, Commun. Math. Phys. 109 (1987)

525–536.[W3] E. Witten, The index of the Dirac operator in loop space, in Elliptic curves and modular

forms in algebraic topology (Princeton, NJ, 1986), 161–181, Lecture Notes in Math.,

1326, Springer, Berlin, 1988.[W4] E. Witten, Quantum field theory and the Jones polynomial, Commun. Math. Phys.

121 (1989) 351–399.[W5] E. Witten, String theory dynamics in various dimensions, Nucl. Phys. B443 (1995)

85-126, [arXiv:hep-th/9503124].[W6] E. Witten, D-Branes and K-Theory, J. High Energy Phys. 12 (1998) 019,

[arXiv:hep-th/9810188].[W7] E. Witten, On flux quantization in M-theory and the effective action, J. Geom. Phys.

22 (1997) 1-13, [arXiv:hep-th/9609122].[W8] E. Witten, Five-brane effective action in M-theory, J. Geom. Phys. 22 (1997) 103-133,

[arXiv:hep-th/9610234].[W9] E. Witten, Duality relations among topological effects in string theory, J. High Energy

Phys. 0005 (2000) 031, [arXiv:hep-th/9912086].

Department of Mathematics, Yale University, New Haven, Connecticut 06511

Current address: Department of Mathematics, University of Maryland, College Park,

Maryland 20742

E-mail address: hsati@math.umd.edu

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Proceedings of Symposia in Pure Mathematics

Landau-Ginzburg Models, Gerbes, and Kuznetsov’sHomological Projective Duality

Eric Sharpe

Abstract. In this talk we briefly outline several recent developments in Landau-

Ginzburg models and associated areas. We begin by discussing recent work onA- and B-twisted Landau-Ginzburg models – certain two-dimensional topologi-cal field theories derived from Landau-Ginzburg models. After briefly outliningsome pertinent work on string propagation on stacks and gerbes, including the

decomposition conjecture describing how strings on gerbes are equivalent tostrings on disjoint unions of spaces, we discuss how certain Landau-Ginzburgmodels on total spaces of bundles over gerbes are equivalent to strings onbranched double covers and noncommutative resolutions thereof, and realize

Kuznetsov’s “homological projective duality.”

1. Introduction

Landau-Ginzburg models, certain non-scale-invariant two-dimensional quan-tum field theories, have historically been a fruitful arena for many issues in stringcompactifications.

Roughly, a Landau-Ginzburg model consists of a string propagating on a space,together with a potential function over that space (which in a supersymmetrictheory is defined by a holomorphic function, known as a “superpotential”).

Historically, most Landau-Ginzburg models considered described strings prop-agating on vector spaces (plus a superpotential), or finite group quotients thereof,rather than a more general space or stack. One reason for this is that many suchexamples are closely related to strings on nontrivial spaces, but perhaps a morehonest reason is that a string on a vector space (plus superpotential) is technicallymuch easier to analyze than a string on a nontrivial space.

Technology is now finally being developed to allow the analysis of Landau-Ginzburg models on nontrivial spaces. This is technically more difficult, but, onecan also get more interesting results.

In this talk we will summarize several recent developments in Landau-Ginzburgmodels and related areas. We will begin by outlining A- and B-type topological

2010 Mathematics Subject Classification. Primary 81T45, Secondary 14D23, 14F05, 53C08,

53D37, 14N35.Key words and phrases. Landau-Ginzburg, gerbes, homological mirror symmetry, stacks.The author was supported in part by NSF grants DMS-0705381, PHY-0755614.

c©0000 (copyright holder)

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Proceedings of Symposia in Pure MathematicsVolume 81, 2010

c©2010 American Mathematical Society

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2 ERIC SHARPE

field theories based on Landau-Ginzburg models. We then study certain examplesof Landau-Ginzburg models defined over total spaces of bundles on gerbes. Ap-plying results from the study of strings on stacks and gerbes, we find that thoseLandau-Ginzburg models are equivalent to strings on branched double covers and,sometimes, noncommutative resolutions thereof, giving a physical realization ofsuch noncommutative spaces as well as new conformal field theories. We concludewith a discussion of how these Landau-Ginzburg models fit into families of physi-cal theories, and give a physical realization of Kuznetsov’s “homological projectiveduality.”

2. A-, B-topological Twists of Landau-Ginzburg Models on NontrivialSpaces

We begin by briefly outlining the construction of the A- and B-model topolog-ical field theories, and their analogues for Landau-Ginzburg models.

