Maximal SupergravityRenataKallosh,Stanford

BasedonN >4supergravity workwithNicolai,Roiban,Yamada,workinprogress

andwithFreedman,Murli,VanProeyen,Yamada JHEP2017

Cosmologyworkwith Ferrara,Linde,Roest,Wrase,Yamadaona-attractormodelsandtargetsforprimordialgravitationalwaves from

M-theory,stringtheory,maximal supergravity,2016-2017

July32017

Hawking75

PlanckandfuturesatelliteCMBmissionsandmaximal supergravity inthesky

RecentprogressinUVpropertiesofperturbative maximal N =8 supergravity anduniversalityofN >4 supergravities

NewB-modetargetsfrommaximalsupersymmetryr between10-2 and10-3

S. W. Hawking“Is the End in Sight for Theoretical Physics: An

Inaugural Lecture” 1980• “AtthemomenttheN=8 supergravity theoryistheonlycandidatein

sight.Therearelikelytobeanumberofcrucialcalculationswithinthenextfewyears whichhavethepossibilityofshowingthatthetheoryisnogood.”

• “Ifthetheorysurvivesthesetests, itwillprobablybesomeyearsmorebeforewedevelopcomputationalmethodsthatwillenableustomakepredictions.”

• “Thesewillbetheoutstandingproblemsfortheoreticalphysicists inthenexttwentyyearsorso.”

1981:3-loopcandidatecounterterms arefound,withunknowncoefficient

2007:Calculationsshowthatthecoefficientinfrontofthe3-loopcountertermvanishes!!!

Is N=8 supergravity all-loop finite? What do we know 37 years later, in 2017?

RecentprogressinobservationalcosmologyRecentprogressinscatteringamplitudes

B-modes • Thomson scattering within local quadrupole

anisotropies generates linear polarization • Scalar modes Æ T, E • Tensor modes Æ T, E, B • Ratio r = ΔT / ΔS • Gravitational waves at LSS

create B-mode polarization • Probes Lyth bound of Inflation • Ekpyrotic models Æ r = 0

Lorenzo Moncelsi

Planck 2015

BICEP2 2014

W. Hu

B>0 B<0

Moriond 22/3/16

Planck XX 2015

BK14 w / 95GHz 2016

NextinCMBcosmology:

Relentlessobservation

IfB-modeswillbediscoveredsoon,r>10-2naturalinflationmodels,axion monodromymodels,a-attractormodels,…,willbevalidatedNoneedtoworryaboutlogscaler

Otherwise,weswitchtologr

October2016

CMB-S4 Science BookFirst Edition

CMB-S4 Collaboration

August 1, 2016

arX

iv:1

610.

0274

3v1

[astr

o-ph

.CO

] 10

Oct

201

6

4 Exhortations

2000 2005 2010 2015 2020

10−4

10−3

10−2

10−1

WM

AP

Planck

CMB−S4

Year

Appro

xim

ate

raw

exp

erim

enta

l sensi

tivity

(µ

K)

Space based experiments

Stage−I − ≈ 100 detectors

Stage−II − ≈ 1,000 detectors

Stage−III − ≈ 10,000 detectors

Stage−IV − ≈ 100,000 detectors

Figure 2. Plot illustrating the evolution of the raw sensitivity of CMB experiments, which scales asthe total number of bolometers. Ground-based CMB experiments are classified into Stages with Stage IIexperiments having O(1000) detectors, Stage III experiments having O(10,000) detectors, and a Stage IVexperiment (such as CMB-S4) having O(100,000) detectors. Figure from Snowmass CF5 Neutrino planningdocument.

1.2.1 Raw sensitivity considerations and detector count

The sensitivity of CMB measurements has increased enormously since Penzias and Wilson’s discovery in1965, following a Moore’s Law like scaling, doubling every roughly 2.3 years. Fig. 2 shows the sensitivity ofrecent experiments, expectations for upcoming Stage-3 experiments, characterized by order 10,000 detectorson the sky, and the projection for a Stage 4 experiment with order 100,000 detectors. To obtain many of theCMB-S4 science goals requires of order 1 µK arcminute sensitivity over roughly half of the sky, which for afour-year survey requires of order 500,000 CMB-sensitive detectors.

To maintain the Moore’s Law-like scaling requires a major leap forward, a phase change in the mode ofoperation of the ground based CMB program. Two constraints drive the change: 1) CMB detectors arebackground-limited, so more pixels are needed on the sky to increase sensitivity; and 2) the pixel count forexisting CMB telescopes are nearing saturation. Even using multichroic pixels and wide field of view optics,these CMB telescopes are expected to field only tens of thousands of polarization detectors, far fewer thanneeded to meet the CMB-S4 science goals.

CMB-S4 thus requires multiple telescopes, each with a maximally outfitted focal plane of pixels utilizingsuperconducting, background limited, CMB detectors. To achieve the large sky coverage and to takeadvantage of the best atmospheric conditions, the South Pole and the Chilean Atacama sites are baselined,with the possibility of adding a new northern site to increase sky coverage to the entire sky not contaminatedby prohibitively strong Galactic emission.

CMB-S4 Science Book

T

T

E

Figure 1: This Figure is taken from [16], it represents a forecast of CMB-S4 constraints in the ns � r planefor a fiducial model with r = 0.01. Here the grey band shows predictions of the sub-class of ↵-attractor models[2, 3, 4]. We have added to this figure a blue circle with the letter T inside it corresponding to a highestpreferred value 3↵ = 7 and the purple one corresponding to the lowest preferred value 3↵ = 1 in a seven-diskgeometry. All intermediate cases 3↵ = {1, 2, 3, 4, 5, 6, 7} are between these two. They all describe the classof ↵-attractor models with V ⇠ tanh2('/

p6↵), so-called quadratic T -models. The quadratic E-models with

V ⇠ (1 � ep

2/3↵')2 tend to be slightly to the right of the T -models, see [2]. We show them as a navy circlewith the letter E inside it.

by requiring that

3↵ = 7 : ⌧1 = ⌧2 = ⌧3 = ⌧4 = ⌧5 = ⌧6 = ⌧7 ⌘ ⌧3↵ = 6 : ⌧1 = ⌧2 = ⌧3 = ⌧4 = ⌧5 = ⌧6 ⌘ ⌧ , ⌧7 = const3↵ = 5 : ⌧1 = ⌧2 = ⌧3 = ⌧4 = ⌧5 ⌘ ⌧ , ⌧6 = ⌧7 = const3↵ = 4 : ⌧1 = ⌧2 = ⌧3 = ⌧4 ⌘ ⌧ , ⌧5 = ⌧6 = ⌧7 = const3↵ = 3 : ⌧1 = ⌧2 = ⌧3 ⌘ ⌧ , ⌧4 = ⌧5 = ⌧6 = ⌧7 = const3↵ = 2 : ⌧1 = ⌧2 ⌘ ⌧ , ⌧3 = ⌧4 = ⌧5 = ⌧6 = ⌧7 = const3↵ = 1 : ⌧1 ⌘ ⌧ , ⌧2 = ⌧3 = ⌧4 = ⌧5 = ⌧6 = ⌧7 = const (4.17)

We illustrate in Fig. 1 the features of ↵-attractor models [2, 3, 4] with the seven-disk geometryusing the recent discussion of B-modes in the CMB-S4 Science Book [16]. We show in Fig. 1predictions of ↵-attractor models with seven-disk geometry in the ns � r plane for N ⇠ 55, forthe minimal value 3↵ = 1 and for the maximal value 3↵ = 7.

