Consistent truncations with massive modes and holography

0.11 MB·English

by  Davide Cassani

FortschrittederPhysik,28January2011
Consistent truncations with massive modes and holography
DavideCassani1,∗,GianguidoDall’Agata1,2,∗∗,andAntonF.Faedo2,∗∗∗
1 DipartimentodiFisica“GalileoGalilei”,Universita`diPadova,ViaMarzolo8,35131Padova,Italy.
2 INFN,SezionediPadova,ViaMarzolo8,35131Padova,Italy.
1
1
Keywords Consistenttruncations,Supergravitymodels,Gauge/gravityduality.
0
2 WereviewthebasicfeaturesofsomerecentlyfoundconsistentKaluza–Kleintruncationsincludingmassive
n modes. Weemphasizethegeneralideasunderlyingthereductionprocedure,thenwefocusontypeIIBsu-
pergravityon5-dimensionalmanifoldsadmittingaSasaki–Einsteinstructure,whichleadstohalf-maximal
a
J gaugedsupergravityinfivedimensions.Finally,wecommentontheholographicpictureofconsistency.

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]
h
t 1 Introduction
-
p
e Inthecontextofthegauge/gravityduality,consistentKaluza–Kleintruncationsof10-and11-dimensional
h
supergravityhaveprovedtobepowerfulsolution-generatingtools. Atruncationofthehigher-dimensional
[
spectrum on some compact manifold is said to be consistent if all solutions of the truncated, lower-

1 dimensionaltheoryarealsosolutionstothefull,higher-dimensionaltheory.Sincetheyprovideaneffective
v
lower-dimensionalmodelwitharestrictednumberofdegreesoffreedom,consistenttruncationsallowto
2
tackleseveralproblemsinasetupwhichismuchsimplerthantheoriginaltheory,guaranteeingatthesame
1
timetheliftingofthesolutions. Asanexample,consistenttruncationsto5dimensionsprovidea conve-
3
5 nient framework for describing the renormalizationgroup flow of the dual 4-dimensionalquantum field
. theories,withthefifthcoordinateplayingtheroleofanenergyscale.
1
0 Afurther,moreformal,motivationtostudyconsistenttruncationsisthattheyrepresentthemaintoolto
1 investigatewhichlower-dimensionalsupergravitiescanbeconnectedtostringtheory.
1 Recently,therehasbeenarevivalofinterestinconsistenttruncations,mainlymotivatedbytheintense
:
v researchactivitytowardsaholographicdescriptionofstronglycoupledcondensedmatterphenomena,such
i assuperconductivityandquantumcriticalpointsexhibitingnon-relativisticscaleinvariance. Alimitation
X
oftheseholographicmodelsisthatmostofthemareadhocconstructionsbuiltinabottom-upapproach,
r
a whileinordertohavefullcontrolonthegauge/gravitycorrespondencearigorousembeddingintostring
theoryisneeded. Toachievethis,oneshouldfindaconsistenttruncationof10-dimensionalsupergravity
tothedesiredlower-dimensionalmodel.Acrucial,non-trivialfeaturerequiredbytheseapplicationsisthat
thetruncationpreservemassiveand/orchargedKaluza–Kleinmodes(seeinparticular[1,2,3,4]).
Here,wewillreviewthebasicideasunderlyingsomerecentlyfoundconsistenttruncationsofhigher-
dimensionalsupergravitywithmassivemodes. Often,theconsistencyofatruncationreliesonsomesym-
metryunderwhichthepreservedmodesareinvariant,andsuchthatthetruncatednon-invariantmodesare
nevergeneratedintheequationsofmotion. Inthecasesofinterestforus,thecompactmanifoldadmitsa
G-structurewhoseintrinsictorsionisalsoG-invariant. Themodespreservedbythetruncationarechosen
tobepreciselyallthesingletsunderG. When,asitwillbethecaseforus,amongtheinvariantfieldsthere
isatleastonespinor,andtheinvariantbosonicfieldscanbereconstructedbytakingappropriatespinorbi-
linears,onecandefineatruncationansatzwhichpreservesafractionofsupersymmetry.Aclearadvantage
∗ cassaniatpd.infn.it – Speaker
∗∗ dallagatatpd.infn.it
∗∗∗ faedoatpd.infn.it
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2 D.Cassani,G.Dall’Agata,andA.F.Faedo:Consistenttruncationswithmassivemodesandholography
inthiscaseisthatonedisposesofapowerfulorganizingprinciple:sincesupersymmetrictheoriesarevery
constrained,justafewdataneedtobeprovidedinordertofullyspecifythetruncatedmodel.