In a quantum field theory, one computes ‘correlation functions,’ closely analo-gous to correlation functions in statistics, in which instead of summing over eventsand weighting by probabilities, one performs some infinite-dimensional integral, andweights by an exponential. Schematically:

〈O1 · · · On〉 =

∫[Dφ · · · ] exp(−S(φ, · · · ))O1 · · · On∫

[Dφ · · · ] exp(−S(φ, · · · )) .

In a topological field theory, there is a nilpotent scalar symmetry, known as theBRST symmetry, and the Oi live in the cohomology of the generator of that sym-metry, called the BRST operator and typically denoted Q. The quantity S is knownas the ‘action.’ For strings propagating on a space or stack X, the action is an in-tegral over the worldsheet Σ of a quantity constructed from maps φ : Σ → X, andvarious Grassmann-valued quantities ψi,ı

± that are sections of√KΣ tensored with

a pullback of some part of the complexified tangent bundle of X. In the analogywith statistics, the exponential exp(−S) acts as an unnormalized probability for

any given set of φ, ψi,ı± .

The action for a Landau-Ginzburg model has the form

S =

∫Σ

d2x(gij∂φ

i∂φj + igijψj+Dzψ

i+ + igijψ

j−Dzψ

i− + Rijkψ

i+ψ

j+ψ

k−ψ

−

+ gij∂iW∂jW + ψi+ψ

j−Di∂jW + ψı

+ψj−Dı∂jW

),

where W : X → C is a holomorphic function known as the superpotential. Thesuperpotential W : X → C is holomorphic (so Landau-Ginzburg models are onlyinteresting when X is noncompact).

A nonlinear sigma model is a special case of a Landau-Ginzburg model, specifi-cally the case that the superpotential W is identically zero. Nonlinear sigma modelsdescribe strings propagating on X; a Landau-Ginzburg model describes somethingslightly more complicated.

For ordinary nonlinear sigma models, there are two topological twists, knownas the A- and B-models (see e.g. [W91]).

The A-model is a topological field theory whose correlation functions turn outto be independent of the complex structure on X. To build the A-model from anonlinear sigma model, we modify the worldsheet spinors ψi

+ and ψı− to be scalars:

ψi+ ∈ Γ(φ∗(T 1,0X)) → χi, ψı

− ∈ Γ(φ∗(T 0,1X)) → χı.

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LANDAU-GINZBURG MODELS, GERBES, AND KUZNETSOV’S H.P.D. 3

The BRST operator Q acts as

Q · φi = χi, Q · φı = χı, Q · χ = 0, Q2 = 0,

and so we identify

χμ ∼ dxμ, Q ∼ d.

The states of this theory are

bμ···νχμ · · ·χν ↔ H ·,·(X).

The B-model is a topological field theory whose correlation functions turn outto be independent of the Kahler or symplectic structure on X. To build the B-model from a nonlinear sigma model, we modify the worldsheet spinors ψı

± to beworldsheet scalars. It is convenient to define

ηı = ψı+ + ψı

−, θi = gij

(ψj+ − ψj

−

),

in terms of which the BRST operator Q acts as

Q · φi = 0, Q · φı = ηı, Q · ηı = 0, Q · θj = 0, Q2 = 0.

We identify

ηı ↔ dzı, θj ↔ ∂

∂zj, Q ↔ ∂,

and so the states are

bj1···jmı1···ın ηı1 · · · ηınθj1 · · · θjm ↔ Hn (X,ΛmTX) .

Next, we shall outline the A- and B-topological twists of Landau-Ginzburgmodels (i.e. nonlinear sigma models with nonvanishing superpotential W ), begin-

ning with the Landau-Ginzburg B-model. We redefine the ψi,ı± in exactly the same

fashion as for the B-model previously, and define η and θ as before. The onlyessential modification is that the action of the BRST operator changes to

Q · φi = 0, Q · φı = ηı, Q · ηı = 0, Q · θj = ∂jW, Q2 = 0.

Specifically, the BRST variation of θj is no longer zero, but rather is proportionalto a derivative of the superpotential. We can identify

ηı ↔ dzı, θj ↔ ∂

∂zj, Q ↔ ∂,

and the states of the theory

b(φ)j1···jmı1···ın ηı1 · · · ηınθj1 · · · θjmare now interpreted as elements of the hypercohomology group

H·(X, · · · −→ Λ2TX

dW−→ TXdW−→ OX

).

Since B-twisted Landau-Ginzburg models over nontrivial spaces are rarely dis-cussed in the literature, let us take a moment to check that the results above cor-rectly generalize results for both the ordinary B-twisted nonlinear sigma models,as well as Landau-Ginzburg models on vector spaces.

(1) First, consider the standard B-model obtained by setting the superpo-tential to zero. In this case, the hypercohomology group above triviallyreduces to H · (X,Λ·TX), reproducing the result discussed earlier for thiscase.

