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The structure of the super-W∞(λ) algebraBergshoeff, E.; Wit, B. de; Vasiliev, M.

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NuclearPhysicsB366 (1991)315—346 _________________

North-Holland N U C LEARPHYSICS B

THE STRUCTURE OF THE SUPER-W~(A)ALGEBRA

E. BERGSHOEFF

TheoryDivision, CERN,CH-1211Geneva23, Switzerland

B. deWIT

Institutefor TheoreticalPhysics,Universityof Utrecht, Princetonplein5, 3584 CCUtrecht,TheNetherlands

M. VASILIEV

TheoreticalDepartment,P.N. LebedevInstitute, Academyof Sciencesof the USSR,117924LeninskyProspect53, Moscow, USSR

Received30 April 1991

We give a comprehensivetreatment of the super-W,,(A) algebra, an extensionof thesuper-Virasoroalgebra that containsgeneratorsof spin s~ ~. The parameterA defines theembeddingof theVirasorosubalgebra.We describehow to obtain thesuper-W,(A)algebrafromthe associativealgebraof superspacedifferential operators.We discuss the structureof thisassociativealgebraand its relation with theso-calledwedgealgebra,in which thegeneratorsforgiven spin are restricted to finite-dimensional representationsof sl(2). From the super-W,~(A)algebraone canobtaina variety of ~ algebrasby consistenttruncationsfor specificvaluesof A.Without truncationthe algebrasare formally isomorphicfor differentvaluesof A. We presentarealizationin termsof the currentsof a supersymmetricbc system.

1. Introduction

Extensionsof the Virasoro algebra play an important role in the study of

conformalfield theorieswith central chargec ~ 1. Theso-calledWN algebrasareextensionsof the Virasoro algebrathat contain generatorsof all conformal spinsfrom s = 2 up to s = N. The first such algebrais the W3 algebraproposedbyZamolodchikov~], which containsa spin-3 field in additionto the spin-2 energy~—momentumtensorassociatedwith the Virasoroalgebra.A characteristicfeatureofthesealgebrasis that they are non-linear:the closureof the operator algebraisachievedbecauseoperatorsof spin higher than N that arise naturally in theoperatorproduct expansions,take the form of normal-orderedproducts of thelower-spinfields. TheseWN algebras,both the bosonic and the supersymmetricversions,havebeen studied from a variety of viewpoints (see for instancerefs.[2,3]).

0550-3213/9l/$03.50©1991 Elsevier SciencePublishersB.V. All rights reserved

316 E. Bergshoeffet al. / Super-W~(A)algebra

It may be possibleto interpret the WN algebrasin the context of a ~-typealgebra.Such an algebrais basedon an infinite numberof higher-spinoperatorswith spin s ~ 1, or 2 andmultiplicity 1. In contradistinctionwith the WN algebras,thesealgebrasarelinear. It is thensuggestiveto view the WN algebrasas a ~,0-type

algebrasubject to certain non-linearconstraints.The W~,-typealgebrasmay alsoarise in an appropriatelarge-N limit of WN [4,5]. This was in fact one of theoriginal motivations for studying the ~~-type algebrasand in ref. [6] two suchalgebraswereconstructed,denotedby ~,. andW1+

The W algebrasarenot only relevantfor conformalfield theory. Forinstance,inrecent studiesof 2-dimensionalgravity by meansof matrix models, it wasshownthat thesquareroot of the partition function is subjectto thevacuumconditionsof

a W algebra.The simplestcaseconcernsthe 1-matrix model where the relevantalgebrais the Virasoro algebra[7]. More generally,for the (N — 1)-matrixmodelwe have WN, but in fact one can show that there are more conditions [8,9].According to ref. [91the squareroot of the partition function is subject to thevacuumcondition of the W1 + ~ algebra.In fact thereis an intricate relationshipbetweenW algebrasand systemsof non-linear integrabledifferential equations[101.Recentlyit was shownfor the KP hierarchythat the Poisson-bracketalgebracorrespondingto the first hamiltonianstructuregeneratesthe W1~~algebra[11].

Justas the Virasoroalgebraarisesas a residualsymmetryupon gauge-fixingthe

two-dimensional diffeomorphism invariance, one may wonderwhether a corre-spondingsituationexists for the W algebras.This leadsto the notion of W gravity.

Field-theoreticrealizationsof W algebrasin which all W symmetriesaregaugedbyintroducing correspondinggaugefields werediscussedin ref. [12].

Certain versionsor subalgebrasof W~,correspondto area-preserving(or sym-plectic) diffeomorphisms.For instance,the so-calledw~andw1 + algebrascorre-spondto the area-preservingdiffeomorphismsof a cylinder [13,61. Thentherearethe so-calledwedgesubalgebras,which follow from restricting the spin-sgenera-tors to (2s — 1)-dimensionalrepresentationsof sl(2).Thesealgebrasplay a role in

the residual invariance group of the light-cone formulation of relativistic(super)membranes[14]. Algebras of this type were also introduced[15—171todescribeinteractingfield theoriesof higher-spingaugefields in 3 + 1 dimensions[18]. Analogousconstructionswereusedin 2 + 1 dimensions[19]. Furthermore,the

area-preservingdiffeomorphismsare relatedto self-dualfour-dimensionalgravity(seefor instanceref. [20] andreferencesquotedtherein).

In this paperwe give a self-containedtreatmentof the supersymmetricW,0-typealgebrasbasedon the so-calledsuper-W~(A)algebra[211.The genericalgebrahasN = 2 supersymmetryand has generatorsof integerand half-integerspin s ~ 1with multiplicity 2, anda single spin-~ generator.It containsan infinite varietyofVirasorosubalgebras.The parameterA specifiesa particularVirasorosubalgebra,andtied to it, a so-calledwedgesubalgebra.Nevertheless,WOO(A) may be regardedas a singlealgebrawritten in aone-parameterfamily of bases,sothat algebraswith

E. Bergshoeffet al. / Super-WO/A) algebra 317

different valuesof A are isomorphic,at leastin the formalsense.In truncationsofthe algebrathis ceasesto be true. Thesetruncationsrequirespecific valuesfor A,so that the choice of the Virasoro subalgebrais fixed. The spin content of allhigher-spingeneratorsas (quasi-)conformaloperatorsis then characterizedwithrespectto this subalgebra.The wedgesubalgebrasdependon A and belongto aone-parameterfamily of inequivalentalgebras[22,23]. Whenincluding the centralcharge,also the Virasorosubalgebrasare inequivalentfor differentvaluesof A.

In many respectsthe situationis thesamefor extensionsof the Virasoroalgebrawith generatorsof spin s ~ 2, such as the super-Virasoroalgebra.For instance,ifone defines the super-Virasoroalgebra for a supersymmetricbc system[24] ofconformal weight A, then the algebrasfor different values of A are isomorphic.Unlike the caseof the W0,(A)algebra,thestructureconstantsof the super-Virasoroalgebra are independentof A, and furthermore, one can directly rewrite thesuper-Virasorocurrentsdefinedfor onevalue of A in terms of thosefor another

valueof A.As mentionedabovetruncationsof the algebracan be performedfor particular

valuesof the parameterA. For instance,in the bosoniccase,A = -~- correspondstothe W1 algebrawhile A = 0 yields the W,~algebraupon truncationto generatorsof spin s �~2. This particular truncationis only possiblefor this value of A. Thealgebrasw,~andw1 are found by a contraction[6]. For the supersymmetriccasewe havethe samesituation.For A = 0 onecantruncateto the superalgebrafoundin ref. [25], which canbe truncatedfurther to an algebrawith N= 1 supersymme-try, while for A = we find the superalgebraexhibitedin ref. [21] andanotheronewithout supersymmetry.The generic super-W~,,(A)algebra contains W00(A)~W~(A+ ~) as its bosonic subalgebra.Within the framework of the super-W,~(A)

algebraonecan convenientlyclarify the nature of the varioustruncationsof thealgebra and elucidate many of the results found previously from a differentperspective.Just as in ref. [21] we define the algebraboth in the context of the

associativealgebraof superspacedifferential operatorsandtheoperatoralgebraofquasi-primarycurrentsof a (supersymmetric)bc system.Wewill explicitly evaluatethe multiplication propertiesof the former. Furthermorewe discussthe relation

with the so-calledwedgealgebrasand makecontactwith previouswork on thesealgebras[6,22,23,26].