In the following, we illustrate how these ideas work in practice by presenting a consistent truncation
of type IIB supergravity on arbitrary squashed Sasaki–Einstein manifolds, leading to gauged = 4
N
supergravityinfivedimensions[5](seealso[6,7,8]forcloselyrelatedwork,and[2,9,10,11]forearlier
supersymmetric results). As we will briefly describe, the resulting 5-dimensional model exhibits quite
remarkablefeatures,suchasthepresenceofbothmasslessandmassivemodes,aswellastensorfieldsdual
tovectorschargedunderanon-abeliangaugegroup. Moreover,theretainedKaluza–Kleinmodescapture
theuniversalpuregaugesectorofthedual4-dimensionalsuperYang–Millstheories.
2 Type IIBsupergravityon squashed Sasaki-Einsteinmanifolds
Aregular(respectively,quasi-regular)Sasaki–EinsteinmanifoldY canbeseenasaU(1)fibrationovera
Ka¨hler–Einsteinbasemanifold(respectively,orbifold)B :
KE
ds2(Y) = ds2(B )+η η, (1)
KE
⊗
whereη isthegloballydefined1-formspecifyingthefibration. All5-dimensionalSasaki–Einsteinmani-
foldsarealsoendowedwithareal2-formJ andacomplex2-formΩ,bothgloballydefined.Thesesatisfy
thealgebraicconstraints
ηyJ =ηyΩ=0, Ω J = Ω Ω = 0, Ω Ω = 2J J = 4vol(B ), (2)
KE
∧ ∧ ∧ ∧
aswellasthedifferentialconditions
dη = 2J , dΩ=3iη Ω. (3)
∧
Therelations(2)implythatthestructuregroupofthe5-dimensionalmanifold,whichgenericallyisSO(5),
isreducedtoSU(2),withtheformsη,J,ΩbeingSU(2)singlets. Theconditions(3)constrainthetorsion
oftheSU(2)structure,whichisrequiredtoalsobeanSU(2)singlet,andconstant.Startingfromtheforms,
the metric on the 4-dimensionalsubspacetransverse to η can be reconstructedby identifyinga complex
structureI withrespecttowhichΩisoftype(2,0),andtakingtheproductJI. WealsohavetheHodge
dualityrelations
η =vol(B ), J = J η, Ω = Ω η. (4)
KE
∗ ∗ ∧ ∗ ∧
Our ansatz for the dimensionalreductionis defined by writing down the most general expressionfor
the metric and the varioustensor fields oftype IIB supergravityin termsof the formscharacterizingthe
Sasaki–Einstein structure. In doing this, we actually consider a class of internal metrics which is more
general than (1): we allow for an overall volume parameter (the “breathing mode”), as well as for a
parameter modifying the relative size of the U(1) fibre with respect to the size of the Ka¨hler–Einstein
base(the“squashingmode”). Hencetheclassofspacesonwhichwearereducingistheoneofsquashed
Sasaki–Einsteinmanifolds.
Specifically,ourtruncationansatzforthe10-dimensionalmetricintheEinsteinframeis[1]:
ds2 =e−32(4U+V)ds2(M) + e2Uds2(BKE) + e2V(η+A) (η+A), (5)
⊗
whereAisa1–formontheexternal5-dimensionalspacetimeM,whileU andV arescalarsonM,which
combinedparameterizethebreathingandthesquashingmodesofthecompactmanifold.