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4 ERIC SHARPE

(2) Next, consider the B-twist of a Landau-Ginzburg model over X = Cn

with superpotential W a quasi-homogeneous polynomial. In this case, thecoefficient sequence of the hypercohomology group is a Koszul resolutionof the fat point dW = 0, so the hypercohomology reduces to

C[x1, · · · , xn]/(dW )

which is the standard result for this case (see e.g. [V91]).

See [GS08a] for more information on B-twisted Landau-Ginzburg models on non-trivial spaces.

Next, let us consider A-twists of Landau-Ginzburg models. Here, the nonlinearsigma model twist does not by itself give a well-defined result. In particular, the A-twist of a nonlinear sigma model makes ψi

+ a worldsheet scalar and ψj− a worldsheet

one-form, but a Landau-Ginzburg model action contains a term∫Σ

ψi+ψ

j−Di∂jW.

If we were to perform the same A-twist as in a nonlinear sigma model, then theterm above would involve integrating a one-form over the worldsheet Σ, which doesnot make sense. Therefore, the A-twist must be something slightly different thanin a nonlinear sigma model.

There are at least two different ways to fix this particular problem.

• One way is to multiply offending terms in the action by another 1-form,e.g. multiply the superpotential W by a holomorphic section of KΣ. Thisis the approach used to solve a closely related problem in [W94].

• Another way is to combine the twist with a U(1) action, resulting infermions coupling to different bundles.

The second method is advantageous for physics, so it is the method that we shallfollow, but, it has the disadvantage that not all Landau-Ginzburg models will admitan A-twist in this prescription – only those such that the space X admits a U(1)action, with respect to which the superpotential is quasi-homogeneous.

Let us assume that such a U(1) action exists. Let Q(ψi) be a set of numbersencoding the quasi-homogeneity of the superpotential under the U(1) action:

W(λQ(ψi)φi

)= λW (φi).

Then, to define this A-twist, we change the bundles to which fermions coupleschematically as follows:

ψ → Γ(original⊗K

−(1/2)QR

Σ ⊗K−(1/2)QL

Σ

)

where

QR,L(ψ) = Q(ψ) +

⎧⎨⎩

1 ψ = ψi+, R

1 ψ = ψi−, L

0 else.

For example, consider the case of a Landau-Ginzburg model over X = Cn,with W a quasi-homogeneous polynomial of degree d. In this case, to perform the

operation above, we would need to make sense of e.g. K1/(2d)Σ . Since on a genus g

worldsheet, c1(KΣ) is not divisible by 2d for d > 1 in general, we have a problem.There are two ways to proceed:

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LANDAU-GINZBURG MODELS, GERBES, AND KUZNETSOV’S H.P.D. 5

• One way is to couple to topological gravity. After doing so, KΣ mustbe modified to take into account punctures and so forth. In this case,on a worldsheet of essentially any genus, one can find a suitable set of

punctures so that K1/(2d)Σ makes sense for d > 1. This is the approach

implicitly followed by [FJR07a, FJR07b].• If we do not couple to topological gravity, then we cannot make sense of

K1/(2d)Σ in general, and so we cannot make sense of the A-twist in this

particular example. This is the route followed in [GS08a, GS08b].

We shall work with the latter case. As a result, there will implicitly be a selectionrule that further restricts twistable examples, beyond merely requiring the existenceof a suitable U(1) action.

Now, let us consider an example which is twistable if one does not couple totopological gravity. Specifically, consider a Landau-Ginzburg model on

X = Tot(E∨ π−→ B

)

(E a holomorphic vector bundle over B) with superpotential W = pπ∗s, p a fibercoordinate1 on E∨, and s a holomorphic section of E . The U(1) action acts asphases on the fibers of the bundle.

It can be shown that correlation functions in this theory match those in anonlinear sigma model on s = 0 ⊂ B. We see this as follows. In prototypicalcases, they can be written schematically as

〈O1 · · · On〉 =

∫M

ω1 ∧ · · · ∧ ωn

·∫

dχpdχp exp(−|s|2 − χpdziDis − c.c. − Fijdz

idzjχpχp).

︸ ︷︷ ︸Mathai−Quillen form

where M is some (compactified) moduli space of holomorphic maps into B. TheMathai-Quillen form is a representative of a Thom class, so

〈O1 · · · On〉 =∫M ω1 ∧ · · · ∧ ωn ∧ Eul(Ns=0/M)

=∫s=0 ω1 ∧ · · · ∧ ωn,

which matches correlation functions in a nonlinear sigma model on s = 0. Thisis not a coincidence, as we shall see shortly.