This paper is organizedas follows. In sect. 2 we introduce the super-W~(A)algebraexpressedin terms of superspacedifferential operators.We discussthemain featuresof this algebraandestablishthe existenceof an anti-automorphismwhich allows us to derive a number of important propertiesof the structureconstants.In sect. 3 we describethe structureof the bosonic associativealgebra,payingparticularattentionto thesl(2) subalgebra.We derive anexplicit expressionfor the multiplication constantsof the associativealgebrafor arbitraryvalueof A.We analyzethe so-calledwedgesubalgebra,which is isomorphicto the envelopingalgebra of sl(2) factored by an appropriateideal. In sect. 4 we extend this

318 E. Bergshoeffet a!. / Super-WJA)algebra

treatmentto the supersymmetriccase.Herethe wedgesubalgebrais isomorphictothe envelopingalgebraof osp(1,2) or osp(2,2), factoredby appropriateideals. In

sect. 5 we constructa representationin termsof a supersymmetricbc system.Thisleadsto central chargesin the algebra.In the genericcasethesechargeshaveamore complicatedstructurethan in the specific examplesexhibited so far. Some

convenientformulaehavebeencollectedin appendixA.

2. Super-W0,(X)

In ref. [21] we presentedthe super-W0~,(A)algebra in terms of superspacedifferentialoperators,by generalizingthe correspondingexpressionsfor thegener-atorsof the super-Virasoroalgebra,

0~

Gr = {z_~1/20 — A(2r — l)Z_r_t/2]O + ~_r+l/2

3 (2.1)

Here, superspacecoordinatesare denotedby z and 0, 8 d/dz, 8/80 is theleft-derivative, and A is the conformalweight. Motivated by theseexpressionsweconstructeda completeset of operatorsin terms of positive powersof derivatives8, arbitrarypowersof z andthe fermionic coordinateandits derivative.A suitablebasis is found by classifying thesegeneratorsaccordingto their maximal powers — 1 of 8, andrequiringthat they transformaccordingto appropriatenondecom-posablerepresentationsof sl(2). Hence s takespositive integervalues(s ~ 1), SO

that bosonicoperatorscarry spin s and the fermionic operatorsspin s — ~. Theexplicit expressionsare

s—i~ (n—s+ 1)

5_i_~al(s,A)z~~8Ii=0

O s—i

+ 0~-~ (n — s + 1)~__1[a~(s,A + — at(s,A)] z~O’,

s—i + 2A s-iL~= — ~ (n—s + 1)

5_1_~a~(s,A)z’~’O’i=0

O s—i+0— ~ (n—s+1)5_1~

i—Os—2A s—1+2A

X 2s—1a’(s,A + fl+ 2s—1 a’(s,A) z’~’O~,

s—i

~ 0 ~ (r — s+ ~ ~ 1a1(s,A)z~’~/2O’

i—O

O s—2+ — ~ (r — s + ~ 2 ~f3~(s,A)z~t+h/2OI, (2.2)

i=O

E. Bergshoeffet aL / Super-W~,(A)algebra 319

wherewe madeuseof the definition

F(a+n)(a)~~ [‘(a) =a(a+1)(a+2)...(a+n—i), with (a)0=i. (2.3)

Furthermore,a’(s,A), a’(s,A) and /3’(s,A) aredefinedby

1 (—2A—s+2)51~a1(s,A)=(5~ ) (s+i)51, (0~i~s—1),

1 (—2A—s+2)511a1(s,A)=(5~ ) (s+i—i)51, (0~i~s—i),

“ 2’ (—2A—s+2)52.~/3’(s,A) = ~ ) . (0~i~s—2). (2.4)I ~5 +

Somepropertiesof thesecoefficientsare discussedin appendixA.One easily verifies that ~ and ~ coincide with the super-Virasoro

generatorsdefined above (cf. (2.1)). Furthermore, apart from spin ~- whereG~~=~ thereis a two-fold degeneracyfor eachspin. In the Neveu-Schwarzsector, where the indices r take half-integer values, L~~?t,~ and G(~)j~,~2

generatethe osp(1,2) algebra.The operators(2.2) then transform accordingtonon-decomposablerepresentationsof osp(1,2), such that (Lv, ~ and~ constitute spin (s,s— ~) and (s,s+ ~) supermultiplets,respec-tively.

The differential operatorsgiven in (2.2) form an associativealgebra,which isisomorphicto a superextensionof a version of the Gl(cx) algebra.On the basisoftheir (anti)commutatorthey define an explicit representationof the super-W~(A)algebra. Special casesof this infinite-dimensionalLie superalgebrahave beenconsideredin the literature.For instance,for A = 0, one canverify from (2.2) and(2.4) that all generatorsare proportionalto eitheroneof thederivatives0 or 0/00,with the exceptionof L~’~and ~ Therefore the algebramay be consistently

restricted* to ~ and the higher-spingenerators~ and ~ with s ~ 2.One is then left with the super-W,,.algebraconstructedin ref. [25]. Of coursethesameargumentappliesto the bosoniccase.The bosonicLie algebraW~(A)canberealizedby the part of the generators~ independentof 00/00. When A = 0

one can consistentlyreducethe W~(A)algebrato the W~,algebraof ref. [6] aftersuppressingthe spin-i operator.For general A such a decouplingof the spin-i

operatordoesnot take place. For instance,the bosonic algebra~ coincides

* Observethat similar truncationsare possiblewheneverA takes negativeinteger or half-integer

values.However, the resulting algebrasno longercontain the Virasoroalgebra.

320 E. BergshoeffetaL / Super-J+~XA)algebra

with the ~ algebraproposedin ref. [6]. Finally the w~andw1~~algebrasmaybe obtainedby a contraction.

It is often convenientto sum over all the modes and to define correspondingdifferential operatorsin termsof commutingfunctions A(s)±(z)and anticommut-ing functions e~s~~(z)

L~= ~j( A)(( —O)~A~~~(z)) 01

O s—i

+ 0— ~ [a~(s,A + — at(s,A)] (( ~ 01,

s—i+2A~~~ — ~ a1(s,A)(( _3)5 tA(s)_(z)) 3’

i~O

05~ s—2A s—i+2A+ ,~ [~s —

1a’(s, A + ~)+ 2s — 1 a’(s,A)J((—3)’A~~~(z))8’,

~ 0~a’(s, A)(( —0)~’ ~)±(z)) 0’

O s—2

±~ ~/3’(s, A)(( ~ 01, (2.5)

where A~~~(z)=~ and ~9~~~(z)= ~~9(S)±zr+5i/2

Important information regardingthe A-dependenceof W~(A)follows from theexistenceof an anti-automorphismwhich is closely related to hermitian conjuga-

tion for superspacedifferential operators.The anti-automorphismis implementedby changingthe orderof thevariousoperators(including theparameterscontainedin (2.5)) and by replacing

0 0Z—*z 0—* —O 0—~—iO ——-~i— (2.6)30 00

suchthat the basic (anti)commutatorsbetweenz and 0 and 0 and 0/00 remainunchanged.Underthis anti-automorphismoperatorschangeaccordingto

~ G~5)i(_)~G~?~EA. (2.7)

This result follows from the identities(A.8)—(A.iO) given in appendixA. Alterna-tively, onecanfirst verify this result for the Virasorogenerators,whichshowsthatthe (quasi-conformal)spin doesnot.changeunderthe anti-automorphism.There-

E. Rergshoeffet a!. / Super-W,jA) algebra 321

fore, to verify the validity of (2.7), it suffices to check the leading terms(i.e. thetermswith the maximal numberof derivatives).

From (2.7) it follows that the WOO(A) structureconstantson the basisspecified in(2.2) or (2.5) are either symmetric or antisymmetricunder the replacementA

— A, dependingon the valuesof the spinsinvolved. To beprecise,the structureconstantsinvolving three~ operators,or one ~ andtwo ~ operators,are symmetric(antisymmetric)underthe changeA —* — A wheneverthe sumoftheir spinsis even(odd), while the structureconstantsthat involve two ~ andone ~ operator, or three L’~~operators,are antisymmetric (symmetric)wheneverthe sumof thethreespinsis even(odd). For the structureconstantsthatinvolve two fermionic operators,we havea similar result. If the third (bosonic)operatoris ~ the structureconstantsaresymmetric(antisymmetric)wheneverthe sum of the spins is odd (even), while, if the third operator is ~ thestructureconstantsaresymmetric(antisymmetric)wheneverthe sumof the spinsiseven(odd).

