Compactifications with S-Duality Twists

We consider generalised Scherk Schwarz reductions of supergravity and superstring theories with twists by electromagnetic dualities that are symmetries of the equations of motion but not of the action, such as the S-duality of D=4, N=4 super-Yang-Mills cou

February1,2008

QMUL-PH-03-09

hep-th/0308133

Compacti cationswithS-DualityTwists

arXiv:hep-th/0308133v2 15 Sep 2003ChristopherM.Hull1,2¨andAybikeC¸atal-OzerDepartmentofPhysicsQueenMary,UniversityofLondonMileEndRd,LondonE14NS,UKDepartmentofPhysicsMiddleEastTechnicalUniversity˙on¨In¨uBulvar Yolu,06531Ankara,TurkeyAbstractWeconsidergeneralisedScherkSchwarzreductionsofsupergravityandsuperstringtheorieswithtwistsbyelectromagneticdualitiesthataresymmetriesoftheequationsofmotionbutnotoftheaction,suchastheS-dualityofD=4,N=4super-Yang-Millscoupledtosupergravity.Thereductioncannotbedoneontheactionitself,butmustbedoneeitheronthe eldequationsoronadualityinvariantformoftheaction,suchasonein

thedoubledformalisminwhichpotentialsareintroducedforbothelectricandmagnetic elds.Theresultingtheoryinodd-dimensionshasmassiveform eldssatisfyingaself-dualityconditiondA～m A.WeconstructsuchtheoriesinD=3,5,7.21

We consider generalised Scherk Schwarz reductions of supergravity and superstring theories with twists by electromagnetic dualities that are symmetries of the equations of motion but not of the action, such as the S-duality of D=4, N=4 super-Yang-Mills cou

1Introduction

Twistedtoroidalcompacti cationsorScherk-Schwarzreductionsareausefulwayofintroducingmassesintosupergravityandstringcompacti cations,generatingapotentialforthescalar elds

[1-19].AtheoryinD+1dimensionswithglobalsymmetryGcanbecompacti edonacirclewith eldsnotperiodicbutwithaGmonodromyaroundthecircle,andthemonodromyintroducesmassesintothetheoryandbreakssomeofthesymmetry.Thepurposehereistogeneralisesuchcompacti cationstothecaseinwhichGisasymmetryoftheequationsofmotiononly,notoftheaction;weshallrefertosuchsymmetrieshereasS-dualities.AstandardexampleisS-dualityin4-dimensions.Theheteroticstringcompacti edtofourdimensionshasaclassicalSL(2,R)symmetrywhichactsthroughelectromagneticdualitytransformationsandsoisonlyasymmetryoftheequationsofmotion.Inthiscase,weconsideracirclereductiontothreedimensionswithamonodromyinSL(2,R).Inthequantumtheory,theSL(2,R)symmetryisbrokentoSL(2,Z)[20]andinthatcasethemonodromymustbeinSL(2,Z)[6].Wegeneralisethistootherdimensions,anddiscussexamplesinD=3,5and7dimensions.

ConsideraD+1dimensionalsupergravitywithaglobalsymmetryG.Anelementgofthesymmetrygroupactsonageneric eldψasψ→g[ψ].ConsidernowadimensionalreductionofthetheorytoDdimensionsonacircleofradiusRwithaperiodiccoordinatey～y+1.Inthetwistedreduction,the eldsarenotindependentoftheinternalcoordinatebutarechosentohaveaspeci cdependenceonthecirclecoordinateythroughtheansatz

ψ(xµ,y)=g(y)[ψ(xµ)](1.1)

forsomey-dependentgroupelementg(y)[6].Animportantrestrictionong(y)isthatthereducedtheoryinDdimensionsshouldbeindependentofy.Thisisachievedbychoosing

g(y)=exp(My)(1.2)

forsomeLie-algebraelementM.Themapg(y)isnotperiodicaroundthecircle,buthasamonodromy

M(g)=expM(1.3)

iManysupergravitytheoriesinD+1=2ndimensionshaveasetofnform eldstrengthsHn

wherei=1,...,rlabelsthepotentials,whichtypicallysatisfyageneralisedself-dualityequationoftheform

ijHn=Qij(φ) Hn(1.4)

whereQijisamatrixdependingonthescalar eldsφand istheHodgedualinD+1dimensions[21].Foranyn,consistencyrequiresthat(Qij(φ) )2=1,sothatif( )2= 1,asin

We consider generalised Scherk Schwarz reductions of supergravity and superstring theories with twists by electromagnetic dualities that are symmetries of the equations of motion but not of the action, such as the S-duality of D=4, N=4 super-Yang-Mills cou