To understand the matching above, we must take a moment to explain therenormalization group. The renormalization group (strictly speaking, a semigroup,not a group) constructs a series of quantum field theories in which each element isan approximation to the previous one, valid at longer distance scales. The effectof the renormalization group is much like starting with a picture and standingfurther and further away from it, to get successive approximations. The final resultmight look very different from the starting point. By repeating this approximationmany times, one constructs a sequence of theories, said to lie along the flow of therenormalization group.

The renormalization group is a very useful, and widely used, idea in quantumfield theory. Unfortunately, as a practical matter, it is usually impossible to followit completely explicitly. Typically, the best one can do is construct an asymptotic

1i.e. a tautological section of π∗E∨.

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6 ERIC SHARPE

series approximation to the tangent vector to the renormalization group at anygiven point in its motion.

The renormalization group is pertinent here because it has the property thatit preserves topological field theory structures. In other words, if two physicaltheories are related by renormalization group flow, then correlation functions ina topological twist of one should match correlation functions in a correspondingtopological twist of the other.

For example, in principle the Landau-Ginzburg model discussed above, on

X = Tot(E∨ π−→ B

)

with W = pπ∗s, flows under the action of the renormalization group to a nonlinearsigma model on s = 0 ⊂ B. This is the basic reason why the A-twisted correlationfunctions in the Landau-Ginzburg model should match those in the nonlinear sigmamodel on s = 0.

This observation can have computational benefits. For example, consider curve-counting in a degree 5 (quintic) hypersurface in P4. To compute A-model correla-tion functions directly, one needs to know the moduli space of curves in the quintic,which can be somewhat complicated. If we apply the idea of the renormaliza-tion group, then we can replace the nonlinear sigma model on the quintic with aLandau-Ginzburg model on

Tot(O(−5) −→ P4

).

In this Landau-Ginzburg model, curve-counting involves moduli spaces of curvesin P4, which are much simpler than moduli spaces of curves in the quintic. In themathematics literature, this is a standard mathematical trick for curve-counting(see e.g. [Kont94]), though its physical realization in Landau-Ginzburg modelsseems to be novel.

So far we have outlined the A- and B-topological twists of Landau-Ginzburgmodels, and talked about how the renormalization group is realized mathematicallyin these examples via a Thom class computation. Something closely analogoushappens in elliptic genus computations. For example, an elliptic genus of a Landau-Ginzburg model on

X = Tot(E∨ π−→ B

)

with superpotential as before, is given by [AndoS09]

∫B

Td(TB) ∧ ch

(Λ−1(TB)⊗ Λ−1(E∨)

⊗n=1,2,3,···

Sqn((TB)C)⊗

n=0,1,2,···Sqn((E∨)C)

⊗n=1,2,3,···

Λ−qn((TB)C)⊗

n=1,2,3,···Λqn((E∨)C)

).

It can be shown [AndoS09] that the elliptic genus above matches the Witten genusof s = 0 ⊂ B, by virtue of a Thom class computation.

In the case of A-twisted correlation functions, we saw the renormalization groupwas realized mathematically via a Mathai-Quillen representative of a Thom form.It is true in general, as in the example above, that something analogous happens

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LANDAU-GINZBURG MODELS, GERBES, AND KUZNETSOV’S H.P.D. 7

in elliptic genera: elliptic genera of Landau-Ginzburg and nonlinear sigma modelsrelated physically by renormalization group flow, are related mathematically byThom forms. This suggests a mathematical interpretation of the renormalizationgroup in twisted theories as a Thom class, possibly arising ultimately from an un-derlying Atiyah-Jeffrey/Baulieu-Singer-type of description [AJ90, Bau-Singer].

3. Landau-Ginzburg Models on Gerbes

3.1. Strings on stacks and gerbes. There are two basic motivations forconsidering stacks in physics: first, the possibility of building new string compact-ifications, new conformal field theories, and second, to better understand certainexisting string compactifications (such as orbifolds).

The first question a physicist must ask is, how does one construct quantumfield theories for strings propagating on stacks? Put another way, how to makesense of strings on stacks concretely?

To do this, we can use the fact that smooth, separated, generically tameDeligne-Mumford stacks of finite type over a field can be presented as a globalquotient [X/G] for X a space and G a group. (G need not be finite, and need notact effectively.) (See for example [Kresch05][theorem 4.4] or [Tot02].) To such apresentation, one associates a “G-gauged sigma model on X” [PSa, PSb, PSc].

The first problem is that such presentations of a given stack are not unique,and those presentations can have very different physics. This problem can be fixedwith the renormalization group: we conjecture that stacks are associated to certainequivalence classes (known as “universality classes”) of renormalization group flowin gauged sigma models.

Analogous issues arise in other relatively recent developments in physics, suchas in the physical realization of derived categories. There, localization on quasi-isomorphisms is realized physically by (boundary) renormalization group flow, andfixes a closely analogous potential presentation-dependence issue.

