N=8Supergravity

RenataKallosh

BasedonRK,1103.4115,1104.5480,Carrasco,RK,Roiban,1108.4390,Chemissany,RK,Or?n,1112.0332,Broedel,Carrasco,Ferrara,RK,Roiban,workinprogress,RK,Or?n,workinprogress.

S.W.Hawking“IstheEndinSightforTheore?calPhysics:An

InauguralLecture”1980•  “AtthemomenttheN=8supergravitytheoryistheonlycandidateinsight.Therearelikelytobeanumberofcrucialcalcula?onswithinthenextfewyearswhichhavethepossibilityofshowingthatthetheoryisnogood.”

•  “Ifthetheorysurvivesthesetests,itwillprobablybesomeyearsmorebeforewedevelopcomputa?onalmethodsthatwillenableustomakepredic?ons.”

•  “Thesewillbetheoutstandingproblemsfortheore?calphysicistsinthenexttwentyyearsorso.”

1981:3‐loopcountertermsarefound,withunknowncoefficient

2007:Calcula?onsshowthatthecoefficientinfrontofthe3‐loopcountertermvanishes!!!

IsN=8supergravityall‐loopfinite?Ifso,whatdoesitmean?

OldWisdom,andwhydidwequitin1981Usingtheexistenceofthecovarianton‐shellsuperspaceBrink,Howe,1979andthebackgroundfieldmethodinQFTonecanusethetensorcalculusandconstructtheinvariantcandidatecountertermsRK;Howe,Lindstrom,1981.Suchgeometriccountertermshaveallknownsymmetriesofthetheory,includingE7(7).Theystartatthe8‐looplevel.Linearizedonesstartatthe3‐looplevel,RK;Howe,Stelle,Townsend,1981Clarifica?onofthe1/8BPS7‐loopcandidate,Bossard,Howe,Stelle,Vanhove,2011.

Tensorcalculus=infiniteprolifera?onofcandidatecounterterms

Newera,newpeople,newcomputers,newrules:neveruseN=8supergravityrules,buildit

fromN=4Super‐Yang‐Mills

•  Light‐conesuperspacecountertermsarenotavailableatanylooporder(predic?onofUVfiniteness).RK2008,2009.Thisiss?llthecasein2012:

•  3‐loopfinitenessfollowsfromE7(7)

Explicitcalcula?ons:noUVdivergencesat3‐and4loops

Bern,Carrasco,Dixon,Johansson,Kosower,Roiban,2007‐2009

Broedel,DixonBeisert,Elvang,Friedman,Kiermaier,Morales,S?eberger

Bossard,Howe,Stelle,2009‐2010

Explana?on?

•  Stringtheory‐>fieldtheory,M.Greene’stalk

Five‐loopprogressiscon?nuingbutnonewresultsyet(asofJanuary3).

TheE7(7)symmetryofN=8supergravitywasdiscoveredbyCremmerandJulia

in1979

Cremmer,Julia1979;Cremmer,Julia,Sherk,1978

DeWit,Nicolai1982;deWit,Freedman,1977

Ifwetrustcon?nuousglobalE7(7)atthe3‐loopquantumlevelwhatisthepredic?onathigherloops?

E7(7)revisited:RK,2011Noether‐Gaillard‐Zuminocurrentconserva?onisinconsistentwiththeE7(7)invarianceofthecandidatecounterterms

E7(7)revisited:Bossard‐Nicolai,2011Yes,NGZcurrentconserva?onisinconsistentwiththeE7(7)invarianceofthecandidatecounterterms.However,thereisaprocedureofdeforma?onofthelineartwistedself‐dualityconstraint,whichshouldbeabletofixtheproblem.ExamplesofU(1)duality.

E7(7)revisited:Carrasco,RK,Roiban,2011Bossard‐Nicolaideforma?onprocedureneedsasignificantmodifica?ontoexplainthesimplestcaseofBorn‐Infelddeforma?onoftheMaxwelltheorywhichconservestheNGZcurrent

Adualitydoubletdependsonfieldsinac?on,F=dAandon

Noether‐Gaillard‐Zuminocurrentconserva?onrequiresthattheac?ontransformsinapar?cularway,INSTEADOFBEINGINVARIANT

Dualitysymmetryrotatesvectorfieldequa?onsandBianchiiden??es

U(1)caseissimple FF̃ +GG̃ = 0

Canonicalexample:Maxwellvs.Dirac‐Born‐Infeld

TopreservetheU(1)NGZcurrentconserva?ononeneedsallpowersofF,onceF4wasadded,oneneedsFn!

x

Givensomequantumgeneratedduality‐invarianttermintheeffec?veac?onofsomeduality‐preservingtheory,isitpossibletoaddhigher‐ordertermstorestoretheinvarianceofthefieldequa?ons?