Lorentzianspaceofdimension4m,thenQ2=

andQisaproductstructure.Inthe

itheorieswewillconsider,theHntransforminanr-dimensionalrepresentationofarigidduality

groupG.Ind=4,N=8supergravity,therearer=562-form eldstrengthstransformingasa56ofthedualitygroupG=E7[22,23].Thesesplitinto28 eldstrengthsF=dAand

= F+...,withQacomplexstructureonR56.Ind=6,N=828dual eldstrengthsF

supergravity,thereare53-form eldstrengthswhichsplitinto5self-dualonesand5anti-selfdualones,andthese10transformasa10ofG=SO(5,5)[24].The103-form eldstrengths iwithi=1,...,10,satisfy(anti)self-dualityconstraintsoftheform(1.4)withQrelatedtoHn

theSO(5,5)-invariantmetric.Ind=8maximalsupergravity,thereisa3-formpotential,andits eldstrengthanditsdualcombineintoanSL(2,R)doublet,satisfyingaconstraintoftheform(1.4)withQ=iσ2.

OurmaininteresthereisinreductionsinwhichthemonodromyM∈Gisasymmetryof

iviatransformationstheequationsofmotionbutnottheaction,actingonthe eldstrengthsHn

involvingHodgeorelectromagneticdualities,sothattheycannotberealisedlocallyonthefundamentaln 1formpotentials.We ndthat(inthecaseinwhichMisinvertible)the eld

isatisfyingtheconstraint(1.4)giverisetorn 1formpotentialsAiin2n 1strengthsHnn 1

dimensionssatisfyingmassiveself-dualityconstraintsoftheform

An 1DAn 1=M(1.5)

whereDisagauge-covariantexteriorderivative, isnowtheHodgedualinDdimensions

∝QM.Suchodd-dimensionalself-dualityconditionswere rstconsideredandthematrixM

in[26]andoftenoccurinodd-dimensionalgaugedsupergravitytheories,andfollowfromaChern-Simonsactionwithmasstermoftheform

ijAi∧ AjL=PijAi∧DAj+M(1.6)

=PM andPijisasuitablychosenconstantmatrix.InthegeneralcaseinwhichMwhereM

isnotinvertible,someofthegauge eldsremainmassless.

Indimensionallyreducingatheorywithatwistthatisasymmetryoftheequationsofmotionandnotoftheaction,oneneedstoreducetheequationsofmotion,nottheaction.However,forthecasesofinterestherethereisadoubledformalism[21]inwhichdualpotentials n 1areintroducedforeachn 1formpotentialAn 1,inwhichthedualitysymmetrybecomesA

],whichissupplementedbyaduality-invariantconstraintthatasymmetryoftheactionS[A,A

intermsofA.ThisdoubledactionandconstraintcanthenbecouldbeusedtoeliminateA

dimensionallyreducedinthestandardwaywithatwistbythedualitysymmetry.Thisgreatlysimpli esthecalculations.

We consider generalised Scherk Schwarz reductions of supergravity and superstring theories with twists by electromagnetic dualities that are symmetries of the equations of motion but not of the action, such as the S-duality of D=4, N=4 super-Yang-Mills cou

Weapplytheseresultstothereductionofsupergravitytheoriesin4,6,8dimensions,givingrisetosupergravitytheoriesin3,5,7dimensionswithmassiveself-dualforms.Thisconstructsnewsupergravitytheoriesinthesedimensionsandgivesahigher-dimensionaloriginfortheoriesin3,5,7dimensionswithChern-Simonsactions.Inparticular,forD=3,Aisavector eldandthisgivesahigherdimensionaloriginfor3-dimensionalgaugedsupergravitytheories,ofthetypediscussedin[27]withChern-Simonsactionsforsomeofthegauge elds.Theplanofthepaperisasfollows.Insection2wereviewtheScherk-Schwarzmechanism,givingtheresultsforthetwistedreductionofgravitycoupledtoscalarsandgaugepotentials,whichareusedinlatersections.Wegiveadetailedanalysisofthegeneralcaseinwhichthemassmatrixisnotinvertible.Insection3wereviewthedoubledformalismof[21].Insection4weperformatwisteddimensionalreductioninthedoubledformalism,andhenceobtainthelagrangianfordimensionalreductionswithS-dualitytwists.Finally,insection5,weapplyourresultstothereductionofsupergravitytheoriesin4,6,8dimensions.