There are various other technicalities, aside from the potential presentation-dependence issue. For example, let us specialize to gerbes, i.e. quotients bynoneffectively-acting groups. One problem with gerbes is that the massless spec-trum computation which gives sensible results for other stacks, for gerbes resultsin multiple dimension zero states. The existence of multiple dimension zero statessignals a violation of one of the foundational axioms of quantum field theory, knownas cluster decomposition.

There is, to our knowledge, a single known loophole: nonlinear sigma mod-els with disconnected target spaces also have multiple dimension zero operators,counting the number of components. In such a case, the quantum field theory isnevertheless well-behaved, despite violating cluster decomposition, because clusterdecomposition has been violated in the mildest fashion possible. We believe thatis what is happening in strings on gerbes.

A little more specifically, we conjecture that strings on gerbes are equivalent tostrings on disjoint unions of spaces. Consider [X/H], where

1 −→ G −→ H −→ K −→ 1

and G acts trivially on X (and is assumed finite). We and the other authors of[HHPSA] conjecture that

CFT([X/H]) = CFT([

(X × G)/K])

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8 ERIC SHARPE

where G is the set of irreducible representations of G, and the K action can beobtained by diagram chasing (or see [HHPSA]). The elements of the disjointunion on the right-hand side above have various B fields2 we have here suppressedfor brevity. We and the other authors of [HHPSA] call this our decompositionconjecture.

For special gerbes known as “banded” gerbes, K acts trivially upon G. In thiscase, the decomposition conjecture reduces to

CFT(G− gerbe on X) = CFT

⎛⎝∐

G

X

⎞⎠

where the various copies of X on the right-hand side above have different B fields.The B field on a copy of X corresponding to a particular element of G, correspond-ing to a map Z(G) → U(1), is determined by the image of the characteristic classof the gerbe:

H2(X,Z(G))Z(G)→U(1)−→ H2(X,U(1)).

There are a number of checks and implications of the decomposition conjectureabove. Very briefly,

• For global quotients by finite groups, one can compute physical quanti-ties known as partition functions (related to elliptic genera) exactly atarbitrary worldsheet genus, and check that the decomposition conjecturemakes correct statements about those partition functions.

• Implies that

KH(X) = twistedKK(X × G)

which can be checked independently.• Implies known facts about sheaf theory on gerbes.• Has nontrivial implications for Gromov-Witten theory of stacks, implica-tions which are currently being checked in [AJT08, AJT09a, AJT09b].

3.2. Landau-Ginzburg models. Now, let us apply the decomposition con-jecture above to some examples of Landau-Ginzburg models. Consider, for example,a Landau-Ginzburg model over

X = Tot(O(−1)⊕2n+2 −→ Pn

[2,2,··· ,2]

)

with superpotential

W =∑a

paGa(φ) =∑ij

φiAij(p)φj

where the Ga(φ) are quadric polynomials in the fiber coordinates φ, and the p’s arehomogeneous coordinates on Pn

[2,2,··· ,2], a Z2 gerbe over Pn. The matrix Aij(p) has

entries that are degree one in the p’s, and is a rewriting of the four quadrics Ga.Writing the superpotential in the form above emphasizes that it is giving a

mass to the φ fields, at least away from the locus detA = 0. Thus, at leastaway from that locus, this theory appears to be that of a nonlinear sigma model onPn

[2,2,··· ,2], a Z2 gerbe, which by the decomposition conjecture, physics will see as

a double cover. A more detailed analysis of the physics [CDHPS, DS, HHPSA]

2On a space Y , the “B field” is an element of H2(Y,U(1)).

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LANDAU-GINZBURG MODELS, GERBES, AND KUZNETSOV’S H.P.D. 9

reveals a ‘Berry phase’ about the locus detA = 0, which seems to imply that thephysics sees a branched double cover of Pn, branched over the degree 2n+ 2 locusdetA = 0, which would make it a Calabi-Yau.

In the cases n = 1, 2, our story ends with the branched double cover, describing(for n = 1) an elliptic curve, and (for n = 2) a K3 surface. However, beginning atn = 3 (for which the branched double cover would be Clemens’ octic double solid),we find something more interesting. In particular, the branched double cover inthis example has singularities, but an analysis of the physics reveals that it doesnot see those singularities, it behaves as if instead the space were smooth.

We believe that in this case, this Landau-Ginzburg model is actually describ-ing a “noncommutative resolution” of the branched double cover worked out byKuznetsov [Kuz1, Kuz2, Kuz3] and others (see for example [Kont98, Soi03,Cos04, VdB1, VdB2, VdB3]).