5 Values of 3↵ in string theory

Here we will show how to derive the 7-disk geometry (4.13) in string theory. We start withthe derivation of non-compact symmetries in string theory following [17], [18]. The toroidal

7

a-attractor models

FutureB-modesatellitemissions

Wellmotivatednewmodelsoriginatinginstringtheory,M-theory,maximalsupergravity

Ferrara,RK,2016,RK,Linde,Wrase,Yamada,2017RK,Linde,Roest,Yamada,2017

Groundbasedexperiments

Newtargets

0.960 0.962 0.964 0.966 0.968 0.970-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

a-attractorslog10 r- ns plane

ns

R2Escher =3a =7,6,5,4,3,2,1

NewB-modetargets:frommaximalsupersymmetrytominimalsupersymmetry

r<0.07

Newinflationarya-attractormodelsdescribeinflationanddarkenergyandsupersymmetrybreaking.TheyprovideB-modetargets forfutureB-modedetectors,withr between10-2 and10-3

E7(7)(R) � [SL(2,R)]7

Ferrara,RK,Linde,Roest,Wrase,Yamada,2016-2017

Maximalsupergravity

• Theoryhas28 =256 masslessstates.• Multiplicityofstates,vs.helicity,fromcoefficientsinbinomialexpansionof(x+y)8 – 8th rowofPascal’striangle

SUSYchargesQa,a=1,2,…,8shifthelicityby1/2

DeWit,Freedman(1977);Cremmer,Julia,Scherk(1978);Cremmer,Julia(1978,1979);DeWit,Nicolai(1982)

• Ungaugedtheory,inflatspacetime

OldWisdomUsingtheexistenceofthecovarianton-shellsuperspace Brink,Howe,1979andthebackgroundfieldmethodinQFTonecanusethetensorcalculusandconstructtheinvariantcandidatecounterterms RK;Howe,Lindstrom,1981.

Suchgeometriccounterterms haveallknownsymmetriesofthetheory,includingE7(7) .Theystartatthe8-looplevel.Linearizedonesstartatthe3-looplevel,RK;Howe,Stelle,Townsend,1981

Tensorcalculus=infiniteproliferationofcandidatecounterterms.Wheretheystartisnotofmajorimportanceiftheyproliferateanyway.

TheE7(7)symmetryofN=8supergravitywasdiscoveredin1979

deWit,Freedman,1977Cremmer,Julia,Sherk,1978DeWit,Nicolai1982

Cremmer,Julia1979

New era,new people,new computers,newrules:neveruseN=8supergravityrules,buildit

fromN=4Super-Yang-Mills

• Light-conesuperspace counterterms inN=8arenotavailable• atanylooporder:predictionofUVfiniteness RK2008,2009

Thisisstillthecasenow,asconfirmedbyL.Brink

• 3-loop-- 6-loopfinitenessfollowsfromE7(7)

On-ShellAmplitudesandExplicitcalculations:noUVdivergencesat3- and4loopsBern,Carrasco,Dixon,Johansson,Kosower,Roiban,2007-2009

Broedel,DixonBeisert,Elvang,Friedman,Kiermaier,Morales,Stieberger

Bossard,Howe,Stelle,2009-2010

ExplanationofUVfinitenessin3-and4loops?

5-loops:asofJuly12017,aftermanyyearsofeffort,BernetalhavefiguredoutanewdoublecopymethodthatallowedthemtoconstructtheN=8supergravity 5loop4pointIntegrand.TheyarecurrentlyintegratingitinD=24/5 toseeifitbehavesthesamewayasN=4sYM theoryornot.Notclearhowlongthiswilltake.

Bjornsson , Green Stringtheory 7loops?

E7(7) wasusedsofarinthecontextofsoftscalars:Adler’szero

Thissoftscalr analysisstartingwithL=7isinconclusivesincetherearemanymorepowersofmomentainhigherloopsandsoftlimitsarenotuseful.

But E7(7)alsoactsonvectors:requirescurrentconservation!

IfwetrustcontinuousglobalE7(7)atthe3- 6loopsquantumlevel,whatisthepredictionathigherloops?

E7(7) revisited:RK,2011Noether-Gaillard-ZuminocurrentconservationisinconsistentwiththeE7(7)invarianceofthecandidatecounterterms

E7(7) revisited:Bossard-Nicolai,2011Yes,NGZcurrentconservationisinconsistentwiththeE7(7) invarianceofthecandidatecounterterms.However,thereisaprocedureofdeformationofthelineartwistedself-dualityconstraint,whichshouldbeabletofixtheproblem.

E7(7) revisited:Carrasco,RK,Roiban,2011Bossard-NicolaideformationprocedureneedsasignificantmodificationtoexplainthesimplestcaseofBorn-InfelddeformationoftheMaxwelltheorywhichconservestheNGZcurrent

Adualitydoublethasthefollowingcomponents

Noether-Gaillard-Zuminocurrentconservationrequiresthattheactiontransformsinaparticularway,INSTEADOFBEINGINVARIANT

DualitysymmetryrotatesvectorfieldequationsandBianchiidentities

and are formally mapped into linear combinations of themselves by a GL(2) transformation. Further requiring thatthe transformed G may be obtained from the action evaluated on the transformed F though eq. (2.6) and that theresulting action is a deformation of Maxwell’s theory L = − 1

4F2 +O(F 4) restricts [4] the possible transformations to

δ

!FG

"=

!0 B

−B 0

"!FG

". (2.9)

In other words, the duality transformation exchanges the Bianchi identity and the equations of motion of the originalLagrangian. The original Lagrangian is self-dual if L and LD have the same functional form. It is easy to check thatMaxwell’s theory, with L(F ) = − 1

4F2, is such a theory.