2ScherkSchwarzReduction

WewillconsiderhereScherk-SchwarzdimensionalreductiononacirclefromD+1toDdimen-sions,withatwistbyanelementofaglobalsymmetryG.Theansatzfordimensionalreductionofageneric eldis(1.1)withy-dependencegivenby(1.2)withmonodromyMgivenby(1.3)intermsofthemass-matrixM.ThemassmatrixMintroducesmassparametersintothetheory,and eldsinnon-trivialrepresentationsofthegroupGtypicallybecomemassivewithmassesgivenintermsofM,orare“eaten”bygauge eldsthatbecomemassiveinageneralisedHiggsmechanism.Inparticular,thescalar eldswillobtainascalarpotentialgivenintermsofM.However,di erentmass-matricescangiveequivalenttheories,andanimportantquestionishowtoclassifytheinequivalenttheories.In[14]itwasshownthatthetheoriesaredeterminedbythemonodromyM,notthemassmatrixM.Tworeductionswithdi erentmassmatrices

′M,M′butthesamemonodromyM=eM=eMgivethesamereducedtheory,providedthe

fullspectrumofmassivestatesiskept,andnotruncationismade.In[6],itwasshownthattheorieswithmonodromiesinthesameGconjugacyclassareequivalent,sothatthetheoriesareclassi edbytheGconjugacyclasses.Inquantumstringtheory,aglobalgroupoftheclas-sicaltheorytypicallybecomesadiscretegaugesymmetryG(Z)[28]andforsuchtheoriesthemonodromymustbeinG(Z),givingquantizationconditionsonthemassparameters,andthedistincttheoriesaredeterminedbythemonodromyM∈G(Z)uptoG(Z)conjugation.ThemassmatrixMgeneratesaonedimensionalsubgroupLofG,whichbecomesagaugesymmetryofthereducedtheory,sothatsuchareductionofasupergravitygivesagaugedsupergravity

[7,8,9,14].

We consider generalised Scherk Schwarz reductions of supergravity and superstring theories with twists by electromagnetic dualities that are symmetries of the equations of motion but not of the action, such as the S-duality of D=4, N=4 super-Yang-Mills cou

Ourmaininterestwillbeinthereductionofsupergravityandsuperstringtheories.ExtendedsupergravitytheoriestypicallyhaveaglobalsymmetryGandthescalarstakevaluesinthecosetspaceG/HwhereGisanon-compactgroupandHisthemaximalcompactsubgroupofG.ThescalarsectorofthetheoryistheninvariantunderthegroupGandthissymmetrytypicallyextendstothefulltheoryforsupergravitiesinodddimensions.Insomeevendimensionaltheories,thesymmetryGextendstoasymmetryoftheequationsofmotiononly,actingthroughdualitytransformationsexchanging eldequationswithBianchiidentitites.

ThetheorycanbeformulatedwithalocalHsymmetryaswellasaglobalGsymmetry.ThescalarsinthecosetspaceG/HcanberepresentedbyavielbeinV(x)∈GwhichtransformsunderglobalGandlocalHtransformationsas

V→h(x)V,V→Vg,

L= 1Thelagrangianish(x)∈Hg∈G(2.7)

sothatδabisaninvariant,butthegeneralisationtoother

representationsisstraightforward.

AnalternativeformulationthatdoesnotinvolveextrascalarsistouseametricKonG/Hinsteadofavielbein,transformingas(forarealrepresentationofG)

K→gTKg(2.9)

SuchametriccanbeconstructedfromthevielbeinasKij=δabVaiVbj,whereiandaarethecurvedand atindicesrespectively.KisinvariantunderlocalHtransformationsashTh=

,wewouldusethehermitianmetric

K=V VtransformingasK→g Kg.)ThelagrangiancanbewrittenintermsofKas

L=1

We consider generalised Scherk Schwarz reductions of supergravity and superstring theories with twists by electromagnetic dualities that are symmetries of the equations of motion but not of the action, such as the S-duality of D=4, N=4 super-Yang-Mills cou

representationofGandconsiderthetheoryinD+1dimensionsandworkwiththemetricKij.Thelagrangianis1TL=R 1+HnK∧ Hn(2.11)2

TheactionisinvariantundertherigidGsymmetry

δA→L 1A,δK→LTKL(2.12)

whereLijisaG-transformationintherrepresentation,andthespacetimemetricisinvariant.Inlatersections,wewillbeparticularlyinterestedinthecaseinwhichD+1=2n,butfornowwewillkeepD,narbitrary.

Forexample,inthecaseG=SL(2,R),H=SO(2),therearetwoscalarsinthetheory,whichwewilldenoteφandχ,whichparametrisethescalarcosetSL(2,R)/SO(2).ThematrixV(inthedoubletrepresentationofSL(2,R))isageneralSL(2,R)matrix,whichcanbegiven,intermsofφandχandanon-physicalscalarθthatparameterisestheSO(2)subgroup,by

V=heφ/2

wherehisanSO(2)matrix

h=

Then

K=eφ e0 χ1 φ (2.13)cosθsinθ sinθcosθe 2φ(2.14)

andthelagrangian(2.11)canbewrittenas

L=R 1 1

2e2φdχ∧ dχ 1+χ χ2 χ1.(2.15)

2eφH2∧ H2 χeφH1∧ H2(2.16)

andisindependentofθ.