Furthermorewe can deduce from (2.7) that ±(—)~L~)~5and

exhibit the samecommutatoralgebra as the original operatorsL~±and ~

Therefore,we concludethat the superalgebrasWOO(A) andW~(+— A) are isomor-

phic.Clearly the value A = ~ is specialin this respect.On the basisof the anti-auto-

morphism one can show that many of the structureconstantswill vanish in thatcase.This enablesa consistentreductionof the algebrawhere oneretainsonly thegenerators~ and ~ with evens, and ~ with odd s. This algebrahasN = 2 supersymmetry.Another consistenttruncation is to retain the generators~ with evens andthe generatorsL

t~~and~ with odd s. Althoughthis isstill a superalgebrawe lost supersymmetryin this truncation.As weshall discussin

sect. 4, the algebra with A = ~ is also related to the symplectonhigher-spinsuperalgebra[15,27]. The above analysisalso appliesto the purely bosonic case.Then the anti-automorphismrelatesthe parametervaluesA and 1 — A so that thevalue A = ~ is special.

The aboveresultscanbe convenientlyexpressedin superfieldnotation. Super-spaceoperatorsare thendefinedin termsof superfieldtransformationparametersQ(s)±(z,0). The operators~ and~ areassociatedwith commutingparam-

eters ~ and ~ and Gt~~~ with anticommutingparameters~ Thesuperspaceoperatorsread

2s—2

L~’~=~ A~(s,~k—O

2s—i

~ A~(s,A)(D25(5~)Dk, (2.8)k=O

322 E. BergshoeffetaL / Super-W,/A)algebra

wherewe madeuseof the superspacederivative

0D=——0O with D2= —0. (2.9)

00

The coefficients A~(s,A) aredefinedby

s—1+2AA?~(s,A)= (—)‘a’(s,A), A~(s,A)= ~ 2s — i

A~t(s,A) = — ~(—)‘f3~(s,A), A~1(s,A) = (—)‘13’(s + i, A), (2.10)

and the superfield parametersQ~±are related to the parametersin (2.5)accordingto

fl~~~(z,0)=A”~(z) +20@~~(z),

Q~~~(z,0)~(s±t)_(z) +0A~~~(z). (2.11)

For future usewe describesomepropertiesof the super-Virasorosubalgebra.For clarity let us first discussthe bosonic case. The Virasoro algebra is thenexpressedin termsof a function A~2kz),

~ = A~2~O+ A(0A2~). (2.12)

For functionssatisfyingO3A~2~= 0, the generators(2.12) definethe sl(2) subalgebra.To elucidatesomeof the characteristicfeaturesof the commutatoralgebra,let usevaluatethe first few terms in the commutatorof ~ and ~ characterizedbyfunctions fl(2)( z) and fl(5)( z),

[L~),L~)] ~ (2.13)