There are several notions of noncommutative geometry appearing in the physicsliterature, see e.g. [SW, RW]. However, this notion is (to our knowledge) dis-tinct from other notions of noncommutative geometry that have so far appeared inphysics.

A noncommutative space, in this context, is defined by its category of sheaves.In the present case, the noncommutative resolution in question is defined by thepair (P3, B) where B is the sheaf of even parts of Clifford algebras associated withthe universal quadric over P3: ∑

a

paGa(φ)

– in other words, the GLSM superpotential. On the noncommutative space sodefined, B acts as the structure sheaf, and other sheaves are sheaves of B-modules.Equivalently, instead of working with (P3, B), we could work with the brancheddouble cover f : Z → P3 together with a sheaf of algebras A for which f∗A = B.One can work with either the branched double cover or (P3, B); that said, it canbe convenient to use the description (P3, B), and so we shall do so here.

Physically, the sheaf B above, and sheaves of B-modules, are interpreted as “D-branes” in the Landau-Ginzburg model. In general, a D-brane is some propagatingmultidimensional object, defined by boundary conditions on an open string plusadditional data confined to the boundary. In the case of a Landau-Ginzburg model,that extra data takes the form of a ‘matrix factorization’: a submanifold S withvector bundles E ,F → S and maps F : E → F , G : F → E such that F G andG F are both W |S times the identity maps on F and E , respectively.

Intuitively, we can see that matrix factorizations in this Landau-Ginzburgmodel match the sheaves of B-modules of Kuznetsov’s noncommutative resolu-tion as follows. Work locally over the P3 (physically, in a Born-Oppenheimerapproximation; mathematically, in families). At a point on P3, the superpotentialis a quadratic polynomial. Physically, such a superpotential has no closed stringmodes, but it is known that one still has D0-branes with a Clifford algebra structure[KapLi]. Here, we have a Landau-Ginzburg model fibered over P3, which givessheaves of Clifford algebras (determined by the universal quadric / superpotential)and modules thereof. Thus, we see that matrix factorizations/D-branes duplicateKuznetsov’s definition of his noncommutative resolution, and so we interpret thismodel as living on the noncommutative resolution.

The picture above is of interest to physicists for a variety of reasons, including:

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10 ERIC SHARPE

• Geometry is being realized physically in a novel fashion. Ordinarily whennonlinear sigma models are constructed as the endpoint of RG flow, theLandau-Ginzburg model in question is realizing a complete intersection,arising as the critical locus of the superpotential. Here, the geometry isbeing realized physically in a very different fashion, via nonperturbativeeffects in a gauge theory.

• This is the first physical realization of a noncommutative resolution (forthis particular notion of noncommutative). In particular, this is a newkind of conformal field theory, not previously analyzed by physicists.

• These Landau-Ginzburg models appear as limit points in moduli spacesof “gauged linear sigma models,” and there give counterexamples to oldunproven lore that all such limit points should be related by birationaltransformations.

More recent work on this subject includes an analysis of “D-brane probes”in these backgrounds. On general principles, the “D-brane probe moduli space”should be a (possibly non-Kahler) small resolution of the singular space, and thatis what is found in [Add09a, Add09b].

4. Homological Projective Duality

Kuznetsov describes [Kuz1, Kuz2, Kuz3] a notion of “homological projectiveduality” (induced on linear sections) that relates different spaces and/or noncommu-tative resolutions, including the noncommutative resolutions above. His exampleshave physical realizations in gauged linear sigma models, which reduce to Landau-Ginzburg models such as the ones above at various endpoints. In more detail,given a gauged linear sigma model, one can apply renormalization group flow toobtain a Landau-Ginzburg model (and sometimes a nonlinear sigma model afterfurther renormalization group flow). By altering parameters in the gauged linearsigma model, one can obtain Landau-Ginzburg models over different spaces, real-izing the duality. We list below several examples of homological projective duals,and corresponding Landau-Ginzburg models:

P3[2, 2] branched double cover of P1,branched over a degree 4 locus

LG model on Tot(O(−2)⊕2 → P3) LG model on Tot(O(−1)⊕4 → P1[2,2])

P5[2, 2, 2] branched double cover of P2,branched over a degree 6 locus

LG model on Tot(O(−2)⊕3 → P5) LG model on Tot(O(−1)⊕6 → P2[2,2,2])

P7[2, 2, 2, 2] nc res’n of branched double cover of P3,branched over a degree 8 locus

LG model on Tot(O(−2)⊕4 → P7 LG model on Tot(O(−1)⊕8 → P3[2,2,2,2]

P2g+1[2, 2] branched double cover of P1,branched over a degree 2g + 2 locus

LG model on Tot(O(−2)⊕2 → P2g+1) LG model on Tot(O(−1)⊕2g+2 → P1[2,2])

Additional examples, including additional noncommutative resolutions, are listedin [CDHPS, DS, HT]. In each case above, the Landau-Ginzburg superpotential

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LANDAU-GINZBURG MODELS, GERBES, AND KUZNETSOV’S H.P.D. 11

can be written in the form

W =∑ij

xixjAij(p)

and so we have omitted it from the tables above.Further examples of the same form are listed in [CDHPS].There are additional examples of homological projective duals physically re-

alized by gauged linear sigma models in [HT, DS], but some of their Landau-Ginzburg models are not as well understood, so we do not list them here.