Wenowreducethelagrangian(2.11)onacirclewithatwistgivenbyamonodromyM=eM∈Gwiththeansatz(1.1).Fortheremainderofthissection,wedistinguishtheD+1-dimensional eldsfromD-dimensionalonesbyahat.ThemetricisinvariantundertheglobalsymmetrygroupsoweusethestandardKaluza-Kleinansatz

ds 2=e2α ds2+e2β (dy+A)2

sothattheEinstein-Hilberttermin(2.11)reducesto

Lg=R 1 1

2

F∧ F.(2.18)(2.17)

We consider generalised Scherk Schwarz reductions of supergravity and superstring theories with twists by electromagnetic dualities that are symmetries of the equations of motion but not of the action, such as the S-duality of D=4, N=4 super-Yang-Mills cou

Here isthescalar eldcomingfromthereductionofthemetric,F=dAandAisthegraviphoton.TheconstantsαandβdependonDandare:

α2=1

4

scalarpotential

[1]: ∧ dK 1)fromD+1dimensionstoDdimensionsgivesascalarkinetictermplusatr(dK

Ls=1

2

Theansatz(2.21)impliese2(D 1)α tr(M2+MK 1MTK) 1(2.24)

n(x,y)=e MyHn(x)+e MyHn 1(x)∧(dy+A)H

n=dA n 1.HeretheD-dimensional eldstrengthsareforthen-form eldstrengthsH

Hn 1(x)=dAn 2 ( 1)n 1MAn 1,

Reductionofthekinetictermgives

TK ∧ n→[e 2(n 1)α HTK∧ Hn+e2(D n)α HTK∧ Hn 1]∧dy HHnnn 1

CollectingtheresultswecannowwritedowntheD-dimensionallagrangianas:

LD=Lg+Lb+LsHn(x)=dAn 1 Hn 1∧A.(2.25)(2.26)(2.27)(2.28)

We consider generalised Scherk Schwarz reductions of supergravity and superstring theories with twists by electromagnetic dualities that are symmetries of the equations of motion but not of the action, such as the S-duality of D=4, N=4 super-Yang-Mills cou

where

Lg=R 1

Ls=

and

Lb= 11221e 2(D 1)α F2∧ F2e2(D 1)α tr(M2+MK 1MTK) 1Te2(D n)α Hn 1K∧ Hn 1(2.29)2

The eldstrengths(2.26)areinvariantunderthefollowinggaugetransformations:

δAn 1=dΛ,δAn 2=( 1)n 1MΛ.(2.30)(2.31)

IfMisinvertible,thesecanbeusedtogaugeAn 2tozerobyperformingthegaugetransfor-mation:

An 1→An 1+( 1)n 1M 1dAn 2.

InthisgaugetheD-dimensional eldstrengthsbecome

Hn=DAn 1=dAn 1 ( 1)nMAn 1∧A

Hn 1=( 1)nMAn 1.(2.33)(2.34)(2.32)

ThenAn 2disappearsfromthetheory,andthetermHn 1∧ Hn 1isamasstermforAn 1.Thedegreesoffreedomrepresentedbyther eldsAn 2havebeenabsorbedbyther(n 1)-form eldsAn 1whichhavebecomemassive.NowHn=DAn 1isagaugecovariantderivativewherethegaugegroupisthesubgroupofGgeneratedbyMandthecorrespondinggauge eldisthegraviphotonA.

NowwewillanalyzethecaseMisnotinvertible.Itisusefultoworkwith atindicesHa=VaiHi,Aa=VaiAi.ThenHa=DAa=dAa+ωabAbwhereωistheconnection1-formωab=Vai(dV 1)ib.ThegroupsGarisinginthesupergravitytheoriesofinteresthereallhaveaG-invariantmatrix whichissymmetricifnisoddandanti-symmetricifniseven

ab=( 1)n 1 ba.

¯a=( 1)abHbandMab=Mac cb.NowonehasUsingthis,weintroduceH

aan 1¯(n 1)b,HnMabA 1=DAn 2 ( 1)(2.35)¯(n)a=DA¯(n 1)a H¯(n 1)a∧A=D A¯(n 1)a(2.36)H

isthecovariantderivativewithconnectionsωabandA.whereDNotethatM=eMandMT 1M= 1sinceM∈Gand isG-invariant.(Forcomplexrepresentations,theconditionisM 1M= 1.)AsaresultthemassmatrixMabsatis es:

MT 1+ 1M=0.

(2.37)

We consider generalised Scherk Schwarz reductions of supergravity and superstring theories with twists by electromagnetic dualities that are symmetries of the equations of motion but not of the action, such as the S-duality of D=4, N=4 super-Yang-Mills cou

From(2.35)and(2.37)itfollowsthatMabisasymmetricmatrixifnisevenandantisymmetricifnisodd:

Mab=( 1)nMba.