where the generatorson the right-hand side are expressedin terms of thefunctions

~~~(z) ~ — (s — ~

~(s_i)(z) =0,

(2) = (s 2) [A(A — i) — ~s(s — 3)jfl(S)(03fl(2))

~(53)( z) = — ~(2A — 1)A(A — 1) [2(0ns) (03A(2)) + (s — 1)A(5)(04A(2))] . (2.14)

E. Bergshoeffet al. / Super-WO/A) algebra 323

Observethat the commutatorof two even-spingeneratorsgives, in general,rise togeneratorsof both evenand odd spin. Furthermorethe structureconstantsareeithersymmetricor antisymmetricunderA -+ i — A, in accordancewith the resultsquoted below (2.7). The function ~s) in (2.14) specifiesthat ~ has spin s.However,the presenceof the lower-spingeneratorsin (2.13)impliesthat L~ is, ingeneral, not a primary operator.Becauseall generatorstransform covariantlyundersl(2), all lower-spincontributionsshoulddependon third- andhigher-orderderivativesof ~ as is confirmedby the explicit resultsquotedin (2.14). In sect.3we shall see how sI(2) puts further restrictions on the derivative terms in thecommutators.

The situationis analogousfor the superalgebras.The super-Virasorogenerators

are thenexpressedin termsof a superfieldQ~2~(z,0)and take theform

~ —fl~2~~D2— ~-(DQ(2~)D — A(D2fl~2)~). (2.15)

For functions (1(2)+ satisfying the condition D5f1~2~~=0, the super-Virasoro

algebrais restrictedto the osp(1,2) subalgebra(this requiresthat the generatorsbedefinedin the Neveu—Schwarzsector).

Againwe may evaluatethe leadingtermsin the commutatorof ~ and ~characterizedby functions u1~2~(z,0) and [1(s)±(z 0), respectively,

{L~,L~±] =L~+.... (2.16)

The leading-spinoperator~ is thencharacterizedby the functions

~ 0) = —(2D2u1t2~— ~-(D12 )(Dfl~~)+ (s —

= —1 2~D2fl~~~—~ + (s — fl(D2[2)u1~~~

(2.17)

Becauseall operatorstransformcovariantlywith respectto osp(i,2), the genera-

tors of non-leadingspin on the right-handside of (2.16) will involve only fifth- orhigher-orderfermionicderivativesof ~(2)+

It is convenientto usea moreuniform notationby introducing also half-integervaluesfor s. The generators~ ± andcorrespondingparameters~ ± are then

written as

= ~ (i(s) =

~ Q(s+1/2).....Q(s)_ . (2.18)

The parameters (s ~. 1) thuscoincideswith the highest-spinvalueof the bosonic

324 E. Bergshoeffet a!. / Super-W~~A)algebra

or fermionic generatorcontainedin ~ For integers the highest-spingeneratoris a bosonic operatorwhile for half-integer s the highest-spingeneratoris a

fermionic operator. For generalintegeror half-integers, the generators~ arenow definedby

2s—2

L~(fl~)= ~ A’(s A)(D252m(i(5))D1 (2 19)

i=O

where the coefficients A’(s, A) coincidewith the coefficients A’~(s,A) defined in(2.10)*,

A’(s,A) =A’~([s],A) (s integer)

A’(s, A) =A’([s], A) (s half-integer). (2.20)

Here andhenceforth[a] denotesthe integerpart of a.For half-integera we thushave[a] = a — ~. Observethat (2.19) is always a bosonicoperator,because(i(s) iscommuting(anticommuting)for integer(half-integer)value of s.

For conveniencewe list the generatorswith s ~ 2,

L~k’~(Q~~)f~(1)

L~/2~((i~3/2~)= (1~3~2~D— 2A(DIl~3/2~),

L~)((i(2))= —I1~2~D2— ~(Dtl~2~)D — A(D2(1~2)) . (2.21)

The commutationrelation of the super-VirasorogeneratorL~1with ~ leads

againto ~ plus generatorsof lower spin (cf. (2.16)).The leading-spingeneratoris thencharacterizedby the function (cf. (2.17))

z, 0) = ~ — ~-(D(i~21)(DQ~~~)+ (s — 1) ( D2Q~2~)(i(s), (2.22)

for both integerandhalf-integervaluesof s.Finally we discuss the anti-automorphism(2.6) in this notation. Under the

automorphismonereplacesthe order of all quantitiesandreplacesthe superspacederivativeandthe anticommutingcoordinateby

D—*iD, 0—*—iO. (2.23)

* A single formula for the coefficientsA’(s, A) is

A’ A — — [sl+1±2s(i+1)[ s — 11 ([(1 + 1)/2] + 2A)[S_t/2][(~÷l)/2]

(s, ) — ( ) [ i/2 ] ([s + i/2])2~_l_[5+I,s]

The modifiedbinomial coefficient is definedin eq. (4.5).

E.Bergshoeffeta!. / Super-WJA) algebra 325

Functions(i(s) thuschangeinto functions (i(s) obtainedby changingthe order ofthe anticommutingparametersand replacing 0 by — i0. Onecanthen derivethatthe generatorschangeaccordingto

L~(f2~~~)—~i652L(j~32~(fl(5)). (2.24)

In orderto provethis onemay employ the following relationfor functionsthat can

be written asmultiple superspacederivativesof other functions

= jIflI2 ( )n(2s±‘~D”~. (2.25)

Here ni2 n — 2[n/2], which is equalto 0 or 1 dependingon whethern is evenorodd, respectively.With the aid of eq. (2.25) it is easyto verify the correctnessof(2.24) for the low-spin generatorslisted in (2.21).

3. The bosonicalgebra

In this sectionwe shall discussthe structureof the bosonicassociativealgebra.The results lead directly to explicit expressionsfor the structureconstantsof the

\~,(A)algebra. In sect. 4 we will generalizetheseresults to the correspondingsuperalgebra.We start by deriving the consequencesof thesI(2) covariancefor theassociativealgebra.Considerthe product of ~ and ~ expressedin terms of

correspondingfunctions fl(s) and fl(t),

5+1—1

~ = u~i L~~’~)(ø:)~’)”)), (3.1)

where the operatorson the right-hand side dependon correspondingfunctions

ø~~7’~which canbe expandedaccordingto

u—i

= ~ F~(i A)(0mfl(5))(0~lA(t)) (3 2)i=O

The sl(2) covarianceof the operators~ fixes the i-dependenceof the coefficientsF~(i,A). To see this one takes the commutatorof both sides of eq. (3.1) withL~

2~(fl~2~),where the correspondingfunction fl(2) satisfies the sI(2) condition= 0. This exerciseleadsto three equationsproportional to fl(2), 3fl(2) and

32fl(2), respectively. While the first two equationsjust ensurethat the coefficientsin(3.2) are z-independentandfix the total numberof derivatives,the third equationyields a recurrencerelation for the coefficients [~(i, A), which determinestheiri-dependenceandshowsthat the rangeof i-valuesin thesum(3.2) is restrictedin awaywhich dependson the valuesof s, t and u. Onemay distinguishthreepossible

326 E. Bergshoeffetal. / Super-WJA) algebra

situations:

I~~:0~i~u—i, u~2s—i, u~2t—i,

I~_: 0~<i~<u—2t, u~<2s—i, u>~2t,

I_k: 2s—i~i~u—i, u>~2s, u ~2t—i, (3.3)

wherewe must take into accountthat 1 ~ u ~ s + t — 1. A solution of the recur-rencerelationwhich automaticallysatisfiesthe aboverestrictionson the rangeof ivalues, is given by

~(s±~u) = F5~(A)E ( )‘( u -1 )(2s - u)~1~(2t- u)j(Olfl(5))(Ou-l-1fl(t)).

(3.4)

The region ~ is characterizedby the fact that the right-hand side of (3.2)containsno multiple derivativesof higherorder thanO2~2fl~~~and O2t2fl~°.Thisin contradistinctionwith I~_(I_±),where no multiple derivativesappearof higherorder than 025_2A(5)(021_2fl(t))andof lower order than~

It turns outthat the functionssatisfying(for all s)

O2~’fl~~~=0 (3.5)

define an infinite-dimensionalassociativealgebra,whosecorrespondingLie alge-bra is a subalgebraof W~(A).This algebrais thusbasedon thegenerators~ with

ml ~ s — 1, correspondingto a wedgein the parameterspacespannedby thevalueof the spin s andthe L

0 eigenvaluen. In thisway the generatorsof given spin arerestrictedto a finite-dimensionalirreducible representationof sl(2). Observealsothat the correspondingdifferential operatorsdefined in (2.2) are regular nearz = 0. The fact that we are restricting ourselvesto finite sl(2) representationsisalreadyan indication that the wedge generatorsmust constitutean associativesubalgebra.Neverthelessit is not so easyto give adirect proofof this. Suchaproofrequires to show that o2s±2t_2u_1~:)~t)_u)= 0, for a function ~ definedaccordingto eq. (3.2) in termsof functions fl(s) and fl(l) that satisfy eq. (3.5).However, the existenceof the wedgesubalgebracan also be establishedon thebasisof the following arguments.Considerpolynomialsof the sl(2) generators,L0and L ~, defined for a given value of A. Becauseof the sl(2) commutationrelations, one may restrict oneself to symmetrizedproducts, which constituteirreducible representationsof sl(2) provided one takesout the trace associatedwith the bilinearsl(2) invariant.However,this traceis proportionalto the identity,as the sl(2) Casimir operatorin the representationthat we are dealing with, is