Note that in every example in the list above (in fact, in every example in[CDHPS]), the dual Landau-Ginzburg models are defined over birational spaces.With this in mind, we and the other authors of [CDHPS] conjecture that [CDHPS]

All examples of homological projective duality, induced on linearsections, are equivalent to Orlov-type [Orlov03, Wal04] equiva-lences between matrix factorizations in Landau-Ginzburg modelsover birational spaces, with compatible superpotentials.

We and the other authors of [CDHPS] also conjecture that in all examples ofgauged linear sigma models, the endpoints of the Kahler moduli spaces are relatedby Kuznetsov’s homological projective duality [CDHPS]. This certainly holds truein the examples here, but is very difficult to check more generally, so we leave it asa conjecture.

5. Summary

In this talk we have summarized several recent developments in Landau-Ginzburgmodels and related areas. We began by outlining A- and B-type topological fieldtheories based on Landau-Ginzburg models, including results from B-twisted Landau-Ginzburg models on nontrivial spaces, and outlined multiple possible A-twists, in-cluding details of one particular notion of A-twist that is of physical interest.

We then briefly outlined how to make sense of strings propagating on stacks andgerbes and various technical issues arising in such notions. For strings on gerbes,we outlined the decomposition conjecture, relating a string on a gerbe to a string ona disjoint union. We then applied the decomposition conjecture to understand thephysics of certain Landau-Ginzburg models on total spaces of bundles over gerbes,which we then interpreted in terms of branched double covers and, sometimes,noncommutative resolutions thereof, which gave us novel physical realizations ofgeometry, as well as new conformal field theories.

References

[Add09a] N. Addington, “The derived category of the intersection of four quadrics,” arXiv:

0904.1764.[Add09b] N. Addington, “Spinor sheaves on singular quadrics,” arXiv: 0904.1766.[AJ90] M. Atiyah, L. Jeffrey, “Topological Lagrangians and cohomology,” J. Geom. Phys.

7 (1990) 119-136.[AJT08] E. Andreini, Y. Jiang, H.-H. Tseng, “On Gromov-Witten theory of root gerbes,”

arXiv: 0812.4477.[AJT09a] E. Andreini, Y. Jiang, H.-H. Tseng, “Gromov-Witten theory of product stacks,”

arXiv: 0905.2258.[AJT09b] E. Andreini, Y. Jiang, H.-H. Tseng, “Gromov-Witten theory of etale gerbes, I: root

gerbes,” arXiv: 0907.2087.

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[AndoS09] M. Ando, E. Sharpe, “Elliptic genera of Landau-Ginzburg models over nontrivialspaces,” arXiv: 0905.1285.

[Bau-Singer] L. Baulieu, I. Singer, “The topological sigma model,” Comm. Math. Phys. 125(1989) 227-237.

[CDHPS] A. Caldararu, J. Distler, S. Hellerman, T. Pantev, E. Sharpe, “Non-birational twistedderived equivalences in abelian GLSM’s,” arXiv: 0709.3855, to appear in Comm.Math. Phys.

[Cos04] K. Costello, “Topological conformal field theories and Calabi-Yau categories,” Adv.Math. 210 (2007) 165-214, math.QA/0412149.

[DS] R. Donagi, E. Sharpe, “GLSMs for partial flag manifolds,” J. Geom. Phys. 58 (2008)1662-1692, arXiv: 0704.1761.

[FJR07a] H. Fan, T. Jarvis, Y. Ruan, “The Witten equation, mirror symmetry, and quantumsingularity theory,” arXiv: 0712.4021.

[FJR07b] H. Fan, T. Jarvis, Y. Ruan, “The Witten equation and its virtual fundamentalcycle,” arXiv: 0712.4025.

[GS08a] J. Guffin, E. Sharpe, “A-twisted Landau-Ginzburg models,” arXiv: 0801.3836, toappear in J. Geom. Phys.