(2.38)Letthedimensionofker(M)bel.NowthematrixMabcanbebroughtintothecanonicalform

Mab=

′′00′′0mαβ(2.39)wheremαβisaninvertible(r l)×(r l)matrixwhichisdiagonalifnisevenandskew-diagonalifnisodd.Herewehavesplittheindicesa→(α,α′)whereαrunsfrom1tolandα′runsfroml+1tor.Similarlythegauge eldsAcanbewrittenintheblockform

A=

Performingthegaugetransformation

′¯(n 1)α′+( 1)n 1(m 1)α′β′DAβ¯(n 1)α′→AAn 2 A′Aαα (2.40)(2.41)

¯(n 1)α′becomemassive,havingeatenther l eldsAα′,whileoneseesthatther l eldsAn 2¯Aαn 2andA(n 1)αbothremaininthetheoryasmasslessgauge elds,withlofeach.The eld

strengthsforthe(n 2)-form eldsin(2.36)become

α′nα′β′¯Hn=( 1)mA(n 1)β′, 1ααHn 1=DAn 2(2.42)

andhencetheterm(2.30)canbewrittenas

Lb=

112′′¯(n)β′¯(n)α′∧ He 2(n 1)α δαβH(2.43)

2′′¯(n 1)β′.¯(n 1)α′∧ Ae2(D n)α (mTm)αβA

Wehavechosenthenormalisationof sothat ac bdδcd=δab.

Thegaugegroup,thecouplingsandthescalarpotentialoftheD-dimensionaltheoryfoundabovearegivenexplicitlyintermsofthemassmatrixM,andtwotheoriesaredistinctifthemonodromiesareindistinctG-conjugacyclasses.ForthecaseG=SL(2,R)therearethreeconjugacyclasses,thehyperbolic,ellipticandparabolicconjugacyclassesandsotherearethreedistinctreductions[6].Thehyperbolic,ellipticandparabolicmonodromymatricesandmassmatricescanbetakentobe:

Mh= e

0m0e m

,Me=

cosmsinm

sinmcosm ,Mp=

Mp= 1m01 .(2.44)(2.45)Mh=m0

0 m,Me=0m

m0

,0m00.

We consider generalised Scherk Schwarz reductions of supergravity and superstring theories with twists by electromagnetic dualities that are symmetries of the equations of motion but not of the action, such as the S-duality of D=4, N=4 super-Yang-Mills cou

Themassmatrixgeneratesaone-parametersubgroupofSL(2,IR)andthissubgroupwillbethegaugegroupinthelowerdimensionaltheory.Thuscompacti cationwithMewillgiveacompactgaugingSO(2)whereascompacti cationwithMhandMpwillgiverisetoSO(1,1)-gaugedlowerdimensionaltheories[7].(Notethatinthespecialcaseinwhichn=3,therewillbeextravectorgauge eldsinDdimensionsfromthereductionofthe2-formgauge elds,andstrictlyspeakingthegaugegroupisISO(2),ISO(1,1)ortheHeisenberggroupfortheelliptic,hyperbolicandparaboliccases,respectively[14].)

TheparabolicmassmatrixMpisnotinvertible,andhasaone-dimensionalkernel,i.e.

′r=2,l=1,sothatαandα′bothtakeonlyonevalueandAa=(A1,A1).Inthiscasethe

matrixmαβin(2.39)isthe1×1matrix( m)andfromthegaugetransformation(2.41)it

¯n 11′eatsthe(n 2)-formA1′andbecomesmassive.Thecanbeseenthatthe(n 1)-formAn 2¯n 11andn 2formA1gauge eldsremainmassless.remainingn 1formAn 2′′

3TheDoubledFormalism

TypicallyaD=2ndimensionalsupergravitytheoryhasaglobalsymmetrygroupGwhichcanberealisedatthelevelof eldequationsbutnottheaction,asGactsonn-form eldstrengthsH=dAthroughelectric-magneticdualitytransformations.InsuchcasesitispossibletoconstructamanifestlyG-invariantlagrangianthatdependsonthepotentialsAanddual

.Thedual eldsareregardedasindependent elds,butthe eldequationsarepotentialsA

todA,keepingsupplementedwithaG-covariantconstraintrelatingthen-form eldstrengthsdA

thenumberofindependentdegreesoffreedomcorrect.Thenewlagrangianisequivalenttotheoriginaloneasthetwoyieldequivalent eldequationswhentheconstraintistakenintoaccount.Inthissectionwewillreviewthisformalism,whichwasintroducedin[21]whereitwascalledthe‘doubledformalism’.Wewill rstconsiderthecaseG=SL(2,IR)andthengivethegeneralcaseinthefollowingsubsection.