In the derivation above, the dual field strength is determined by eq. (2.6) and is not an independent field. Sinceduality transformations (2.9) mix the field strength and its dual, it is convenient to interpret G as an independentfield and relate it to F by introducing constraint equations as we discuss in section III.

For theories with nv vector fields the strategy for constructing the dual Lagrangian is unchanged. The equations ofmotion and the Bianchi identities remain of the form (2.8) but are invariant under a much larger set of transformations:

δ

!FG

"=

!A BC D

"!FG

", (2.10)

AT = −D BT = B CT = C (2.11)

Here A,B,C,D are the infinitesimal parameters of the transformations, arbitrary real n× n matrices and the trans-formations (2.10) generate the Sp(2nv,R) algebra. For more general theories, when scalar fields are present, we wouldalso include a δφ(A,B,C,D).

Consistency of the duality constraint can be expressed as requiring that the Lagrangian must transform underduality in a particular way, defined by the Noether-Gaillard-Zumino (NGZ) identity [2]. The NGZ current conservationrequires universally4 that for any duality group embeddable into Sp(2nv,R)

δL =1

4(G̃BG+ F̃CF ) . (2.12)

This leads to the NGZ identity since the variation δL(F,φ) can be computed independently using the chain rule andthe information about δF and δφ.

For example, in the case of a U(1) duality (2.9),

A = D = 0, C = −B , (2.13)

we see that eq. (2.12) reduces to δL = 14 (G̃BG− F̃BF ). Taking into account that in the absence of scalars

δL(F ) =∂L(F )

∂FµνδFµν =

1

2G̃BG , (2.14)

the NGZ identity which follows from (2.12) requires that

1

2G̃BG =

1

4(G̃BG− F̃BF ) . (2.15)

In this case the NGZ identity simplifies to the following relation

FF̃ +GG̃ = 0 . (2.16)

4 Here we discuss theories with actions depending on the field strength F but not on its derivatives. When derivatives are present, ananalogous relation is given by a functional derivative over F of the action, see Appendix A.

4

and are formally mapped into linear combinations of themselves by a GL(2) transformation. Further requiring thatthe transformed G may be obtained from the action evaluated on the transformed F though eq. (2.6) and that theresulting action is a deformation of Maxwell’s theory L = − 1

4F2 +O(F 4) restricts [4] the possible transformations to

δ

!FG

"=

!0 B

−B 0

"!FG

". (2.9)

In other words, the duality transformation exchanges the Bianchi identity and the equations of motion of the originalLagrangian. The original Lagrangian is self-dual if L and LD have the same functional form. It is easy to check thatMaxwell’s theory, with L(F ) = − 1

4F2, is such a theory.

In the derivation above, the dual field strength is determined by eq. (2.6) and is not an independent field. Sinceduality transformations (2.9) mix the field strength and its dual, it is convenient to interpret G as an independentfield and relate it to F by introducing constraint equations as we discuss in section III.

For theories with nv vector fields the strategy for constructing the dual Lagrangian is unchanged. The equations ofmotion and the Bianchi identities remain of the form (2.8) but are invariant under a much larger set of transformations:

δ

!FG

"=

!A BC D

"!FG

", (2.10)

AT = −D BT = B CT = C (2.11)

Here A,B,C,D are the infinitesimal parameters of the transformations, arbitrary real n× n matrices and the trans-formations (2.10) generate the Sp(2nv,R) algebra. For more general theories, when scalar fields are present, we wouldalso include a δφ(A,B,C,D).

Consistency of the duality constraint can be expressed as requiring that the Lagrangian must transform underduality in a particular way, defined by the Noether-Gaillard-Zumino (NGZ) identity [2]. The NGZ current conservationrequires universally4 that for any duality group embeddable into Sp(2nv,R)

δL =1

4(G̃BG+ F̃CF ) . (2.12)

This leads to the NGZ identity since the variation δL(F,φ) can be computed independently using the chain rule andthe information about δF and δφ.

For example, in the case of a U(1) duality (2.9),

A = D = 0, C = −B , (2.13)

we see that eq. (2.12) reduces to δL = 14 (G̃BG− F̃BF ). Taking into account that in the absence of scalars

δL(F ) =∂L(F )

∂FµνδFµν =

1

2G̃BG , (2.14)

the NGZ identity which follows from (2.12) requires that

1

2G̃BG =

1

4(G̃BG− F̃BF ) . (2.15)

In this case the NGZ identity simplifies to the following relation

FF̃ +GG̃ = 0 . (2.16)

4 Here we discuss theories with actions depending on the field strength F but not on its derivatives. When derivatives are present, ananalogous relation is given by a functional derivative over F of the action, see Appendix A.

4

✓@µF̃µ⌫ = 0@µG̃µ⌫ = 0

◆0

=

✓A BC D

◆✓@µF̃µ⌫ = 0@µG̃µ⌫ = 0

◆

F=dA G̃µ⌫ ⌘ 1

2

@L@Fµ⌫

Whydualitysymmetryiscalledhidden symmetry

A new story of N >4supergravities

TheproofoftheAbsenceofAnomaliesin1-loopamplitudesofN >4supergravities

Freedman,RK,Murli,VanProeyen,Yamada 2017

1-loopanomaliesinN=4areassociatedwithUVdivergenceat4loopCarrasco,RK,Roiban,Tseytlin,1303.6219.Bernetal,1309.2498

Theabsenceof1-loopanomaliesinamplitudesinN=5 supergravityagreeswithUVfinitenessof4loopsinN=5 supergravity, establishedbyBern,Davies,Dennen

N=5 supergravity is UV finite at 4 loops2014,ComputationbyBern,Davies,Dennen

Therewasnoexplanationfor3yearsStep1.

Step2

Aformalismforallanomaly-freeN =5,6,7,8isavailableusingsymplectic sectionsdevelopedbyFerraraetalinthecontextoftheblackholeattractorsandtheiruniversality

Wenowusethisuniversality forperturbativeUVproperties

In particular, the �-dependent terms break duality current conservation [4], [5] at the level

�2. Some new terms of the order �2 have to be added to fix the problem, which in turn moves

to �3 etc. The current conservation is restored with an infinite number of higher order terms,

which also have an infinite number of higher derivatives, [6],[7].