E. Bergshoeffet aL / Super-l4~(A)algebra 327

equalto

C2(sl(2))wL~—~{L1,L_j=A(A—1). (3.6)

From the structureof the generators~ and the sl(2) assignmentsof thesymmetrizedtracelesspolynomials*, one concludesthat the wedge generatorscoincidewith the generatorsof the envelopingalgebraof sl(2), factoredover theideal generatedby C2(sI(2))— A(A — 1). This confirms the requiredresult that thewedgegeneratorsconstitutean associativealgebra.The multiplication propertiesof this algebrawill only dependon A throughthe CasimiroperatorvalueA(A — 1).

We denotethe Lie algebrabasedon the wedge generatorsby ~. As explainedin the previoussection, ~ and ~ are isomorphic.In thisparticular case,this is

obvious in view of the fact that the multiplication propertiesof the associativealgebradependonly on A(A — 1). Observethat the generatorL

t1~’~is just propor-

tional to the identity, so that it may be droppedfrom the~Lie algebra .~. (Thecorrespondingcentral extensionfor ~A is trivial.) However, it doesbelongto theassociativealgebrabasedon productsof the wedgegenerators.The Lie algebras

~ havebeenencounteredpreviously in a different context.Sometimeago [22,23]it was found that both the (2 + 1)-dimensional higher-spinalgebrahs(i, 1) that is

usedfor the constructionof interactinghigher-spingaugetheories,andthe algebraof area-preservingdiffeomorphismsof thetwo-dimensionalhyperboloidsdiff(S”),belong to the one-parameterfamily ~ of infinite-dimensional algebras.Theprecisecorrespondenceis given by hs(i,1) =~~‘I/4 and sdiff(S

t”) ~ Further-

more, it was shown that differentvaluesof A with A ~ -~- alwaysleadto inequiva-lent algebras[22,23]. This is consistentwith thefact that in our representation,thewedgegeneratorsfor given valueof A cannot,in general,be expressedin termsofthe wedgegeneratorscorrespondingto someother valueof A.

It is possibleto view the WO~(A)as an extensionof the .~ algebraby includingthe generatorsoutsidethe wedge. The samepoint of view can be taken withrespectto the associativealgebra.According to (3.3) the constantsF~(A)associ-

atedwith theregion~ arecompletelydeterminedby the wedgealgebra,whereasthe regionsI~_and I_~are only relevant for the extensionoutside the wedge.The possibility for extendingthe .~ algebrasfor generalvalue of A was already

discussedbefore [6,26] (note that the relevanceof algebrasof differential opera-tors to the solution of this problem was emphasizedin ref. [26]). The questioniswhether such extensionsare possiblewithout introducing generatorsof negativespin. An additional restriction imposed in the work of ref. [6] is, that the Liealgebra exhibits a certain grading according to which the commutatorof twoeven-spingeneratorscanonly give rise to even-spingenerators.As it turns out [21],

* Alternatively, onemaystart from the simple identity L~±= (L~)~ = d~’,andgeneratesimilar

identities by applying infinitesimal sl(2) transformations.Theseidentitiessufficeto reachthesameconclusion.

328 E. Bergshoeffet aL / Super-WJA) algebra

the W0,(A) algebraonly satisfiesthis requirementfor A = -~ and,provided that onesuppressesthe spin-i generator,for A = 0, or 1. We will return to this pointat theendof this section.

Now we turn to a more explicit evaluationof the constantsF~(A).From theproductrule (3.1) one canestablishthe following set of relations,

~0(i) 0(s -1-i) 0(k -j) 9(i +j - k)(k l)(_)l+iai(sA)aJ(t A)

s+t—i s+(—u—-i—k

= ~ 0(k)0(s+t—u—i—k)F~(A)a’~(s+t—u,A)u=i 1=0

X 0(s — 1 — i —1)0(u — s + i + 1)( —

~ (s + t — U — 1 — k ) ( ~ — 1)(2s — u)~_5+~±,(2t—

(3.7)

where i and k are arbitrary integersand 0(n) = 1 for n ~ 0 and zerootherwise.The equations(3.7)are notall independent;someof them are relatedby sl(2).

In order to solve (3.7) and find an expressionfor F~(A),one sums over iincluding a sign factor (— )‘. Then, after shifting the summationparameter1 toi — i, oneusesthe equation

(3.8)

for integern > 0. This equationexpressesthe fact that (1 — 1)~= 0 if n ~ 0. Afterapplicationof the Gausssummationtheorem(cf. eq. (A.7)) on both sides of theresultingequation,onefinds

F’~ A — — ~+1(2s+2t—2u—i)!st( )( ) (2s+2t—u—2)!

s—i t—1

x ~ ~(i+j—s — t +u + 1) a’(s,l —A) a’(t,A) . (3.9)i—0 1=0

This expressionfor F~(A)satisfiesthe relation

F~(A)= F,~(i— A), (3.10)

in accordwith the bosonicversion of the anti-automorphism(2.6).

E. Bergshoeffet aL / Super-WOJA)algebra 329

Performingoneof the summationsin (3.9) onecan removethe Kroneckerdeltaandobtain the result

(2s+2t—2u—1)!FU(A\ — _________________St” ~ (2s+2t—u—2)!

~ t—i \ (2A—s)1 (—2A—t+2)~1~ 311)~ i Ru — i—i) (2s—1—i)1 (2t — u + i)~_1_1 ‘

where the boundariesof the summationdomainare implied by the zeroesof thetwo binomial coefficients. Dependingon the valuesfor s, t and u, the sum runsover s, t, u or s + I — u terms andcan be expressedin the form of a generalizedhypergeometricseries.

It is instructive to evaluatesomeof the explicit expressionsfor the constantsF~(A).A straightforwardcalculationgives for u = 1,2,3,

s— 2AF5~(A)=~0(s—2)0(t—2)+~0(s—2)5(t—1)s—i

t — 2 + 2A+-~0(t—2)8(s—i) ~

16A(A — 1) + 3Fs~(A)=~0(s_2)0(t_2){1_(2s_3)(2t_3)(2s+2t_5)}

4A(A—i) s(s—1) 2A+-~0(s—3)~(t—i) + — _____

(2s—3)2 (2s_3)2 2s—3

4A(A—1) t(t—i) 2(1—A)+~0(t—3)~(s—1) + — . (3.12)

(21_3)2 (2t—3)2 2t—3

Clearly, the evaluationof theseexpressionsbecomesvery cumbersomefor largevaluesof u. The termswith the Kroneckerdeltascorrespondto the regionsI~_andI_÷indicatedin (3.3). The other termsbelongto the region~ andcoincidewith the multiplication constantsof the wedgesubalgebra.As mentionedprevi-

ously, thesecoefficientscan oniy dependon A through the combinationA(A — 1).By virtue of eq. (3.10), thesecoefficientsmust thereforebe symmetricunder theexchangeof s and t. This exchangeproperty,which is manifestin the resultslistedin (3.12), can also be derived directly for the wedgesubalgebra.Namely, one

330 E. Bergshoeffeta!. / Super-WJA)algebra

considersan anti-automorphisminducedby the substitutionL~2~—* — L~2~accom-paniedby a reversal in the order of the operatorsin productsof the Lt2~.Thisoperationleavesboth the sl(2) commutationrelationsand the Casimir operatorunchanged.As the spin-sgeneratorsinsidethe wedgearebuilt from symmetrizedproductsof the sl(2) generators,they changeunder the anti-automorphismby afactor ()s_i This suffices to show that the constantsF~(A)are symmetric in sand t insidethe wedge.

The structureconstantsfor the WOO(A) algebracan be expressedas in (3.4) butnow with coefficientsf~(A)equalto

f5~(A)= F~(A)+ (_)uP~(A)= F5~(A)+ ( )uFu(1 — A). (3.13)

In thelast equationwemadeuseof eq.(3.10).Insidethewedgeonethusfinds thatthe structureconstantsvanishfor odd valuesof u, irrespectiveof the value for A.For the full W,,~(A)algebrathis phenomenonalso takes place for A = ~, as ismanifestin eq. (3.13).Actually, thereis onemorevaluefor A wherethis is also thecase,namelyfor A = 0 (or, equivalently,for A = 1), provided that we excludethespin-i generator. In that case all the generators~ are proportional to aderivative,andwe maywrite

~ (s~2). (3.14)

It turns out that the differential operatorI~changesunder the anti-automor-

phism(2.6) in the following way*

j(s).....~()SL(s). (3.15)

This allows oneto show that

F~(0)=F1~(0) (s,t~2). (3.16)

From the expressionfor ~s3) in (2.14)we canverify that A = 0, ~, 1 representtheonly caseswhere the exchangepropertyholds also outsidethe wedge.Hence, forthesevaluesof A one can truncatethe Lie algebrato the even-spingenerators.One then obtains the algebrasdenoted by ~1+=)/2 and W/2 [6]. For the

supersymmetricalgebra a similar phenomenontakesplace, as we shall discusslater.

4. The superalgebra

We now apply the analysisof the previoussection to the associativesuperalge-bra basedon the generators~ where s takes integerand half-integervalues

