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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. 7 2 Copyrightlinewillbeprovidedbythepublisher ] 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 Copyrightlinewillbeprovidedbythepublisher 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 ∧ Copyrightlinewillbeprovidedbythepublisher fdpheaderwillbeprovidedbythepublisher 3 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]. Copyrightlinewillbeprovidedbythepublisher 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 | − | Copyrightlinewillbeprovidedbythepublisher fdpheaderwillbeprovidedbythepublisher 5 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”. References [1] J.Maldacena,D.MartelliandY.Tachikawa,Commentsonstringtheorybackgroundswithnon-relativisticcon- formalsymmetry,JHEP0810(2008)072[arXiv:0807.1100[hep-th]]. [2] J.P.Gauntlett, S.Kim, O.VarelaandD.Waldram, Consistentsupersymmetric Kaluza–Kleintruncationswith massivemodes,JHEP0904(2009)102[arXiv:0901.0676[hep-th]]. [3] S.S.Gubser,C.P.Herzog,S.S.PufuandT.Tesileanu,SuperconductorsfromSuperstrings,Phys.Rev.Lett.103 (2009)141601[arXiv:0907.3510[hep-th]]. [4] J. P. Gauntlett, J. Sonner and T. Wiseman, Holographic superconductivity in M-Theory, Phys. Rev. Lett. 103 (2009)151601[arXiv:0907.3796[hep-th]]. [5] D.Cassani,G.Dall’AgataandA.F.Faedo,TypeIIBsupergravityonsquashedSasaki-Einsteinmanifolds,JHEP 1005(2010)094[arXiv:1003.4283[hep-th]]. [6] J.T. Liu, P.Szepietowski andZ.Zhao, Consistent massivetruncations of IIBsupergravity onSasaki-Einstein manifolds,Phys.Rev.D81,124028(2010)[arXiv:1003.5374[hep-th]]. [7] J.P.GauntlettandO.Varela,UniversalKaluza-KleinreductionsoftypeIIBtoN=4supergravityinfivedimen- sions,JHEP1006,081(2010)[arXiv:1003.5642[hep-th]]. [8] K.Skenderis, M.TaylorandD.Tsimpis,AconsistenttruncationofIIBsupergravityonmanifoldsadmittinga Sasaki-Einsteinstructure,JHEP1006(2010)025[arXiv:1003.5657[hep-th]]. [9] D.CassaniandA.K.Kashani-Poor,ExploitingN=2inconsistentcosetreductionsoftypeIIA,Nucl.Phys.B817 (2009)25[arXiv:0901.4251[hep-th]]. Copyrightlinewillbeprovidedbythepublisher 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 Table1 Mass eigenstates of the type IIB fieldsin our truncation on the supersymmetric AdS5×Sasaki–Einstein5 background, andtheirdualsuperfieldoperators. SincethevacuumhasN = 2supersymmetry,thefieldfluctuations organizeinN = 2multiplets. Weprovidethemasseigenstatesenteringineachmultiplet,togetherwiththeirmass eigenvaluesm2,theconformaldimension∆ofthecorrespondingdualoperators,andthedualsuperfieldsaccommo- datingthesingleoperators. Themasseigenvalues areevaluated choosing thefluxk = 2, whichyieldsaunit AdS radius.Thevectorsb1, c1, A+aJ1 acquireamassviaaStu¨ckelbergmechanism(thenotationistheoneof[5]). [10] A.BuchelandJ.T.Liu,GaugedsupergravityfromtypeIIBstringtheoryonY(p,q)manifolds,Nucl.Phys.B771 (2007)93[arXiv:hep-th/0608002]. [11] J.P.GauntlettandO.Varela,ConsistentKaluza-KleinReductionsforGeneral SupersymmetricAdSSolutions, Phys.Rev.D76(2007)126007[arXiv:0707.2315[hep-th]]. [12] A.K.Kashani-Poor,NearlyKaehlerReduction,JHEP0711(2007)026[arXiv:0709.4482[hep-th]]. [13] D.CassaniandA.F.Faedo,Asupersymmetricconsistenttruncationforconifoldsolutions, Nucl.Phys.B843 (2011)455[arXiv:1008.0883[hep-th]]. [14] I.Bena,G.Giecold,M.Grana,N.HalmagyiandF.Orsi,SupersymmetricConsistentTruncationsofIIBonT(1,1), arXiv:1008.0983[hep-th]. [15] G.Papadopoulos,A.A.Tseytlin,Complexgeometryofconifoldsandfive-branewrappedontwosphere,Class. Quant.Grav.18 (2001)1333-1354.[hep-th/0012034];M.Berg,M.Haack,W.Mueck,Bulkdynamicsinconfin- inggaugetheories,Nucl.Phys.B736 (2006)82-132.[hep-th/0507285]. [16] I. R. Klebanov and M. J. Strassler, Supergravity and a confining gauge theory: Duality cascades and chiSB- resolutionofnakedsingularities,JHEP0008(2000)052[arXiv:hep-th/0007191];J.M.MaldacenaandC.Nunez, TowardsthelargeNlimitofpureN=1superYangMills,Phys.Rev.Lett.86,588(2001)[arXiv:hep-th/0008001]. [17] G. Dall’Agata, C. Herrmann and M. Zagermann, General matter coupled N = 4 gauged supergravity in five dimensions,Nucl.Phys.B612(2001)123[arXiv:hep-th/0103106]. [18] J.Scho¨nandM.Weidner,GaugedN=4supergravities,JHEP0605(2006)034[arXiv:hep-th/0602024]. [19] I. Bah, A. Faraggi, J. I. Jottar and R. G. Leigh, Fermions and Type IIB Supergravity On Squashed Sasaki- EinsteinManifolds,arXiv:1009.1615[hep-th].J.T.Liu,P.SzepietowskiandZ.Zhao,Supersymmetricmassive truncationsofIIbsupergravityonSasaki-Einsteinmanifolds,Phys.Rev.D82(2010)124022[arXiv:1009.4210 [hep-th]]. [20] H. Samtleben, Lectures on Gauged Supergravity and Flux Compactifications, Class. Quant. Grav. 25 (2008) 214002[arXiv:0808.4076[hep-th]]. [21] L.J.Romans,NewCompactificationsOfChiralN=2D=10Supergravity,Phys.Lett.B153(1985)392. Copyrightlinewillbeprovidedbythepublisher