We will now study dualities in N � 5 supergravities, [19], [20]. In the case of N = 4

duality symmetry is anomalous, [10], [11] but N � 5 are anomaly-free, [14] and they have only

gravity supermultiplets, no matter. These theories contain in the bosonic sector the metric,

a number nv of vectors and m of (real) scalar fields, see Table 1. The relevant classical vector

and scalar part of action has the following general form:

Lvec = ihN̄

⇤⌃

F�⇤

µ⌫ F�⌃|µ⌫ �N⇤⌃

F+⇤

µ⌫ F+⌃|µ⌫i+

1

2grs(�)@µ�

r@µ�s . (2.2)

where grs(�) (r, s, · · · = 1, · · · ,m) is the scalar metric on the �-model described by the scalar

manifold Mscalar of real dimension m and the vectors kinetic matrix N⇤⌃

(�) is a complex,

symmetric, nv ⇥ nv matrix depending on the scalar fields, see Table 1. F±⇤ are self-dual and

anti-self dual combinations of the vectors field strength’s, details in Appendix A.

Table 1. Scalar Manifolds of N � 4 Extended Supergravities

N Duality group G isotropy H Mscalar nv m

4 SU(1, 1)⌦ SO(6) U(4) SU(1,1)U(1)

6 2

5 SU(1, 5) U(5) SU(1,5)S(U(1)⇥U(5))

10 10

6 SO?(12) U(6) SO?(12)

U(1)⇥SU(6)

16 30

7, 8 E7(7)

SU(8)E7(7)

SU(8)

28 70

In the table, nv is the number of vectors and m is the number of real scalar fields. In all the

cases the duality group G is embedded in Sp(2nv,R).

The formalism of symplectic sections [19], [20] allows a better way to study duality

symmetry of extended supergravities for the case of a general N . The details are in Appendix

A. Instead of a metric in the vector space, N⇤⌃

(�), in eq. (2.2) one can introduce duality

doublets, symplectic section depending on scalars of the theory

f⇤

AB

h⇤AB

!(2.3)

so that the kinetic matrix N can be written in terms of the subblocks f , h, and turns out to

be, symbolically N = h f

�1

N⇤⌃

= h⇤AB (f�1)AB

⌃

(2.4)

– 5 –

coset spaceG

H

What E7(7) doesforN=8

SO?(12) doesforN=6

doesforN=5SU(1, 5)

G knownas`groupsoftypeE7’

Brown,Duffetal,Ferrara,RK

1969, 2009, 2011

Universality

E7(7) invariant

isnowgeneralizedto

G-invariant,E7invariant:

forexample,inN=5 casewherethecomputationsareavailable,dualitysymmetryisG=SU(1,5)

RK,Nicolai,Roiban,Yamada,workinprogress

Hidden Symmetries of N > 4 Supergravities

Strategy:findthereasonwhyBHSUVdivergencefor N >4isabsent

E7(7) SO?(12) SU(1, 5)

L=N-1

N =4,L=3

N =5,L=4

N =8,L=7

FoundUVfinite

FoundUVfinite

D=4Boussard,Howe,Stelle,Vanhove prediction2011:thisfamilyisUVdivergentsincethereisacandidatecounterterm whichissupersymmetricanddualityE7-invariant

???

Bernetal,2013

Bernetal,2014

N =4iscompromisedduetoU(1)anomaly,whichkicksinatL=4However,N =5doesnothaveanomalies,whyitisUVfinite,despitetheexistenceofthe`legitimate’counterterm BHSVcandidate?

2(N�2)

Zd

4x d

4(N�1)✓ (�1ij

�̄N ij)↵�̇ (�1kl

�̄Nkl)↵�̇

ButE7-invarianceofthecounterterm breaksNGZcurrentconservation,theactionhastobedeformedtorestoreduality

Theanswerturnedouttobesimple:hiddenE7-dualitysymmetries,

E7(7) SO?(12) SU(1, 5)

arenotsymmetriesoftheaction

and are formally mapped into linear combinations of themselves by a GL(2) transformation. Further requiring thatthe transformed G may be obtained from the action evaluated on the transformed F though eq. (2.6) and that theresulting action is a deformation of Maxwell’s theory L = − 1

4F2 +O(F 4) restricts [4] the possible transformations to

δ

!FG

"=

!0 B

−B 0

"!FG

". (2.9)

In other words, the duality transformation exchanges the Bianchi identity and the equations of motion of the originalLagrangian. The original Lagrangian is self-dual if L and LD have the same functional form. It is easy to check thatMaxwell’s theory, with L(F ) = − 1

4F2, is such a theory.

In the derivation above, the dual field strength is determined by eq. (2.6) and is not an independent field. Sinceduality transformations (2.9) mix the field strength and its dual, it is convenient to interpret G as an independentfield and relate it to F by introducing constraint equations as we discuss in section III.

For theories with nv vector fields the strategy for constructing the dual Lagrangian is unchanged. The equations ofmotion and the Bianchi identities remain of the form (2.8) but are invariant under a much larger set of transformations:

δ

!FG

"=

!A BC D

"!FG

", (2.10)

AT = −D BT = B CT = C (2.11)

Here A,B,C,D are the infinitesimal parameters of the transformations, arbitrary real n× n matrices and the trans-formations (2.10) generate the Sp(2nv,R) algebra. For more general theories, when scalar fields are present, we wouldalso include a δφ(A,B,C,D).

Consistency of the duality constraint can be expressed as requiring that the Lagrangian must transform underduality in a particular way, defined by the Noether-Gaillard-Zumino (NGZ) identity [2]. The NGZ current conservationrequires universally4 that for any duality group embeddable into Sp(2nv,R)

δL =1

4(G̃BG+ F̃CF ) . (2.12)

This leads to the NGZ identity since the variation δL(F,φ) can be computed independently using the chain rule andthe information about δF and δφ.

For example, in the case of a U(1) duality (2.9),

A = D = 0, C = −B , (2.13)

we see that eq. (2.12) reduces to δL = 14 (G̃BG− F̃BF ). Taking into account that in the absence of scalars

δL(F ) =∂L(F )

∂FµνδFµν =

1

2G̃BG , (2.14)

the NGZ identity which follows from (2.12) requires that

1

2G̃BG =

1

4(G̃BG− F̃BF ) . (2.15)

In this case the NGZ identity simplifies to the following relation

FF̃ +GG̃ = 0 . (2.16)

4 Here we discuss theories with actions depending on the field strength F but not on its derivatives. When derivatives are present, ananalogous relation is given by a functional derivative over F of the action, see Appendix A.

4

Bossard-Nicolai-Carrasco-RK-Roiban deformationprocedurecausedbyUVdivergenceisrequired

theyflipequationsofmotionintoBianchiidentity:dualitycurrentconservation

WehavenowconstructedthecompleteN >4supergravitybosonicactionwithdualitycurrentconservation.ThecandidateUVdivergenceisaddedtotheclassicalactionandinfinitenumberofderivativesispresentedwhichrestoretheE7currentconservation.