* This implies that L~can bewritten as a sum of L~’’t, L3~, etc., definedfor A =

E. Bergshoeffet aL / Super-WJA)algebra 331

with s ~ ~. We begin by requiring the covarianceof the algebrawith respecttoosp(i,2). The associativealgebrais definedby specifyinghow the productof twogenerators,say ~ and ~ decomposesin terms of the generators.Fromdimensionalarguments,using the parametrization(2.19), we may againwrite theproductas

s+t—1

~ = u~i ~ (4.1)

where,in contrastwith eq.(3.1), s, t and u may takeboth integerandhalf-integervalues. The generatorson the left-hand side dependon correspondingfunctions

(i~5kz,0)and (ittkz,0). The generatorson the right-handside dependon func-tions ~ 0), which canbewritten in the form

2u—2= ~ F~(iA)(Dt(i(5))(D2u_2_1(i(t)) (4.2)

i—0

We stressthat the summationindex i takesonly integervalues.Theosp(i,2) covarianceis implementedby taking the commutatorof bothsides

of eq. (4.1) with the super-Virasorooperator L~2k(it2~),where the correspondingfunction (i(2) satisfiesthe osp(i,2) condition D5(i~2~= 0. We thusfind five equa-tions. Three are identically satisfied(those are just the conditions that require

to be of the form quoted in (4.2)),while the equationproportionalD4A~2~follows from the oneproportionalto D3fl~2~.The latter yieldsa recurrencerelationfor the coefficientsF~(i,A), which determinestheir i-dependenceandshows,justas in the bosoniccase,that the rangeof i-values in the sum(4.2) is restrictedin acertainway. Again threepossiblesituationscanbe distinguished:

I~~:0~i~2u—2, 2u~4s—2, 2u~4t—2,

I~: 0 ~i ~ 2u —4! + 1, 2u ~ 4s—2, 2u ~ 4!— 1,

I_k: 4s—3~i~2u—2, 2u >~4s—1, 2u~4t—2. (4.3)

A solutionof the recurrencerelationwhich incorporatestheserestrictionson therangeof i values, is given by

2u—2U) = ~( A) ~ ( )[l/2J+21(5+U)[ i/2 ] ([2s — u])[~_i—i/2J±I2u~sI2u—2—iJ~

X ([2t — u])[~/2]±I2~I2I~I2(D’(i

t~~)(D2u2_i(i(t)) , (4.4)

where the notation ni2 was introducedin sect. 2. We also introduceda slightly

332 E. Bergshoeffet a!. / Super-14<JA) algebra

modified versionof the binomial coefficient,

[a — [a]!Lb = [b]![a-b]L (4.5)

It is easyto verify that for integervaluesof s, t and u (4.4) reducesto (3.4) if werestrictthe sumin (4.4) to evenvaluesof i.

The region ~ is characterizedby the fact that the right-hand side of (4.4)containsno multiple derivativesof higher order than D4~4(i~~~and D4t4(itt~.This in contradistinctionwith I~_(I_k), where no multiple derivativesappearof higher order than D4~4(i~~~(D4’4(i~°) and of lower order thanD4t3(i(t)(D453(i(5)). The generatorscorrespondingto functions (i(s) satisfying(for all s)

D4~3(i~~~=0, (4.6)

form an infinite-dimensionalsubalgebra.Its multiplication rules are governedbythe constantsF~(A)belongingto the region~ This algebrawill be called thesuperwedgealgebra,because,just as before in the bosonic case,thesegeneratorsbelongto a wedgein the parameterplanespannedby thevalueof thespin andtheL

0 eigenvalue.In termsof the bosonicand fermionic generatorsdefinedin (2.2),the superwedgegeneratorsconsistof the bosonicgenerators~ with ni ~ s — 1and the fermionic generators~ with ri ~s — ~. The generatorsare thusrestrictedto finite-dimensionalirreduciblerepresentationsof osp(i,2)*.

In order to provethat the superwedgegeneratorsconstitutean infinite-dimen-sionalassociativealgebra,we makeuseof the fact that thesegeneratorsspantheenvelopingalgebraof osp(i,2) factoredby certainideals.This envelopingalgebrais basedon functionsof the osp(i,2) generators,L ±~, L0 and G ±1/2 ~Sincethe sl(2) generatorsL ± 1 and L0 can be expressedin termsof anticommuta-tors of the superchargesGa we may restrict ourselvesto functions of the latter.Here we employ a two-componentnotation where Ga with a = 1,2 denotes(Gi/2, G_1/2). The superchargesdo not commute,

[GU,G~] =(Q+fl~.~, (4.7)

where Q is somebosonicoperator,whichbelongsto the envelopingalgebra.Fromthe osp(i,2) commutationrelationsonefinds that

{Q,Ga}=0. (4.8)

* We remindthereaderthat for the generators(2.2) s wasdefinedto bean integer.Furthermore,we

note that the superwedgealgebracan only be definedfor the Neveu—Schwarzsector.In termsofthe functions A(5)±(z)and �~~~(z)(4.6) readsazslA(5)±_~

2s—2g(s)± 0, so that A~’~’andare polynomialsin z of degree2s—2 and 2s — 3, respectively.

E. BergshoeffetaL / Super-WOO(A) algebra 333

As noneof the sl(2) commutationrelationsimposesfurther conditionson Ga and

Q, the enveloping algebra of osp(l,2) is isomorphic to the associativealgebrageneratedby Ga and Q. It turns out the Q is directly related to the osp(i,2)Casimiroperator,

C2(osp(1,2))=L~— ~(L1, L1) + ~[GI/2,G_l/2] = — ~. (4.9)

In the representationat hand the Casimiroperatoris equalto

C2(osp(1,2))=A(A—~). (4.10)

In this way we establishthe existenceof the Klein operator K, which in thisrepresentationis equal to K = 1 — 200/00. Indeed K satisfiesK

2 = 1 and anti-commuteswith 0 and 0/00. (An analogousrealizationof K wasproposedin ref.[17] in termsof Clifford algebras.)It is related to the operatorQ by

Q= 2(A — ~)K. (4.11)

The enveloping algebra of osp(1,2) factored over the ideal generatedbyC

2(osp(1,2)) — A(A — ~) is thus isomorphicto the algebrageneratedby the Ga andK with K

2 = 1. Its elementscanbewritten as [23]

F(Ga,K) = ~ (FO~!.ak+KFI~I.ak)GaI...Gak, (4.12)k=0where the coefficients are symmetric in sl(2) indices a

1 .. . ak. From the sl(2)assignmentswe may concludethat thesefunctions coincidewith (linear combina-

tions of) the generatorsin the wedge.Thereforethesegeneratorsconstituteanassociativealgebra which is isomorphic to the enveloping algebra of osp(l,2)

factoredby an ideal.In generalthe superwedgealgebraadmitsN = 2 supersymmetry,aswasalready

observedin ref. [25] for the specialcaseof A = 0 andin ref. [16] for the caseA =

The reasonis that the algebracontainstwo kinds of supersymmetrycharges,G~2~~.

Togetherwith L~~?t,~ and ~ they generatethe osp(2,2) algebra.The twosuperchargesare relatedby the Klein operatoraccordingto

G~~~=GaK=~KGa. (4.13)

Thereforethe superwedgealgebrais also containedin the envelopingalgebraofosp(2,2), factoredby certain idealsrelatedto conditionssuch as (4.10), (4.13) andK2= 1.

For A �~ we may choosea basis for the superwedgealgebraconsistingof thesymmetrized polynomials Gai . . . Gak and QGak. It is thenclear that the product

334 E. Bergshoeffeta!. / Super-WJA)algebra

structureof the algebradependson A through the value of the Casimiroperator(4.10), just as in the bosonic case.The anti-automorphism(2.6)—(2.7) is inducedhere by the substitutionsGa ~ ~Ga,Q ~4 Q, wherewe simultaneouslyreversetheorder of all operators Ga and Q in (4.12). Under this operationthe osp(i,2)

commutation relations, the Casimir operator (4.10) and the commutator(4.7)remainunchanged.

As mentionedpreviously the value A = 3 is special.In that caseit follows fromeqs.(4.9)—(4.i0) that Q

2 = 0. As a result the elementsproportionalto Q form anideal of the envelopingalgebraof osp(i,2) which can be factoredout. Thereforeone has the option of restricting the superwedgealgebra to the K-independent

part of (4.12), so that one is dealing with N = 1 supersymmetryonly. ThecorrespondingN = 1 Lie algebrais isomorphicto the so-calledsymplectonhigher-spin superalgebra[15]. Two other truncations,basedon the anti-automorphism(2.6)—(2.7),amountto retainingthe generators~ with s evenand ~ with sodd and the generators~ ± with s either evenor odd. In the case A = 3 themultiplication rules within the wedge algebra often take a simpler form. Forinstance,onecaneasilydeducethat F~(3)vanishesinsidethewedgefor half-integervaluesof u.

Beforecontinuingthe discussionof the full algebra,we note that the existenceof the anti-automorphismimplies the following relationfor the constantsF~(A),

F~(A)= ()2UFu(1A) (4.14)