RegardingtheformfieldsoftypeIIBsupergravity,letusforinstanceconsidertheNSNS2-formB. We
takethegeneralexpansion
B = b +b (η+A)+bJJ +Re(bΩΩ), (6)
2 1
∧
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where b is a 2-form, b a 1-form, bJ a real scalar, and bΩ a complex scalar on M. The other type IIB
2 1
formfieldsareexpandedalongthesamelines. Finally,thedilatonφandtheRamond-RamondscalarC
0
areassumedtobeindependentoftheinternalcoordinates.
Thefactthatthesystemofdifferentialforms η, J, Ω isclosedunderthevariousoperationsappearing
{ }
inthehigher-dimensionalequationsofmotion(exteriorderivative,wedgeproduct,Hodgestar)ensuresthe
consistencyofthetruncation. Indeed,onecanplugthetruncationansatzinthe10-dimensionalequations
of motion, and check that they reduce to 5-dimensionalequations; in particular, the dependence on the
internalcoordinatesdropsout. Inparallel,onereducesthetypeIIBactionbyperformingtheintegralover
theinternalspace (careisneededinimplementingtheself-dualityconstraintontheRamond-Ramond5-
formF ,seediscussionin[5]).Thenoneverifiesthattheresulting5-dimensionalactionprovidesprecisely
5
the5-dimensionalequationsthatwereobtainedbyreducingthe10-dimensionalequations.Thisprovesthe
consistencyofthetruncation.
Asafinalremark,wenotethatanansatzexpressedintermsoftheinvariantforms η, J, Ω ofageneric
{ }
SU(2)structureonthecompactmanifoldwouldingeneralnotbeenoughforhavingaconsistenttruncation.
Indeed,whilethese formsaresingletsunderthestructuregroup,theirderivatives,providingtheintrinsic
torsionoftheSU(2)structure,wouldgenericallycontainnon-singletcontributions,andthiswouldapriori
spoilconsistency.Hencethesimplicityof(3),namelythefactthatonlysinglets(withconstantcoefficients)
appearontherighthandside,isasessentialas(2). BesidesthecaseofSasaki–Einsteinstructuresinodd
dimension,therearefurtherexamplesofgeometriescharacterizedbyaG-structurewhoseintrinsictorsion
isalsoG-invariant,sothatatruncationansatzexpressedintermsoftheG-invarianttensorsisconsistent:
forinstance,the7-dimensionalweak-G manifolds[2],the6-dimensionalNearly-Ka¨hlermanifolds[12],
2
aswellasthespecialholonomymanifolds,whoseG-structurehasvanishingtorsion.
Asituationinwhichtheconditionsjustdescribedarefulfillediswhenthecompactmanifoldisacoset
space / .Inthiscase,thestructuregroupcanbeidentifiedwith ,andtheansatzbasedonthesingletsof
G H H
thestructuregroupcanberephrasedasanansatzinvariantundertheleft-actionof . Consistenttruncations
G
of supergravityon coset spaces have been studied in [9, 13]. In [13] (see also [14]), a specific example
of5-dimensionalSasaki–Einsteinmanifoldwasconsidered,namelytheT1,1 = (SU(2) SU(2))/U(1)
×
coset space (also known as the base of the conifold). The consistent truncation outlined above can in
thiscase besubstantiallyenhanced,andprovidesthesupersymmetriccompletionofamorelimitednon-
supersymmetrictruncationonT1,1[15],containingseveralphysicallyrelevantconifoldsolutions,suchas
theKlebanov–StrasslerandtheMaldacena–Nun˜ezones[16].
3 Gauged = 4 supergravitydescription
N
The5-dimensionalmodelstemmingfromtheproceduredescribedabovecanbeunderstoodintheframe-
work of gauged = 4 supergravity[17, 18]. In this section we present just its most relevantfeatures,
referringto[5]foNrtheexpressionofthecompletebosonicaction.1 Actually,thankstotheconstraintsdic-
tatedbyhalf-maximalsupersymmetry,inordertocompletelydescribethemodeloneneedstospecifyjust
a few data, namelythe numberof vectormultipletsand the embeddingtensor describinghow the gauge
groupisembeddedintotheglobalsymmetrygroup.Letusdiscusstheminturn.