Carrasco,Kallosh,RR

focusonnonlinearE&M‐typedualitywepresentedanalgorithmicconstruc?onoftheBIac?onbyrequiringduality

‐‐Twistedlinearself‐duality

Structureofthedeforma?on;nonlineartwistedself‐duality

Mustbemanifestlydualityinvariant,accordingtoBossard,Nicolai

I(T, T !)

Butwhatisit?ForexampleforBI?

StartwithIandconstructnonlineartwistedself‐dualityequa?on

Makeanansatzfortheac?on

Theconstruc?on Carrasco,RK,Roiban

Easytorecoveranydesired/knownac?onexhibi?ngdualityThereisaninfinitesupply

Gibbons,Rasheed,1995;Gaillard,Zumino,1997,…

Courant‐Hilbertdiffeq

FortheBorn‐Infeldac?ononeneeds:

Remarkablycomplicatedsourceofdeforma?on,isthereareasonforit?

Hereweknewwhatwewantedtoobtainandusingthatinforma?onweconstructedtoallorders

Butifwedonotknowtheac?on,howdowefind?

I

II

NewU(1)dualityinvariantmodels,unknownbeforeChemissany,RK,Or?n,``Born‐InfeldwithHigherDeriva?ves,'’1112.0332

Thefirstquar?ccorrec?ontoMaxwellwasknownfromopenstringtheoryandD3‐branequantumcorrec?ons:(dF)4Andreev,Tseytlin1988,Shmakova;DeGiovanni,Santambrogio,Zanon,1999;Green,Gutperle,2000

(!!)4(s2 + t2 + u2)tµ1!1µ2!2µ3!3µ4!4Fµ1!1(p1)Fµ2!2(p2)Fµ3!3(p3)Fµ4!4(p4)

t(8) ! tµ1!1µ2!2µ3!3µ4!4Green‐Schwarz8‐tensor,1982

Chemissany,deJong,deRoo,2006:U(1)NGZcurrentisconservedat(dF)4level,butbrokenathigherlevels.Howtoaddd4nF2n+2torestoretheNGZcurrentconserva?on?

UsingCarrasco,RK,Roibanalgorithmwefoundarecursiveanswer

Born‐Infeldmodelwithhigherderiva?vesandNGZcurrentconserva?on

Recursionrela?ongeneratesn‐orderfromthepreviousones: [µ!] ! a

Algorithmforanac?onatanyorderin d4nF2n+2

d4F4

d8F6

Forexample

etc

Newformofthereconstruc?veiden?tyforU(1)duality

S(F ) = 14!

!d4x d!FG̃(!)

Ifweknowthemanifestlyinvariantsourceofdeforma?on,weknowandweknowtheac?on,whichpreservestheNGZcurrentconserva?oninallordersin

I(T, T !)

G̃(!)

!

!xx

Caseofopenstring(dF)4correc?ons

Istring = ! t(8)abcd["µT!+ a"µT" b"!T

!+ c"!T" d +1

2"µT

!+ a"µT !+ b"!T" c"!T" d]

Thiscorrespondstothefollowing4‐pointamplitudeOncewehaveiden?fiedamanifestlyU(1)invariantsourceofdeforma?on,thecompleteac?onwithallhigherderiva?vetermsfollowsfromthealgorithm

S(1) =!

24

!d4x t(8)abcd"µF

a"µF b"!F c"!Fd

U(1)dualityandN=2supersymmetryBroedel,Carrasco,Ferrara,RK,Roiban

Superfieldac?onreconstruc?veiden?ty

Newclassofmodels!NGZcurrentconserva?on:allpowersofW’sintheac?on.Werecoveredallmodelsdiscoveredbeforeandaninfiniteclassofnewones.