3.1G=SL(2,IR)Case

Considerthefollowinglagrangianin2ndimensionswithneven

11e2φdχ∧ dχ χFn∧Fn(3.46)L= 22

HereFn=dAn 1andφandχarescalar elds.The eldequationsofthislagrangianhaveanSL(2,IR)S-dualityinvariance(forevenn)actingonFthroughelectromageneticdualitytransformations,aswenowdiscuss.

De ninganewn-formGby

Gn=δL

We consider generalised Scherk Schwarz reductions of supergravity and superstring theories with twists by electromagnetic dualities that are symmetries of the equations of motion but not of the action, such as the S-duality of D=4, N=4 super-Yang-Mills cou

thelagrangian(3.46)canbewrittenas

L=

strengthFnare

dFn=0

dGn=d( e φ Fn χFn)=0

whichcanbecombinedas

dHn=0

whereHnistheSL(2,IR)doublet

Hn= 12F∧G(3.48)whereKisasin(2.15).TheBianchiidentityandtheequationofmotionforthen-form eld(3.49)(3.50) Fn

Gn.(3.51)

The eldequationsaremanifestlySL(2,R)invariant,buttheF∧Gterminthelagrangian(3.48)isnotinvariant.However,aninvariantlagrangiancanbeconstructedasin[21]ifthe

nsothatGn=dA n,which eldequationdGn=0issolvedbyintroducingadualpotentialA

icanbecombinedwithAntoformanSL(2,R)doublet,with eldstrengthsHngivenby

Hn= dAn ndA .(3.52)

ThenthenaturalSL(2,IR)invariantlagrangianis

L′=1

4ijHnKij∧ Hn.(3.53)

whichisoftheformconsideredintheprevioussection.

n 1areindependent elds,sothatthenumberofn 1Forthisaction,bothAn 1andA

formdegreesoffreedomhasbeendoubled.Tohalvethemagain,forevennthisactioncanbesupplementedbytheSL(2,R)covariantconstraint[21]

ijHn=Jij Hn(3.54)

whereJistheSL(2,IR)matrix

Jij= ikKkj.

Here istheSL(2,IR)invariantmatrix

= (3.55)01

10

.(3.56)

We consider generalised Scherk Schwarz reductions of supergravity and superstring theories with twists by electromagnetic dualities that are symmetries of the equations of motion but not of the action, such as the S-duality of D=4, N=4 super-Yang-Mills cou

NotethatthematrixJin(3.55)satis esJ2=

RIJFnI∧ FnJ

2 1

δFI.

n

sothatthelagrangiancanbewrittenas

L=1(3.58)

We consider generalised Scherk Schwarz reductions of supergravity and superstring theories with twists by electromagnetic dualities that are symmetries of the equations of motion but not of the action, such as the S-duality of D=4, N=4 super-Yang-Mills cou

ij= ab(V 1)ia(V 1)jb,wherei,j=1,...,2kareindicesforthe2krepresentationofG,satisfying ij=( 1)n 1 ji.

IwithGI=dA ItoformAsbeforeweintroducepotential eldsAn 1nn 1

Hn= dAInI dAn (3.62)

transforminginthe2krepresentationofG.ThenthesystemcanbedescribedbytheG-invariantlagrangian1L′=

.(3.67)

sothattheconstraint(3.64)isconsistentasfor2n-dimensionalLorentzianspace-time Hn=( 1)n 1Hn.Itwasshownin[21]thatthe eldequationsfrom(3.63)areequivalenttothosefrom(3.57)togetherwiththeconstraint(3.64).

4ReductionwithDualityTwist

Thetheorywithlagrangian(3.57)hasaglobalsymmetryGoftheequationsofmotionwhichactsviadualitytransformations.InthissectionwewilldimensionallyreduceonacirclefromD+1=2ntoDdimensionswithatwistthathasmonodromyMinG.ForsomechoicesofmonodromyMinG,thisisinfactasymmetryoftheactionandthisisastandardScherk-Schwarzreduction,asinsection2.Ifitisonlyasymmetryoftheequationsofmotion,thenweusethedoubledformalismofsection3withlagrangian(3.63)supplementedbytheconstraint(3.64).Thelagrangian(3.63)isofthesameformas(2.11),sotheScherk-Schwarzreductionoftheactionproceedsasinsection2.Thisissupplementedbytheconstraintsarisingfromthe

We consider generalised Scherk Schwarz reductions of supergravity and superstring theories with twists by electromagnetic dualities that are symmetries of the equations of motion but not of the action, such as the S-duality of D=4, N=4 super-Yang-Mills cou

dimensionalreductionof(3.64).The eldequationsin2n 1dimensionsarethenthosefromthereducedactiontogetherwiththereducedconstraints,andwegoontoseekanactionin2n 1dimensionsthatgivesboththeconstraintsandthereduced eldequations.