News RK,Nicolai,Roiban,Yamada,workinprogress

IfthereisalocalUVdivergenceatsomelooplevel,weaddthecorrespondingLagrangian totheclassicalactionwithanewindependentcouplingl, andaddhigherorderinl termstorestoretheE7-dualitycurrentconservation,brokenbytheUVdivergence.

Thiswillproducethenewdeformedactionwithallhigherderivatives.Intwo-vectorgraviton-scalarsector theresultwefindissimpleforallN >4usinguniversalnotation

together with its complex conjugate. If instead of using the H-covariant constraint we would

like to use the G-covariant one, we can multiply the equation on f�1 so that

G+

µ⌫ ⇤

� (f�1h)⇤⌃

F+⌃

µ⌫AB = 0 ) G+

µ⌫ ⇤

�N⇤⌃

F+⌃

µ⌫AB = 0 . (3.2)

A non-vanishing deformation in the rhs of these equations, which would also be Lorentz

and H-covariant was presented in eq. (5.7) in [7]. It can be derived, following the proposal in

[6] to use the manifestly duality invariant source of deformation. In this case it depends on

a duality doublet, the independent (F,G), i. e. the classical twisted self-duality constraint

(3.2) is not valid

I = �T�AB � T̄�AB = �(h

⌃ABF�⌃ � f⌃

AB G�⌃

)�(h̄AB⇤

F+⇤ � f̄⇤AB G+

⇤

) (3.3)

It leads to a constraint of the type given in eq. (5.7) in [7]

T+

AB + ��T�AB = 0 (3.4)

where the H covariant di↵erential operator � is defined in eq. (2.9). For the consequent

analysis it is convenient to switch to a G-covariant form of equations

(f�1)AB⇤

⇣T+

AB + ��T�AB

⌘= 0 , (3.5)

which will give us the following (we skip indices, they are easy to restore)

[G+ �N F+ +X(G� �NF�)]⇤

= 0 , (3.6)

and the complex conjugate is

[G� � N̄ F� + X̄(G+ � N̄F+)]⇤

= 0 , (3.7)

Here the di↵erential operators X and X̄ are

X = �f�1�f X̄ = �f̄�1�̄f̄ (3.8)

We may substitute G� from (3.7) into (3.6) and we get

G+

⇤

=h(1�XX̄)�1[X(N � N̄ )F� + (N �XX̄N̄ )F+]

i

⇤

(3.9)

This can be integrated to produce the deformed action, so that the derivative of the action

over F+ will produce the value of G+ in (3.9).

Ldef

= �iF+(1�XX̄)�1X(N � N̄ )F� � iF+(1�XX̄)�1(N �XX̄N̄ )F+ + h.c., (3.10)

– 7 –

together with its complex conjugate. If instead of using the H-covariant constraint we would

like to use the G-covariant one, we can multiply the equation on f�1 so that

G+

µ⌫ ⇤

� (f�1h)⇤⌃

F+⌃

µ⌫AB = 0 ) G+

µ⌫ ⇤

�N⇤⌃

F+⌃

µ⌫AB = 0 . (3.2)

A non-vanishing deformation in the rhs of these equations, which would also be Lorentz

and H-covariant was presented in eq. (5.7) in [7]. It can be derived, following the proposal in

[6] to use the manifestly duality invariant source of deformation. In this case it depends on

a duality doublet, the independent (F,G), i. e. the classical twisted self-duality constraint

(3.2) is not valid

I = �T�AB � T̄�AB = �(h

⌃ABF�⌃ � f⌃

AB G�⌃

)�(h̄AB⇤

F+⇤ � f̄⇤AB G+

⇤

) (3.3)

It leads to a constraint of the type given in eq. (5.7) in [7]

T+

AB + ��T�AB = 0 (3.4)

where the H covariant di↵erential operator � is defined in eq. (2.9). For the consequent

analysis it is convenient to switch to a G-covariant form of equations

(f�1)AB⇤

⇣T+

AB + ��T�AB

⌘= 0 , (3.5)

which will give us the following (we skip indices, they are easy to restore)

[G+ �N F+ +X(G� �NF�)]⇤

= 0 , (3.6)

and the complex conjugate is

[G� � N̄ F� + X̄(G+ � N̄F+)]⇤

= 0 , (3.7)

Here the di↵erential operators X and X̄ are

X = �f�1�f X̄ = �f̄�1�̄f̄ (3.8)

We may substitute G� from (3.7) into (3.6) and we get

G+

⇤

=h(1�XX̄)�1[X(N � N̄ )F� + (N �XX̄N̄ )F+]

i

⇤

(3.9)

This can be integrated to produce the deformed action, so that the derivative of the action

over F+ will produce the value of G+ in (3.9).

Ldef

= �iF+(1�XX̄)�1X(N � N̄ )F� � iF+(1�XX̄)�1(N �XX̄N̄ )F+ + h.c., (3.10)

– 7 –

Atl=0 thisactionisclassical.

WefindthatthisactionwithE7-currentconservationhasnosupersymmetricembedding!

Deformedaction,followingideasinRK,Bossard-Nicolai2011

SU(N )symmetryoftheonshellamplitudeisbrokendowntoSO(N )

RK,Nicolai,Roiban,Yamada,workinprogress

WhySU(N )isbrokentoSO(N )?

A+A+

A A

h++ h�� h++ h��

A+A+

h++ h�� h++ h��

Figure 1. Graphs contributing to the amplitude A(AA+(p1)A

B+(p2)h++(p3)h++(p4)h��(p5)h��(p6)).

The light-cone superfield �(p, ⌘) has now the following simple form:

�(p, ⌘) = h̄(p) + ⌘a a(p) + ⌘abB

ab(p) + ⌘abc�abc(p) +

+⌘abcd�abcd(p) + ⌘̃abc�̄abc(p) + ⌘̃abB̄ab(p) + ⌘̃a ̄a(p) + ⌘̃h(p) . (5.6)

Here ⌘a1...an ⌘ 1

n! ⌘a1 ...⌘a2 and ⌘̃a1...an ⌘ ✏a1...anb1...bn�8 ⌘b1...bn�8 ,. This superfield has helicity

+2 and mass dimension -4. When p2 = 0, it is an on shell superfield in terms of which the

amplitudes for the on shell S-matrix are defined.