WelearnedhowtogethigherorderinFtermstorestoretheNGZcurrentconserva?onforU(1)duality

•  Born‐Infeldtypemodels,knownbeforebutnowwefollowanewprocedure•  N=1supersymmetricBItypemodels,knownbeforebutnowwefollowanewprocedurewhichissupposedtoworkfortheBorn‐InfeldN=8supergravity

•  Born‐Infeldtypemodelswithhigherderiva?ves,NEW!•  GenericN=2supersymmetricBItypemodels,Broedel,Carrasco,Ferrara,RK,Roiban,NEW!

LessonforN=8supergravity:tobeabletoconstructallhigherorderinFtermswithderiva?ves,onceweaddthe4‐vectorcounterterms,weneedtoproduceN=8Born‐Infeldtypesupergravity:tallorder!

Canwesaymoretoday?

NewclassofE7(7)invariantsandUVproper?esofN=8

WorkinprogresswithTomasOr?n

•  WeconstructnewmanifestE7(7)invariantsusingtheoctonionicnatureoftheE7(7)andthestructureoftheBIU(1)dualityinvariantmodelwithhigherderiva?ves.•  OurnewE7(7)invariants,basedonaJordantriplesystem,generalizetheCartan‐

Cremmer‐Juliaquar?cinvariant.Theymaybeusefulforamplitudesinsteadoftheblackholehorizonarea.Theydonotdependonscalars.

•  WeshowthatthecorrespondingE7(7)invariants,requiredfortheBossard‐Nicolaideforma?onprocedure,areinconsistentwiththeUVcounterterms.

•  TheclassofE7(7)invariantsconstructedfromtheSU(8)covarianttensorsmay,or

maynotprovidethesourceofdeforma?on,dependingonvectorsaswellasscalars.Theymay,ormaynotleadtoBorn‐InfeldN=8supergravity.Thisremainstobeseen!

ManifestE7(7)andLocalSU(8)?•  InthecandidatecountertermsweusedthebuildingblockswhicharelocalSU(8)tensors,“blind”underE7(7),scalarsin

•  ApossibilitytohavelocalSU(8)symmetrysimultaneouslywiththemanifestlyE7(7)symmetricabelianU(1)gaugesymmetryreliesheavilyonthelineartwistedself‐dualityforgraviphotons•  Toconstruct

•  withlocalSU(8)graviphotons,manifestE7(7)covariant56U(1)’sisthechallenge!

I(T , T̄ )

T+AB = 0

T+AB != 0

E7(7)

SU(8)

ThemanifestE7(7)invariantsarerare!Thefundamental56ofE7(7)isspannedbythetwoan?symmetricrealtensors(F,G)

xxx

CartanQuar?cE7(7)Invariant

Cremmer‐JuliaQuar?cE7(7)Invariant

GroupsoftypeE7,1969R.Brown,JournalfurdiereineundangewandteMathema?k,Vol236,79

•  GroupsoftypeE7aregroupsoflineartransforma?onsleavinginvarianttwomul?linearforms:oneisskew‐symmetricbilinear

•  andtheotherisasymmetricfour‐linearform

•  HeretheisthetripleproductdefinedbythecubicJordanalgebra

•  AnexplicitexpressioninGunaydin,Koepsell,Nicolai,2000

•  ReviewinDuffetal,0809.4685

{x, y}

q(x, y, z, w) = {T (x, y, z), w}

T (x, y, z)

J ! q(x, x, x, x) wasusedfortheblackholesandquantuminforma?on

NewE7(7)invariants

J(8) ! J[µ1!1][µ2!2][µ3!3][µ4!4] = q(xµ1!1 , yµ2!2 , zµ3!3 , wµ4!4)

t(8)[µ1!1][µ2!2][µ3!3][µ4!4]J[µ1!1][µ2!2][µ3!3][µ4!4] ! t(8) · J(8)Relevantfortheamplitudes!Insteadofelectricandmagne?cchargeswehavehelicityamplitudes

RK,Or?n

(p, q) ! (Fµ! , Gµ!)

f(s, t, u) t(8) · J(8)

WhenE7(7)isreducedtoU(1),ournewinvariantsdescribetheopen

stringcorrec?ons

(!!)4(s2 + t2 + u2)tµ1!1µ2!2µ3!3µ4!4Fµ1!1(p1)Fµ2!2(p2)Fµ3!3(p3)Fµ4!4(p4)

N=4YMN=8TwistorsOctonions

S.W.Hawking,anop?mist!

“Thesewillbetheoutstandingproblemsfortheore?calphysicistsinthenexttwentyyearsorso.”

OnN=8supergravityin1980

Stephen