4.1DimensionalReductionintheDoubledFormalism

Thelagrangian(3.63)inthedoubledformalismisofthesameformas(2.11),butwithanextrafactorof1/2inthenormalisationofthegauge eldkineticterm.TheScherk-Schwarzreductionofthelagrangian(2.11)wasalreadydiscussedinsection2,whereweshowedthatityieldsthelagrangian(2.28)inDdimensions.Itfollowsthatthereductionof(3.63)shouldgive(2.28)butnowwith(2.30)dividedbytwotogive:

Lb= 1

4Te2(D n)α Hn 1K∧ Hn 1(4.68)

Justasthelagrangian(3.63)shouldbesupplementedbytheD+1dimensionalconstraint(3.64)inordertogivethecorrectD+1dimensional eldequations,theDdimensionallagrangian(2.28)with(2.29),(4.68)shouldbesupplementedbytheconstraintwhichisobtainedbythedimensionalreductionof(3.64).InthissectionwewilldescribethereductionoftheD+1-dimensionalconstraint(3.64).NotethatitisG-covariant,sotheydependenceofthe eldsintheansatz(1.1)cancelsoutinthereduction.

Usingtheansatz(2.20),(2.21)theD+1dimensionalconstraint(3.64)reducestotheD-dimensionalconstraint:

Hn=eγQ Hn 1(4.69)

whereQisasin(3.65),Kisgivenby(3.66)andwehavede nedγ≡2(D n)α .Asaresult,then-form eldstrengthsaredualtothen 1-form eldstrengths.Theconstraint(4.69)canberewrittenusing atindicesas

nbc¯¯(n)a=eγδab (DAbH(n 2)+( 1)MA(n 1)c).(4.70)

Foranuntwistedreduction(i.e.onewithM=0,sothatitisastandardreduction)thisconstraintcanbeusedtoeliminatethe2kpotentialsAn 1sothatthetheorycanbewrittenintermsofthe2kpotentialsAn 2(oralternativelythepotentialsAn 2canbeeliminatedandthetheorywrittenintermsoftheAn 1,ormoregenerallyintermsofspotentialsAn 2and2k spotentialsAn 1).InthetwistedcasewithinvertibleM,onecangotothegaugeinwhichthe eldsAin 2aresetzero,aswasdiscussedinsection2.Inthisgaugethe eldstrengthsHnandHn 1aregivenin(2.33)and(2.34)sothatthedualitycondition(4.69)is:

An 1DAn 1=( 1)neγM

(4.71)

We consider generalised Scherk Schwarz reductions of supergravity and superstring theories with twists by electromagnetic dualities that are symmetries of the equations of motion but not of the action, such as the S-duality of D=4, N=4 super-Yang-Mills cou

=QM.Thisisamassiveself-dualityconditionforthe2kpotentialsAn 1.SuchwhereM

self-dualityconditionsinodddimensionswereintroducedin[26].Theself-dualityconstraint(4.71)impliesthemassive eldequation(suppressingnon-linearterms)

2An 1+... D DAn 1=e2γM(4.72)

2.However,theconstraint(4.71)halvesthenumberofwithmassmatrixproportionaltoM

degreesoffreedomofamassiven 1form eld.

Itisinstructivetocheckthenumberofphysicaldegreesoffreedom.Inddimensionsa

2masslesspformgauge eldAphascddegreesoffreedom,wherecsppisthebinomialcoe cient

csp=(s)!

We consider generalised Scherk Schwarz reductions of supergravity and superstring theories with twists by electromagnetic dualities that are symmetries of the equations of motion but not of the action, such as the S-duality of D=4, N=4 super-Yang-Mills cou

¯(n 1)α′,halvingthenumberImposingtheconstraintimposesself-dualityonthemassive eldsA

¯(n 1)αtoAα,sothathalfofthemcanbeeliminated(e.g.ofdegreesoffreedom,andrelatesAn 2¯(n 1)α,orA¯(n 1)αcanbeeliminatedleavingAα).ThusoneAαcanbeeliminatedleavingAn 2n 2

2n 2isleftwithkcn 1degreesoffreedom,asrequired.

The eldequationsfromtheD-dimensionallagrangian(2.28)with(2.29),(4.68)aresup-plementedbytheDdimensionalconstraint(4.69).Thisimpliesthatthe eldstrengthsHnandHn 1in(2.29)arenotindependentbutarerelatedviathedualitycondition(4.69).Notethatifthisconstraintwereappliedtotheaction,itwouldmakethegauge eldkineticterm(4.68)vanish.Thiswastobeexpectedasthetwistedself-dualitycondition(3.64),fromwhichthedualitycondition(4.69)isobtained,impliesthevanishingofthegaugekinetictermintheD+1-dimensionaldoubledlagrangian(3.63).Thusitisimportantthatone rstvariestheactionandthenimposestheconstraint(4.69).