5.2 SU(N ) symmetry of the on-shell amplitudes is broken down to SO(N )

Since in this section we treat the coupling � as independent of the gravitational coupling,

to test the on shell symmetry properties of the deformed action (B.18) it su�ces to analyze

tree-level amplitudes with �-dependent vertices. Thus, slightly rewriting eq. (B.18) to make

the Feynman rules manifest and keeping terms with the minimum number of gravitons, we

analyze the O(�2) amplitudes of the action (here we set = 1 since it is not relevant for our

discusssion)

Lb.p. =� 1

2(F+)2 � 1

2(F�)2

� 8�(@µ1AA⌫1)P+

µ1⌫1⇢1�1

X⇢1�1;⇢2,�2;⌘1⇣1;⌘2⇣2h⌘1⇣1h⌘2⇣2P�µ2⌫2⇢2�2

(@µ2AA⌫2)

� 4�2(@µ1AA⌫1)P+

µ1⌫1⇢1�1

X⇢1�1;⇢2,�2;⌘1⇣1;⌘2⇣2h⌘1⇣1h⌘2⇣2P�⇢2,�2;�2,�2�AB

(@↵1AB�1)P

+

↵1�1�1�1

X�1�1;�2,�2;✓1⌧1;✓2⌧2h✓1⌧1h✓2⌧2

+ h.c. (5.7)

Here P± are projectors onto the self-dual and anti-self-dual components of 2-tensors

P±µ1⌫1⇢1�1

=1

4

��µ1⇢1 �

⌫1�1

� �⌫1⇢1�µ1�1

⌥ i✏µ1⌫1⇢1�1

�(5.8)

and Xhh is the leading term in the expansion of � in the metric fluctuations.

– 11 –

N >4

A+A+

A A

h++ h�� h++ h��

A+A+

h++ h�� h++ h��

Figure 1. Graphs contributing to the amplitude A(AA+(p1)A

B+(p2)h++(p3)h++(p4)h��(p5)h��(p6)).

The light-cone superfield �(p, ⌘) has now the following simple form:

�(p, ⌘) = h̄(p) + ⌘a a(p) + ⌘abB

ab(p) + ⌘abc�abc(p) +

+⌘abcd�abcd(p) + ⌘̃abc�̄abc(p) + ⌘̃abB̄ab(p) + ⌘̃a ̄a(p) + ⌘̃h(p) . (5.6)

Here ⌘a1...an ⌘ 1

n! ⌘a1 ...⌘a2 and ⌘̃a1...an ⌘ ✏a1...anb1...bn�8 ⌘b1...bn�8 ,. This superfield has helicity

+2 and mass dimension -4. When p2 = 0, it is an on shell superfield in terms of which the

amplitudes for the on shell S-matrix are defined.

5.2 SU(N ) symmetry of the on-shell amplitudes is broken down to SO(N )

Since in this section we treat the coupling � as independent of the gravitational coupling,

to test the on shell symmetry properties of the deformed action (B.18) it su�ces to analyze

tree-level amplitudes with �-dependent vertices. Thus, slightly rewriting eq. (B.18) to make

the Feynman rules manifest and keeping terms with the minimum number of gravitons, we

analyze the O(�2) amplitudes of the action (here we set = 1 since it is not relevant for our

discusssion)

Lb.p. =� 1

2(F+)2 � 1

2(F�)2

� 8�(@µ1AA⌫1)P+

µ1⌫1⇢1�1

X⇢1�1;⇢2,�2;⌘1⇣1;⌘2⇣2h⌘1⇣1h⌘2⇣2P�µ2⌫2⇢2�2

(@µ2AA⌫2)

� 4�2(@µ1AA⌫1)P+

µ1⌫1⇢1�1

X⇢1�1;⇢2,�2;⌘1⇣1;⌘2⇣2h⌘1⇣1h⌘2⇣2P�⇢2,�2;�2,�2�AB

(@↵1AB�1)P

+

↵1�1�1�1

X�1�1;�2,�2;✓1⌧1;✓2⌧2h✓1⌧1h✓2⌧2

+ h.c. (5.7)

Here P± are projectors onto the self-dual and anti-self-dual components of 2-tensors

P±µ1⌫1⇢1�1

=1

4

��µ1⇢1 �

⌫1�1

� �⌫1⇢1�µ1�1

⌥ i✏µ1⌫1⇢1�1

�(5.8)

and Xhh is the leading term in the expansion of � in the metric fluctuations.

– 11 –

TheSU(N )breakingtotal6-pointamplitudedoesnotvanishiftheE7currentisconservedandviceversa:

l l l2

eithersupersymmetryorE7currentconservation

Namelyonemayconsiderthesituationwherethedeformationparameterl duetoL-loopUVdivergenceisk-dependent

�2 =c24L

✏2Wetriedtofindanintermediateloopamplitudedivergences,whichwouldrestoreSU(N )andsupersymmetrybrokenbyaUVdivergence.Ifwewouldfindarestorationofsupersymmetryinthisapproach,itmightindicatethatourapproachwithanindependentparameterofdeformationneedtobereconsidered.ButsofarwedidnotfoundarestorationofSU(N )inthisapproachandnoindicationoftheloophole.

OnonehandournewresultprovidesthefirstexplanationofN=5,L=4UVfiniteness,ontheotherhanditappearstobeverystrong.IncreasingthelooporderLamountstoinsertionofafunctionofMandelstamvariablesf(s,t,u)intoa4-pointamplitude.

ThisinsertiondoesnotinterferewithbreakingofSU(N ) toSO(N )anditappearsthatthisincompatibilityofE7symmetrywithsupersymmetryrequirestheabsenceofallloopUVdivergencesforN >4supergravities.

Beforetakingthislineofargumentseriously,inviewofitsimportance,wehavemadeanattempttofindanotherapproachtotheissueinsteadofstudiesthedeformedactionanditsamplitudeswithk-independentparameterl

� =c2L

✏

Wehavecomputedtheon-shellbosonicamplitudeswhichresultfromthedeformedactionwithE7-dualitycurrentconservation.Thedeformationparameter l isindependentongravitationalcouplingk

Wefoundthattheon-shellbosonicamplitudesinevitablybreakSU(N )symmetrytoSO(N )

Suchamplitudesdonothaveasupersymmetricembedding

IfE7-symmetryandsupersymmetryhavenoanomalies,theonlywaytovalidatebothofthemistohaveUVfiniteness!OnlyinsuchcasethereisnoneedfordeformationwithhigherderivativetermsN=5fourloopsisthe“experimentalpoint”supportingthispicturewhichpredictsUVfinitenessofN >4supergravities.

Summary:

TheargumentappearstobevalidforanylooporderLforN >4NootherexplanationofN=5L=4isavailable!

RK,Nicolai,Roiban,Yamada,workinprogress

However,theissueofk-dependentl requiresmorestudies

Thesymmetryofthemaximal supergravity protectsit,togetherwithmaximalsupersymmetry,fromUVdivergences.