To determinethe constantsF~(A)we proceedas in sect. 3. First we substitute

(4.4) into (4.1) and evaluate the term proportional to (D2~21fl~~~)~ This leadsto a generalizationof (3.7), which we sum over i

after including a factor P. To distinguishthe imaginaryunit from the index i, wedenoteit by i. After usingeq.(3.8) the result reads

2s—2 2t—2 1i/2]~ 0(k —j)0([i/2] — [(k —j)/2J) ~1k ~ 21 A’(s, A)A~(t,A)

i=0 j=0 R i,/ ~

xI’(mk —j — 112 + ( — )~iii

2Ik i12}

s+t— i

= ~ 0(k)0(2s+2t—2u—2—k)F~(A)A”~(s+t—u,A)u=1

2u—2

x ~ ( — )[i/2]+2i(s+U)12s_2..i[tL1]

x ([2s — u])[U_i_~/2]+I2UI2I2U_2_II2([2t —

{~(s+ t — u — 1 — 3k) + ((— )25_2_l — ~) 6(s + t — u — — 3k)),

(4.15)

E. Bergshoeffeta!. / Super-WOIA) algebra 335

where i, j, k alwaystakeintegervalues,while the parameterss, 1, u canbe integer

or half-integer.

After shifting the summationindex i on the left-handside andusing

A2~~(s,A)= ~k (i + 1 — [S])k([5] + i)kA2i(s, A),(2A + ~)k(i + i)k

A2~~t(s, A) = (_)k (i + 2— [s + +I)k([~+ 31 + i)kA2i+1(s A), (4.16)(2A + i + i)k(i + l)k

we canperformthe sumover i with the aid of the Gausssummationtheorem(cf.eq. (A.7)). Also on the right-handside this theoremcan be usedto perform the

i-summation.The result takesthe following form

2s—2 2t—2

~ 8([i/2] — [(k —j)/2] )A’(s, A)A3(t, A)( — )[~_~2]

i=0 j=0

(2A — [s])[s_i_j/2]x) ii+11

21k—i—112~ (2A +

(2A — [s —

3])[5-1-i/2]

+ [i/2] + l)[51j/2]~i2 (1k -j 112 + (-)31k _Jm2}}

s+t— 1

= ~ 0(k)0(2s+21_2u_2_k)Ak(s+t_u,A)u=i

(2s+2t—2u —2+[u])!x125_2 F~(A)

(2s+21—2u —1)!

/ 1—u \12u±i12]12u12 2s ‘~ 12u12 I I Ix{[2s_u] +i(—) [LI—U] I I I\2s+2t—u—2J J

+ t — u — 1 — 3k)

+r[Lsu] + — ~2s+2t_u_2)2uj2 [2t u]12u12( 1 — u 12u+iIs]

2s “1 ‘I2 2x[(—1) —,j 6(s+t—u— ~— ‘k)j. (4.17)

336 E. BergshoeffetaL / Super-l~VjA)algebra

We now multiply this equationby ( I)2s — 2[( — )2s + i] andtakeits imaginarypart.This cancelsthe term proportionalto [( )2s — 1] on the right-handside of (4.17)and leadsdirectly to an expressionfor F~(A),

(2s+ 2t — 2u — 2)!FUIA\ — f \[5+t—U—1] ___________________

~‘ ‘~ ‘ (2s+2t—[u]—3)!

2s—2 21—2

x E 8([i/2] — [s+t—u —1— 3jJ)A’(s,A)A-’(t,A)1—0 j=0

(2A — [s])[5_l_~/2}x It+1I2i2s+2t—2u—j—1I2

(—2A — [s] + 2)[5_1_t/2]

(2A — [s — 31)[5—1—j/2] . 2s+t+ I1I2

(—2A — [s —3] + 1)l5_11/2]

x{I2s+2t_2u_J_1I2+(_)ui2s+2t_2u_JI2}}. (4.18)

Using the relation(A.20) derivedin appendixA, the aboveresultcanbewritten as

(2s+2t—2u—2)! 2s—22t—2F~(A)= (_)[s+t_U_1] + 2t — [u] —3)! E 8(i +j — 2s— 2t + 2u + 2)

XA’(s, 3— A)AJ(t, A)( )2s+21(s±tu) (4.19)

This result, which satisfies relation (4.14), is very similar to the correspondingresult (3.9) for the bosonic algebra.As the qualitativefeaturesarealso the same,we refrain from giving further details here.We shouldnote, however, that theF~(A)satisfy a numberof identities owing to the N = 2 supersymmetry.Theseidentitiesarenot manifestlysatisfiedby the representation(4.19), but do not takea particularlyilluminating form.

Finally, we canexpressthe structureconstantsfor the W~,(A)algebrain termsofthe constantsF~(A).Again the result of the commutatorscan be written in the

form (4.4), with the constantsF~(A)replacedby

f~(A)=F~(A)+ ()[_u]+4(s±u)(t+u)Fl,1(3A), (4.20)

where we made use of eq. (4.14). For A = 3 the structure constantsvanishwhenever[—u]+4(s+u)(t+u) is odd. This shows that the Lie algebracan be

E. BergshoeffetaL / Super-WJA) algebra 337

consistentlytruncatedto the generatorswith s = 2n or s = 2n — 3, where n is apositive integer, in accordwith previousargumentsbasedon the anti-automor-phism (2.6).The second(non-supersymmetric)truncationis not directly obviousinthis supersymmetricformulation. Anothertruncation,which can be implementedfor A = 0, 3, shall be discussedin sect.5.

5. Currents

It is possible to define a field-theoretic representationof the super-W~(A)algebrain termsof an operatorproductexpansionfor quasi-primaryfield opera-tors. For specific valuesof A such representationshavebeendefinedin termsoffree bosonor fermion fields [25,28, 29]. For generalvalueof A such a representa-tion can be basedon two conformal superfields,a commutingfield B and an

anticommutingfield C, with conformal weights A and 3 — A, respectively.Whensubjectedto their field equations,thesefields decomposeaccordingto B(z,0) =

/3(z) + Ob(z)and C(z, 0) = c(z) + Oy(z). Since 0 hasweight — 3, we find that b,c, /3 and y haveconformalweights A + 3, —A + 3, A and —A + 1. The supersym-metric action equals[241

1 — 1 — —

S= —fd2zd2OBDC=_fd2z{1307+boc). (5.1)

The generatorsof the super-W~(A)algebra are then related to the following

conservedcurrents

~(s)+( z) = ~ a’(s, A)O~’ ‘{(O’/3)y) + La’(s, A + 3)o~1~’{(O’b)c},

s—1+2A s-IV~~~(z)= — 2 — 1 ~ a’(s, A)O~’’{(0’/3)y}

i=0

s—2A s-i+ ~ a’(s, A +

1=0

s—I s—2

Q~5~(z)= ~ a’(s A)O~1’{(O’f3)c}R ~ f3’(s A)052’{(O’b)y} (5 2)1=0 1=0

Using

1 1y(z)/3(w) z—w’ c(z)b(w) ‘~ z—w’ (5.3)

338 E. Bergshoeffet al. / Super-WJA)algebra

with all other independentcontractionsvanishing, one can then evaluatetheoperatorproducts of thesecurrents.To establish the correspondencewith thepreviousrepresentationof the algebra,one considersthe productsof the currentswith some of the fields. For instance, the operator product expansionsofV~~~(z)/3(w)andV~~~(z)b(w)takethe form

s—i 11’~~~(z)/3(w) ~ a’(s,A)0~’’ 0’/3(w)

1=0 z—w

s—I 1 \

V~~~(z)b(w) ~ a’(s,A +3)0_t_t — )O’b(w). (5.4)

1=0 z w

The commutatorof B(z,0) with the charge

Vt~= ~~dz A(~(z)V~~~(z), (5.5)

then leadsdirectly to the differentialoperator~ definedin (2.5) actingon thesuperfieldB(z,0).

It is instructiveto write out explicitly someof the lower-spincurrents,

J/(2)+ (Of3)y — AO(/3y) + (Ob)c — (A + 3)O(bc),

Q~~=(O/3)c—2A0(/3c)~by,

~ f3y + bc,

V~= —2A/3y+(1—2A)bc,

(5.6)

The currents(Q(2)+ V~2~)reproducethe N = 1 super-Virasoroalgebra(cf. (2.1)).

The currents(Q~,s)+~At5~~) and~ Q~~) constitutenon-decomposablerep-resentationsof osp(1,2). In superfieldnotationthe currents(5.2) takethe form

1~~~(z,0)=3Q~(z) +0V~~~(z) (s~2),

W~(z,0) = Q~t)+(z)+

l4’~~~(z, 0) = l/,k~~~(z)— 0Q~~~(z), (5.7)

and can be expressedin terms of the superfields B(z,0) and C(z,0). Thenormalizationis suchthat the operatorproductexpansionof the currentswith the

E. Bergshoeffet a!. / Super-WJA)algebra 339

B, C superfieldsreproducesthe superspacedifferential operatorsgiven in eq.

(2.19). For the lower-spincurrents(5.6), this leadsto

W~2~(z,0)= AB(D2C) — 3(DB)(DC) — 3(1 — 2A)(D2B)C,

~~z,0) = —2AB(DC)+ (1— 2A)(DB)C,

W5(z,0)=BC. (5.8)

The correspondingexpressionsfor currents of arbitrary spin will be presentedshortly.

It is well known [24] that the supersymmetricbc systemactually has a N = 2

superconformal invariance. Indeed the currents (V, Q(2)±,Vt2~)define an

N = 2 super-Virasoroalgebra.In the Neveu—Schwarzsector it containsa finiteosp(2,2) subalgebra,which was already discussedin sect. 4, generatedby thecurrent componentscorrespondingto (Lv, G~,G~

2,~ L~). All cur-rents thusfit into N = 2 supermultipletswith respectto this osp(2,2) subalgebra.The resultingcombinationsare

(V(s)—,Q(s+ ~±, ~ 1)+) s = 1,2,3,...,

(5.9)

where(Q~),VW+) constitutesa so-calledN = 2 scalarmultiplet.In this field-theoreticrepresentationthe equivalenceof WOO(A) and W~,,(3— A) is

related to the interchangeof the two superfieldsB and C. To be morespecific,considerthe effect of the replacements

f3—’ —ig2c,

y—s —e1b, b—~ie2y, (5.10)

where ~ and e2 are real anti-commutingparameters*.Using the resummationformulae(A.8)—(A.10) given in the appendix,we canprovethat the substitutionof(5.10) into the expressionsfor the currentsyields

y5(s)±...~ ±(—)5a

12V~~5,

Q(5)±.~(— )~a12Q~)~_A~ (5.11)

where we introduced the notation a12= ~ The above result can easily beverified for the low-spin generatorslisted in (5.6).

* In the purelybosoniccaseoneinterchangesthefields f3 and y.

340 E. Bergshoeffet a!. / Super-WJA)algebra