[GS08b] J. Guffin, E. Sharpe, “A-twisted heterotic Landau-Ginzburg models,” arXiv:

0801.3955, to appear in J. Geom. Phys.[HHPSA] S. Hellerman, A. Henriques, T. Pantev, E. Sharpe, M. Ando, “Cluster decompo-

sition, T-duality, and gerby CFT’s,” Adv. Theor. Math. Phys. 11 (2007) 751-818,hep-th/0606034.

[HT] K. Hori, D. Tong, “Aspects of non-Abelian gauge dynamics in two-dimensional N =(2, 2) theories,” JHEP 0705 (2007) 079, hep-th/0609032.

[KapLi] A. Kapustin, Y. Li, “D-branes in Landau-Ginzburg models and algebraic geometry,”JHEP 0312 (2003) 005, hep-th/0210296.

[Kont94] M. Kontsevich, “Enumeration of rational curves via torus actions,” pp. 335-368in The moduli spaces of curves (Texel Island, 1994) (R. Dijkgraaf, C. Faber, G.van der Geer, eds.), Progress in Math. 129, Birkhauser, Boston-Basel-Berlin, 1995,hep-th/9405035.

[Kont98] M. Kontsevich, “Course on non-commutative geometry,” ENS, 1998, lecture notesat http://www.math.uchicago.edu/~arinkin/langlands/kontsevich.ps.

[Kresch05] A. Kresch, “On the geometry of Deligne-Mumford stacks,” pp. 259-271 in Algebraicgeometry (Seattle, 2005), Proc. Sympos. Pure. Math. 80, part 1, Amer. Math. Soc.,Providence, RI, 2009.

[Kuz1] A. Kuznetsov, “Homological projective duality,” Publ. Math. Inst. Hautes EtudesSci. 105 (2007) 157-220, math.AG/0507292.

[Kuz2] A. Kuznetsov, “Derived categories of quadric fibrations and intersections ofquadrics,” Adv. Math. 218 (2008) 1340-1369, math.AG/0510670.

[Kuz3] A. Kuznetsov, “Homological projective duality for Grassmannians of lines,”math.AG/0610957.

[Orlov03] D. Orlov, “Triangulated categories of singularities and D-branes in Landau-Ginzburgmodels,” Proc. Steklov Inst. Math 246 (2004) 227-248, math.AG/0302304.

[PSa] T. Pantev, E. Sharpe, “Notes on gauging noneffective group actions,”hep-th/0502027.

[PSb] T. Pantev, E. Sharpe, “String compactifications on Calabi-Yau stacks,” Nucl. Phys.B733 (2006) 233-296, hep-th/0502044.

[PSc] T. Pantev, E. Sharpe, “GLSMs for gerbes (and other toric stacks),” Adv. Theor.Math. Phys. 10 (2006) 77-121, hep-th/0502053.

[RW] D. Roggenkamp, K. Wendland, “Limits and degenerations of unitary conformal fieldtheories,” Comm. Math. Phys. 251 (2004) 589-643, hep-th/0308143.

[Soi03] Y. Soibelman, “Lectures on deformation theory and mirror symmetry,” IPAM, 2003,http://www.math.ksu.edu/~soibel/ipam-final.ps.

[SW] N. Seiberg, E. Witten, “String theory and noncommutative geometry,” JHEP 9909(1999) 032, hep-th/9908142.

[Tot02] B. Totaro, “The resolution property for schemes and stacks,” math/0207210.[VdB1] M. van den Bergh, “Three-dimensional flops and noncommutative rings,” Duke

Math. J. 122 (2004) 423-455.

248

LANDAU-GINZBURG MODELS, GERBES, AND KUZNETSOV’S H.P.D. 13

[VdB2] M. van den Bergh, “Non-commutative crepant resolutions,” pp. 749-770 in Thelegacy of Niels Henrik Abel: the Abel bicentennial, Oslo, 2002, Springer, 2004.

[VdB3] M. van den Bergh, “Non-commutative quadrics,” arXiv: 0807.3753.[V91] C. Vafa, “Topological Landau-Ginzburg models,” Mod. Phys. Lett. A6 (1991) 337-

346.[Wal04] J. Walcher, “Stability of Landau-Ginzburg branes,” J. Math. Phys. 46 (2005) 082305,

hep-th/0412274.

[W91] E. Witten, “Mirror manifolds and topological field theory,” pp. 121-160 in Mir-ror Symmetry I, ed. S.-T. Yau, Amer. Math. Society, International Press, 1998,hep-th/9112056.

[W94] E. Witten, “Supersymmetric Yang-Mills theory on a four-manifold,” J. Math. Phys.35 (1994) 5101-5135, hep-th/9403195.

Physics Department, Robeson Hall (0435), Virginia Tech, Blacksburg, VA 24061

E-mail address: ersharpe@vt.edu

249

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