Incosmologicalsettingthemodelswithmaximalsupersymmetryconsistentlytruncatedtominimalsupersymmetryarebasedonthesamesymmetryofthemodulispace

E7(7)(R) � [SL(2,R)]7

E7(7)

Ithasasubgroupwith7Poincaredisksofunitsizea = 1/3

AtleasttillL=7,aseverybodyagrees,ormaybeevenforalllooporders

Duff,Ferrara2006

Prepared for submission to JHEP

D3 Induced Geometric Inflation

Renata Kallosh,

aAndrei Linde,

aDiederik Roest,

band Yusuke Yamada

a

aStanford Institute for Theoretical Physics and Department of Physics, Stanford University, Stan-ford, CA 94305, USA

bVan Swinderen Institute for Particle Physics and Gravity, University of Groningen, Nijenborgh4, 9747 AG Groningen, The Netherlands

E-mail: kallosh@stanford.edu, alinde@stanford.edu, d.roest@rug.nl,yusukeyy@stanford.edu

Abstract: Effective supergravity inflationary models induced by anti-D3 brane interactionwith the moduli fields in the bulk geometry have a geometric description. The Kählerfunction carries the complete geometric information on the theory. The non-vanishingbisectional curvature plays an important role in the construction. The new geometricformalism, with the nilpotent superfield representing the anti-D3 brane, allows a powerfulgeneralization of the existing inflationary models based on supergravity. They can easilyincorporate arbitrary values of the Hubble parameter, cosmological constant and gravitinomass. We illustrate it by providing generalized versions of polynomial chaotic inflation, T-and E-models of ↵-attractor type, disk merger. We also describe a multi-stage cosmologicalattractor regime, which we call cascade inflation.

Ferrara,RK,Linde,Roest,Wrase,Yamada

V(') = ⇤+m2 tanh2'p3↵

Anilpotentsuperfield S representingtheanti-D3braneinabackgroundofCalabi-Yau moduli

Cosmologicalmodelbuildingparadise

R2=3a

ThemodelsarerelativelysimpleiftheT-modulihavehyperbolicgeometryofacombinationofPoincaredisks.Inhalf-flatgeometryvariablestheKahler functionis

TheKahler functionisinvariantunder

inversionandscalingpartoftheMobiussymmetry

Ti !1

Ti, Ti ! a2 Ti

Inflaton shiftsymmetryisbrokenonlyviainteractionwiththeanti-D3brane,viathenilpotentS-fieldgeometry

GSS̄(Ti, T̄i)SS̄

G���S=0

= �1

2

X

i=1

log

h(Ti +

¯Ti)2

4Ti¯Ti

1

m43/2

iTessellation

Ferrara,RK,Linde,Roest,Wrase,Yamada

1. StartwithM-theory,orStringtheory,orN=8supergravity

2. PerformaconsistenttruncationtoN=1supergravityind=4witha7-diskmanifold

3.5 Seven-disk merger model

Finally, we briefly discuss the possible merger of several disks. Consider for instance,

G = logW 20 � 1

2

7X

i=1

log

(1� ZiZi)2

(1� Z2i )(1� Z

2i )

+ S + S + GSSSS, (3.32)

GSS=

1

W 20

(3W 20 +V). (3.33)

corresponding to seven disks with ↵i = 1/3. The scalar potential is

V = ⇤+

m2

7

X

i

|Zi|2 + M2

7

2

X

1ij7

⇣(Zi + Zi)� (Zj + Zj)

⌘2, (3.34)

and the last term gives the dynamical constraint �i = �j where we have defined canonicalfields as Zi = tanh

�i+i✓ip2

. During inflation at �i = �j ='p7, the scalar potential reads

V(') = ⇤+m2tanh

2 'p14

, (3.35)

in terms of the canonically normalized inflaton field.

The axionic directions are stabilized at their origin, and their masses are given by

m2✓i = 2(m2

+ 2W 20 )�

1

7

m2

7 + 6 cosh

r2

7

'

!cosh

�4 'p14

. (3.36)

The first two constant part dominate the mass and the remaining negative part is suppressedduring inflation. At the minimum, the mass of the axions becomes m2

✓i=

17m

2+ 4W 2

0 andis still positive.

For real directions {�i}, the following canonical mass eigenbasis is useful, ' =

1p7

P7i=1 �i,

and �i =1p8�i

((7� i)�i � �i+1 · · ·� �7). The inflaton is ' and moduli �i are stabilized attheir origin with the mass

m2�i

=

1

7

2m2

+ 4M2 �m2cosh

r2

7

'

!cosh

�4 'p14

. (3.37)

As the two disk models, the mass of the moduli �i becomes small, and when 4M2 <

m2cosh

q27', they becomes tachyonic. At the minimum ' = 0, the inflaton and moduli

mass are given by

m2� =

1

7

m2, m2�i

=

1

7

m2+

4

7

M2. (3.38)

Note that SUSY breaking takes place at the minimum; GS = 1 andq

GSGSSGS =

p3W0.

Here again we see the advantage of using the new geometric class of models comparative tothe earlier version of the seven-disk model in Ref. [24] where we only studied an inflationarystage.

– 17 –

corresponding to the merger of seven disks of unit size

3.5 Seven-disk merger model

Finally, we briefly discuss the possible merger of several disks. Consider for instance,

G = logW 20 � 1

2

7X

i=1

log

(1� ZiZi)2

(1� Z2i )(1� Z

2i )

+ S + S + GSSSS, (3.32)

GSS=

1

W 20

(3W 20 +V). (3.33)

corresponding to seven disks with ↵i = 1/3. The scalar potential is

V = ⇤+

m2

7

X

i

|Zi|2 + M2

7

2

X

1ij7

⇣(Zi + Zi)� (Zj + Zj)

⌘2, (3.34)

and the last term gives the dynamical constraint �i = �j where we have defined canonicalfields as Zi = tanh

�i+i✓ip2

. During inflation at �i = �j ='p7, the scalar potential reads

V(') = ⇤+m2tanh

2 'p14

, (3.35)

in terms of the canonically normalized inflaton field.

The axionic directions are stabilized at their origin, and their masses are given by

m2✓i = 2(m2

+ 2W 20 )�

1

7

m2

7 + 6 cosh

r2

7

'

!cosh

�4 'p14

. (3.36)

The first two constant part dominate the mass and the remaining negative part is suppressedduring inflation. At the minimum, the mass of the axions becomes m2

✓i=

17m

2+ 4W 2

0 andis still positive.