Theexpectationofhaving =4supersymmetryisfirstofallmotivatedbythegravitinoansatz. Type
N
IIB supergravitycontains two Majorana–Weylgravitini of the same chirality Ψα , where α = 1,2, and
M
M is a 10-dimensionalspacetime index. To define the truncation ansatz for these fields, we exploitthe
factthattheSU(2)structureconditionimpliestheexistenceoftwogloballydefinedspinorsζ1,ζ2 onthe
internalmanifold,beingonethechargeconjugateoftheother. Thesearerelatedtotheformsη, J, Ωvia
appropriatespinorbilinears. Weusethetwospinorstoexpandthe5-dimensionalspacetimecomponents
ofthe10-dimensionalgravitinias
Ψα = ψα1 ζ1+ψα2 ζ2. (7)
µ µ ⊗ µ ⊗
1 Adetailedstudyoftheinclusionofthefermionicsectorhasbeenperformedin[19].
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4 D.Cassani,G.Dall’Agata,andA.F.Faedo:Consistenttruncationswithmassivemodesandholography
The5-dimensionalfieldsψα1, ψα2 canthenbecombinedintofourgravitiniψi,i = 1,...,4,satisfying
µ µ µ
thesymplectic-Majoranacondition
ψ (ψi)†γ0 =Ω (ψj)TC, (8)
µi ≡ µ ij µ
where Ω is the USp(4) invariant form while C is the charge conjugation matrix. This provides the
ij
gravitinocontentof =4supergravity.
N
Turningtothe5-dimensionalbosonicsector,onecanverifythatitorganizesin =4multiplets,with
N
allthecouplingsrespectingsupersymmetry.Forinstance,byusingasolvableparameterizationoneverifies
thatthetargetmanifoldofthescalarsigma-modelistheprescribedhomogeneousspace
SO(5,n)
=SO(1,1) , (9)
scal
M × SO(5) SO(n)
×
with n = 2, which correspondsto a model with two vector multiplets. The countingand the couplings
of the vector fields agree with this, though one has to take into account some complications due to the
gauging. Indeed,whileungauged = 4supergravityin5dimensionscontainseightvectorfields,inour
N
modelwefindfourvectorsandfour2-forms. ThelatterareseenasthePoincare´dualsofthemissingfour
vectors,thedualizationbeingrequiredbythegaugingathand.
In order to fully specify the gauging, one computes the embedding tensor mapping the gauge group
generatorsintothegeneratorsofthedualitygroup,whichinthepresentcaseisSO(1,1) SO(5,2). This
×
determinesthevariousadditionalcouplingsinthelagrangianwithrespecttotheungaugedcase,including
the scalar potential, as well as the fermionicshifts appearingin the supersymmetrytransformations(for
the embedding tensor formalism we refer to [20] and references therein). In our case, the embedding
tensorhascomponentsf =f andξ =ξ ,wheretheindicesM,N,P =1,...7runin
MNP [MNP] MN [MN]
thefundamentalofSO(5,2). Thesecanbedeterminedbystudyingthe gauge-covariantderivativeofthe
scalars,andwefind
f =f =f = f = 2,
125 256 567 157
− −
ξ = 3√2, ξ =ξ = ξ =ξ = √2k. (10)
34 12 17 26 67
− − −
The higher-dimensional origin of the f-components is found in the geometric flux associated with the
non-closure of the form η on the internal manifold, while ξ can be traced back to the non-closure of
34
Ω. The remainingnon-zeroξ-componentsare proportionalto the constantk parameterizingthe internal
Ramond-Ramond5-formflux,Fflux =kJ J η.
5 ∧ ∧
By studyingthe commutationrelationsof the generatorsidentified by the embeddingtensor, one can
inferthatthegaugegroupis givenby theproductof U(1)with the three-dimensionalHeisenberggroup.
The10-dimensionaloriginofthisgaugesymmetryisfoundinpartinthereparameterizationinvarianceof
thespacetimeandinpartintheshiftsymmetryofthetypeIIBformfields.
Finally,wenoticethattakingthelimitofvanishingfluxes(dη = dΩ = k = 0),weobtainaconsistent
truncationoftypeIIBsupergravityonK3 S1 toungauged = 4supergravitycoupledtotwo vector
× N
multiplets.