The abovesubstitutioncan also be applied to productsof currents.First note

that the contractions(5.3) changeunder the substitutions(5.10) into —a12(z—

wY’. For a product of two currentsone makesthe substitution (5.10) in onecurrentandthe samesubstitution,butnow with ~ and e2 replacedby e3 and e4,respectively,in the secondcurrent.This then leadsto the samerestrictionson thestructureconstantsas found before.

One can show that a truncatedversion of the operatoralgebramust exist forA = 0, 3. For instance,for A = 0 the conformalweightsof b and c areequalto 3while the conformalweights of /3 and y are equal to 0 and 1, respectively*.Inthe operator expansionone can then replace‘y by 04, where 4 has conformalweight 0. The truncationcannow be implementedby identifying b with c and /3with 4. To seethe effectof this identification we rewrite the currents(5.2) as

s—I s—i

= ~ ã’(s,A)(O’/3)(O5~’y)+ ~ ã’(s,A + 3)(O’b)(O~’’c),1=0 1=0

s—1+2A~~ - -

= — ‘~1 1 ~ a’(s, A)(O’$)(O~’’y)L.S1 i=0

s—2A~’+ ~.ã

1(s A+3)(O’b)(O~~’c)2s — 1 ~

s—i s—2Q~s)±(z) = ~ ~‘(s, A)(O’p)(35 ~‘c) T ~ ~‘(s, A)(O’b)(0~2’y) (5 12)

1=0 1=0

Herewe madeuseof the resummationformulae (A.11)—(A.13) given in appendixA; the coefficientsa’(s, A), ~‘(s, A) and /3’(s, A) aredefinedin (A.14). Substituting

b = c, /3 = 4 and y = O4, one verifies, using (A.15), (A.19) and the fact that~o(so)o for s>1, that Q~5~=0for s even and Q~5~=0for s>i odd.Furthermoreone uses(A.21) to show that all operatorsJ/(s)± with odd 5> 1vanish as well. The operatorswith s = 1 require specialattention. While V~1~vanishesidentically for A = 0, the remainingoperatorsare given by V°~=404

and Q(i) = 4w. Theseare the only operatorsthat dependon 4 andnot exclusivelyon derivativesof 4. Therefore,the s = 1 generatorsare never generatedin theoperatorproductsof the s> 1 generators(actually,thesegeneratorsareno longerconserved).Consequentlywemay consistentlyreducethe operatoralgebrato the

s> 1 operators,so that we are left with the operators y(s)± and Q(s) withs = 2,4,... and Q(s~with s = 3,5 This algebra,which exhibits N = 1 super-symmetry,coincideswith the super-Wr~~/

2algebraof ref. [25]. Thebosonicsubalge-

*FOr A = ~ the weightsof b, c, f3 and y are equal to 1, 0, ~ and ~, respectively,so that thesame

argumentscan be applied.

E. BergshoeffetaL / Super-W~jA)algebra 341

bra of super-W,,,/2 equalsW~/2~ \~I±~)/2’ where W,,/2 and \~I+a)/2 are thetruncated versions of W~.and W1~ discussedin ref. [6]. Observe that theidentification of b with c and y with 0/3 causesadditional factors of 2 inthe operatorproductexpansions.However,thesefactorscan againbe re-absorbedinto the definition of the currentsso that the structureconstantsof the algebraremain the same.Of course,the central chargewill be different whenwe reducethe numberof degreesof freedom.

As alreadydiscussedseveraltimes, thereexist two other truncationsof super-

W,,,(A) for A = 3. In that casethe conformalweightsof b, c, /3 andy areequalto 3,3-, 3- and3, respectively.The remaininggeneratorsare~ with s = 2,4, . . . ,

with s = 1,3,... and the fermionic operatorsQ(s)±with s restricted to eithers = 1,3,... or to s = 2,4

The field-theoreticrealization allows an explicit determinationof the centralextensionof super-W(A).Our startingpoint are the following expressionsfor the

currents,which follow from (5.7) and (5.12),

151 152— ~ ii’(s A)(0’B)(O~~~C)— — ~ /~‘(s A)(O’DB)(O~~

2DC)1—0 i=0

s—i + 2A s-I— ~ ~i(~ A)(0’B)(0~’~DC)

S 1=0

s—2A Si+ ~ ã’(s A + 3)(O1DB)(0~’~C) (5 13)

S 1=0

The operatorproductof the superfieldsB(z,0) and C(z,0) is equalto [24]

012(5.14)

z12

with 0I2=01~02and z12=z1—z2+0102.NotethatD1z12=—012 and D1012=i.The central chargesare defined by the coefficient of the leading term in theoperatorproduct expansionof two currents,

W~~~(z,,0t)W~’~(z2,02)~c~(s,t; A) + ...

(z12)

~ 012(zi2)

W(~~(z1,0~)W~t~(z

2,02) c~(s,I; A) 1 ~ + (5.15)(z12)

342 E. Bergshoeffet a!. / Super-WJA)algebra

It satisfiesthe condition

c~(s,t;A) = (_)S+lc±±(t,s;A) = _cil(t,s; 3—A),

c~(s,I; A) = (—)~~‘~c~(t,5; A) = —c~(t,5; 3—A). (5.16)

A straightforwardcalculation,usingthe two-point functionsgiven in (5.14),yieldsexplicit expressionsfor the centralcharge.For instance,we find

s—i 1—2

c~(S,t; A) = 3( _)5_i ~ ~ (s +1 — i—i)! (t + i —j — 2)!~’(s,A)f~~(t,A)1=0 j=0

s—2 t—I~ (s+j—i—2)!(t+i—j—1)!f.~’(s,A)&~(t,A),

1=0 j=0

(5.17)

andonemayobtain similar expressionsfor c±~ and c~.Note that the superwedgealgebrais an anomaly-freesubalgebra.To show this,

one multiplies the operator-products(5.15) with the correspondingparameters(i(s)±(z, 0), integratedalong an appropriateloop in holomorphicsuperspace.Afterwriting thefirst term in the operator-expansionas a multiplesuperspacederivativeof (5.14) and performinga suitable integrationby parts, one can show that itvanishesby virtue of eq. (4.6).

Forcomparison,weevaluatethe centralchargefor the super-Virasorosubalge-bra. Using (5.17)we find

c~(2,2;A)= —3(1—4A). (5.18)

This resultcoincideswith that of ref. [24].

We thank J. GoereeandC.N. Pope for stimulating discussionsand the CERNTheoryDivision andthe Institutefor TheoreticalPhysicsin Utrechtfor hospitalityextendedto usduring the courseof this work.

Appendix A

For conveniencewe list someuseful formulaerelatedto the coefficientsa’(s,A),a’(s, A) and f3’(s,A), definedin (2.5). Thesecoefficientsarenot independent,and

E. Bergshoeffet a!. / Super-WJA) algebra 343

we note the relations

2(s—1) -

a’(s,A) = + ~a’(~,A), (A.1)

- (2A+s—2)(s+i—i) -/3’(s, A) = (2A + i)(2s 3) a(s — 1,A). (A.2)

Furthermore

(~+l—S)k (5+~)k -at+k(s,A) = (j + 1)~ (2A + ~)ka(5~A)~ (A.3)

(~+l—S)k (S+~—l)k -

a’~(s,A) = (i + l)k (2A + ~)k a’(s, A), (A.4)

(~+2—S)k (S+~)k

I3~+k(SA)= (i + l)k (2A +i + l)k~~’~ (A.5)

In sumswe oftenencounterexpressionsthat involve hypergeometricseries,suchas the Gaussfunction

~ ‘~ “ 1b’ k

2F1[a,b;c;z]= ~ ~a)k~ Ikf (A.6)k=0 (C)k k.

For a or b equalto somenegativeinteger,this becomesa finite series.The Gausssummationtheoremstatesthat (seefor instanceref. [30])

F(c)F(c—a —b)2F1[a, b; c; 1] = F(c — a)F(c — b) (A.7)

The aboveresultscanbe usedto obtain the following resummationformulae,

s—i s—I

~ a’(s,A)O’((O~~’A)B)= ~ (— )‘a’(s 1 — A)(O~~’A)(O’B) (A 8)1=0 i=0

~a’(s, A)8’((O~~’A)B)= L (—)~~‘a’(s,3— A)(O~’’A)(O’B), (A.9)

s~2 A)O’((O~2~)B)= s~2( - )s~2~l/3~(~ 3~A)(O52~)(0’B). (A.10)

344 E. BergshoeffetaL / Super-W,.,(A) algebra

Another set of coefficients ã’(s,A), ~‘(s, A) and ~‘(s, A) are defined by theexpansion

s—i s—i

~ a’(s,A)O~~’((O’A)B)= ~ a’(s A)(O’A)(0~~’B) (A 11)i=0 i=0

s—i s—i

~ a’(s,A)O~~’((O’A)B)= ~ ã’(s A)(O’A)(O~~’’B) (A 12)1=0 i=0

s—2 s—2

~ j3’(s, A)0~2’((O’A)B)= ~ f~’(s,A)(0’A)(0~2’B) (A 13)i=0 i=0

Usingagaineq. (A.7) thesecoefficientscanbe evaluated,

ã’(s, A) = (s — 1) (s)5_i (2A — s)~(2— 2A — s)5_i_1,

&~(s,A)= 2(5—1) (s)5_, (2A —s+ i)~(2— 2A —s)sij,

~~(sA) = (s 2) () (2A —s + i)~(22A 5)s-2-i. (A.14)I (5)s—2

They satisfy the identities

ã’(s,A) = (_)5_lds_1_t(s,1 —A),

~‘(s, A) = (_ )5i~s_i_I(5, 3—A),

f’(s, A) = (_ )s_

2jjs_2_z(

5 3 — A). (A.15)

Threeother identitiesthat involve coefficientsfor different valuesof A, are

- i+i+2A s—i—2A --

ã’~(s,A)= ~+ ~ s—i +A 13’(s,A + 3), (A.16)

— 3/3’(s, A) = a’(s, A + 3) — a’(s,A), (A.17)

s—2A s—i+2A13’(s+ i,A) = 2s—1a’(s,A +3)— 2s—1 a’(s,A). (A.18)

E. Bergshoeffet aL / Super-WJA)algebra 345

From eq.(A.16) we derive

~ i(~) = ~‘(s, 3), (A.19)

anidentity that we usedin sect.5. From eqs.(A.17)—(A.18)we find a relationforthe coefficientsA’(s,A) that we used in the derivationof (4.19)

2A+[s]—1 12512 -

A2’~(s3—A)+A2’(s 3—A) = (2A [s] 1) A2’(s, I — A). (A.20)

Finally we note two more identities, which follow directly from the definition

(A.14)

ã5’(s,O) = ( — ~ã’(s, 0),

~ 3) = (— )~ã’(s,3). (A.2i)

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