For real directions {�i}, the following canonical mass eigenbasis is useful, ' =

1p7

P7i=1 �i,

and �i =1p8�i

((7� i)�i � �i+1 · · ·� �7). The inflaton is ' and moduli �i are stabilized attheir origin with the mass

m2�i

=

1

7

2m2

+ 4M2 �m2cosh

r2

7

'

!cosh

�4 'p14

. (3.37)

As the two disk models, the mass of the moduli �i becomes small, and when 4M2 <

m2cosh

q27', they becomes tachyonic. At the minimum ' = 0, the inflaton and moduli

mass are given by

m2� =

1

7

m2, m2�i

=

1

7

m2+

4

7

M2. (3.38)

Note that SUSY breaking takes place at the minimum; GS = 1 andq

GSGSSGS =

p3W0.

Here again we see the advantage of using the new geometric class of models comparative tothe earlier version of the seven-disk model in Ref. [24] where we only studied an inflationarystage.

– 17 –

During inflation

3.5 Seven-disk merger model

Finally, we briefly discuss the possible merger of several disks. Consider for instance,

G = logW 20 � 1

2

7X

i=1

log

(1� ZiZi)2

(1� Z2i )(1� Z

2i )

+ S + S + GSSSS, (3.32)

GSS=

1

W 20

(3W 20 +V). (3.33)

corresponding to seven disks with ↵i = 1/3. The scalar potential is

V = ⇤+

m2

7

X

i

|Zi|2 + M2

7

2

X

1ij7

⇣(Zi + Zi)� (Zj + Zj)

⌘2, (3.34)

and the last term gives the dynamical constraint �i = �j where we have defined canonicalfields as Zi = tanh

�i+i✓ip2

. During inflation at �i = �j ='p7, the scalar potential reads

V(') = ⇤+m2tanh

2 'p14

, (3.35)

in terms of the canonically normalized inflaton field.

The axionic directions are stabilized at their origin, and their masses are given by

m2✓i = 2(m2

+ 2W 20 )�

1

7

m2

7 + 6 cosh

r2

7

'

!cosh

�4 'p14

. (3.36)

The first two constant part dominate the mass and the remaining negative part is suppressedduring inflation. At the minimum, the mass of the axions becomes m2

✓i=

17m

2+ 4W 2

0 andis still positive.

For real directions {�i}, the following canonical mass eigenbasis is useful, ' =

1p7

P7i=1 �i,

and �i =1p8�i

((7� i)�i � �i+1 · · ·� �7). The inflaton is ' and moduli �i are stabilized attheir origin with the mass

m2�i

=

1

7

2m2

+ 4M2 �m2cosh

r2

7

'

!cosh

�4 'p14

. (3.37)

As the two disk models, the mass of the moduli �i becomes small, and when 4M2 <

m2cosh

q27', they becomes tachyonic. At the minimum ' = 0, the inflaton and moduli

mass are given by

m2� =

1

7

m2, m2�i

=

1

7

m2+

4

7

M2. (3.38)

Note that SUSY breaking takes place at the minimum; GS = 1 andq

GSGSSGS =

p3W0.

Here again we see the advantage of using the new geometric class of models comparative tothe earlier version of the seven-disk model in Ref. [24] where we only studied an inflationarystage.

– 17 –

3a=7example

DeSitterexit

Thescalarpotentialdefininggeometryis

r ⇡ 10�2

Ferrara,RK,Linde,Roest,Wrase,Yamada

1. StartwithM-theory,orStringtheory,orN=8supergravity

2. PerformaconsistenttruncationtoN=1supergravityind=4witha7-diskmanifold

3.5 Seven-disk merger model

Finally, we briefly discuss the possible merger of several disks. Consider for instance,

G = logW 20 � 1

2

7X

i=1

log

(1� ZiZi)2

(1� Z2i )(1� Z

2i )

+ S + S + GSSSS, (3.32)

GSS=

1

W 20

(3W 20 +V). (3.33)

corresponding to seven disks with ↵i = 1/3. The scalar potential is

V = ⇤+

m2

7

X

i

|Zi|2 + M2

7

2

X

1ij7

⇣(Zi + Zi)� (Zj + Zj)

⌘2, (3.34)

and the last term gives the dynamical constraint �i = �j where we have defined canonicalfields as Zi = tanh

�i+i✓ip2

. During inflation at �i = �j ='p7, the scalar potential reads

V(') = ⇤+m2tanh

2 'p14

, (3.35)

in terms of the canonically normalized inflaton field.

The axionic directions are stabilized at their origin, and their masses are given by

m2✓i = 2(m2

+ 2W 20 )�

1

7

m2

7 + 6 cosh

r2

7

'

!cosh

�4 'p14

. (3.36)

The first two constant part dominate the mass and the remaining negative part is suppressedduring inflation. At the minimum, the mass of the axions becomes m2

✓i=

17m

2+ 4W 2

0 andis still positive.

For real directions {�i}, the following canonical mass eigenbasis is useful, ' =

1p7

P7i=1 �i,

and �i =1p8�i

((7� i)�i � �i+1 · · ·� �7). The inflaton is ' and moduli �i are stabilized attheir origin with the mass

m2�i

=

1

7

2m2

+ 4M2 �m2cosh

r2

7

'

!cosh

�4 'p14

. (3.37)

As the two disk models, the mass of the moduli �i becomes small, and when 4M2 <

m2cosh

q27', they becomes tachyonic. At the minimum ' = 0, the inflaton and moduli

mass are given by

m2� =

1

7

m2, m2�i

=

1

7

m2+

4

7

M2. (3.38)

Note that SUSY breaking takes place at the minimum; GS = 1 andq

GSGSSGS =

p3W0.

Here again we see the advantage of using the new geometric class of models comparative tothe earlier version of the seven-disk model in Ref. [24] where we only studied an inflationarystage.

– 17 –

corresponding to the dynamical situation where out of 7 disks of unit size only one is evolving, the other 6 are stabilized.

During inflation

3a=1example

DeSitterexit

Thescalarpotentialdefininggeometryis

V(') = ⇤+m2 tanh2'p2 r ⇡ 10�3

V = ⇤+m2|Z1|2 +M2i=7X

i=2

(Zi + Z̄i � C)2

CMB satelliteproposals

All targeting σ(r) ~ few 10-4

PIXIE(NASA MIDEX~2023)

LiteBIRD (JAXA, ~2025)

LiteCORE(ESA M5 ~2026-2030)

???

S. W. Hawking, an optimist!

“Thesewillbetheoutstandingproblemsfortheoreticalphysicists inthenexttwentyyearsorso.”

OnN=8supergravity in1980

Stephen

2017

Maybewehavestartednailingitdown