4 The holographicpicture
Inthissectionweputourconsistenttruncationintheperspectiveofthegauge/gravitycorrespondence.
Westartfromthe5-dimensionalscalarpotential,whichreads
= 12e−134U−23V +2e−230U+34V + 9e−230U−83V−φ bΩ 2
V − 2 | |
(11)
+9e−230U−83V+φ cΩ C0bΩ 2+ e−332U−83V(cid:2)3Im(cid:0)bΩcΩ(cid:1)+k(cid:3)2,
2 | − |
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wherecΩ is a complexscalar arisingfromthe Ramond-Ramond2-formpotential. Studying onefinds
V
twoextrema.ChoosingtheRRfluxk =2,theseare
U = V = bΩ = cΩ = 0, witharbitraryφ, C , (12)
0
and
2 eiθ+φ/2
e4U =e−4V = , bΩ = , cΩ =bΩτ, τ (C +ie−φ), (13)
0
3 √3 ≡
where φ, C and θ are moduliof the solution. The cosmologicalconstant Λ is negativein both
0
≡ hVi
cases(Λ = 6forthefirstandΛ = 27 forthesecond),hencewehaveAnti-deSittervacua. Thefirst
− − 4
extremumhas = 2supersymmetry,andliftstothestandardAdS Sasaki–Einstein solutionoftype
5 5
N ×
IIBsupergravity. Thesecondextremumisnon-supersymmetric,andcorrespondsto anAdS solutionof
5
typeIIBsupergravitywithsquashedinternalmetric,originallyfoundin[21].
Bystudyingthemassspectrumofthefieldfluctuationsaboutthesebackgrounds,onecanimplementthe
standardAdS/CFT dictionaryand deducethe anomalousdimensionsof the dualCFT operators. Forthe
non-supersymmetricextremumwefindsomeirrationalconformaldimensions;sincethesquaremassesare
allpositive,thevacuumisstableatleastwithrespecttothemodeskeptinthetruncation. Theresultsfor
thesupersymmetricvacuumaresummarizedintable1. Bystudyingthedualspectrum(forinstanceinthe
well-knowncasesofS5andT1,1),weseethatwearekeepingjustflavorsinglets,whicharebuiltinterms
of the gaugesuperfieldW in the = 1 super Yang-Millstheory. Since this is a fermionicsuperfield,
α
N
wecanconstructjustafinitenumberofnon-vanishingcombinations,whichpreciselymatchthedegreesof
freedomofthegravitymodel.WeconcludethatourtruncationdescribesthelargeN limitoftheuniversal
gaugesectorof =1superYang–Millstheoriesin4dimensions.
N
Asnoticedin[8],onthefieldtheorysidetheconsistencyofourtruncationtranslatesintothefactthat
thesetofoperatorsappearingintable1isclosedundertheoperatorproductexpansion(atleastinthelarge
N limit). Itwouldbeinterestingtoexploretowhatextentthisisageneralfeatureofconsistenttruncations
withafieldtheorydual.
Acknowledgements TheauthorsaresupportedinpartbytheERCAdvancedGrantno. 226455,“Supersymmetry,
Quantum Gravity and Gauge Fields” (SUPERFIELDS) and by the Fondazione Cariparo Excellence Grant “String-
derivedsupergravitieswithbranesandfluxesandtheirphenomenologicalimplications”.
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6 D.Cassani,G.Dall’Agata,andA.F.Faedo:Consistenttruncationswithmassivemodesandholography
=2multiplet fieldfluctuations m2 ∆ dualoperators
N
A 2aJ 0 3
gravity − 1 Tr(W W )+...
α α˙
g 0 4
µν
bΩ icΩ 3 3
universalhyper − − Tr(W2)+...
φ, C 0 4
0
b , c 8 5
1 1
massivegravitino aΩ 9 5 Tr(W2W )+...
2 α˙
b , c 16 6
2 2
U V 12 6
−
A+aJ 24 7
massivevector 1 Tr(W2W2)+...
bΩ+icΩ 21 7
4U +V 32 8
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