Itisstraightforwardtoverifythatthe eldequationsderivedfromLbforthepotentialsAn 1areconsistentwiththeDdimensionalconstraint(4.69).Aftersomecomputationone ndsthattheconditionforconsistencyisthatthemassmatrixMshouldsatisfytheequation(2.37).

4.2LagrangianforReducedTheory

Theodddimensionalmassiveself-dualitycondition(4.71)canbeobtainedfromaChern-Simonsactionoftheform(1.6),aswenowshow.InthecaseinwhichMisinvertible,theD-dimensionalconstraint(4.71)followsfromthefollowinglagrangian:

L′b=1

We consider generalised Scherk Schwarz reductions of supergravity and superstring theories with twists by electromagnetic dualities that are symmetries of the equations of motion but not of the action, such as the S-duality of D=4, N=4 super-Yang-Mills cou

exceptfortheKaluza-Klein eld .Then

δκLb=

=

= 1eγeγδκ∧ Hn δκ

δκ1δκeγ M∧ An 1 ∧ Hn 1δκM∧ An 1(4.77)M∧ An 1=δκL′b.

establishesthatthetwolagrangiansLandL′havethesame eldequationsforthescalar elds.Inthesecondlinewehaveimposedtheconstraint(4.71).In

the

third

lineweusedthesymmetrypropertiesofthematrices andK,thefactthat isG-invariantandalsothatδK = MT 1 KM= MTKM.This.Thelastequalityin(4.77)holdsbecausePMδκ

Inordertochecktheequivalenceofthe eldequationsforthemetric,itisusefultonotethefollowingrelation:

δ(i)αµ1···µn 1(j)β g( nHnHnµ1···µn 1+1

δgαβ=

2eγeγδ1¯kl(n 1)A(k)ασ1···σn 2A(l)β gM (n 1)(n 1)σ1···σn 2

δgαβ(4.79)

¯kl=KijMiMj=(MTKM)kl.Theequivalenceofthe eldequationswherewehavede nedMkl

fortheKaluza-Klein eld arealsoeasilychecked.

AsaresultwehaveanewD-dimensionallagrangianwhichyieldstheD-dimensional eld

1

2equationsandalsotheconstraint:LD=R 1

+1e 2(D 1)α F2∧ F2

1(4.80)

4tr(DK∧ DK 1)

2γ bac bPab[( 1)n 1Aan 1∧DAn 1+eMcAn 1∧ An 1],(4.81)

We consider generalised Scherk Schwarz reductions of supergravity and superstring theories with twists by electromagnetic dualities that are symmetries of the equations of motion but not of the action, such as the S-duality of D=4, N=4 super-Yang-Mills cou

a=M iVa(V 1)j.NotethatonehaswherePab=Pij(V 1)ia(V 1)jb=( 1)acMcbandMbjib

Pab=Pcd ca db=( 1)n 1Mab.WhenMisnotinvertible,Aαn 1dropsoutfromthislagrangian,

¯(n 1)α′:whichisnowjustalagrangianforA

L′b1=1

4¯(n)α∧ H¯(n)β 1e γδαβH

2

Thenthetotallagrangianis¯(n 1)α∧ DA¯(n 1)β.eγδαβDA(4.84)

L′D=Lg+Ls+L′b1+L′b2(4.85)

whereLgandLsareasin

(2.29).Itisstraightforwardtoshowthatthesegivetheright eldequations,byanargumentsimilartothatintheinvertiblecaseabove.

4.3G=SL(2,IR)Case

InthissubsectionwewillconsiderthecaseG=SL(2,IR).InthiscasethematricesKand areasin(2.15)and(3.56).TherearethreedistinctreductionscorrespondingtothethreeconjugacyclassesofSL(2,IR)asdiscussedinsection2.Themassmatricesrepresentingthethreeconjugacyclassesaregivenin(2.45).NowwewillgivethereducedlagrangiansforeachmassmatrixMe,MhandMp.

Me:

Therearetwomassive,(n 1)-formsinthetheorywhichwewillcallA1andA2.ThisisanSO(2)-gaugedtheorysinceMegeneratestheSO(2)subgroupofSL(2,IR).(Ifn=2,thereareadditionalgauge eldsandthegaugegroupisISO(2).)Thisistheonlycasethetheoryhasastableminimumofthepotential[14].Theglobalminimumofthepotentialisatχ=φ=0.Thelagrangianis:

We consider generalised Scherk Schwarz reductions of supergravity and superstring theories with twists by electromagnetic dualities that are symmetries of the equations of motion but not of the action, such as the S-duality of D=4, N=4 super-Yang-Mills cou

LD=R 1

+112e 2(D 1)α F2∧ F2(4.86)

4tr(DK∧ DK 1) 2e2(D 1)α m2[sinh2φ+χ2(2+e2φ(2+χ2))] 1.

Mh:

Therearetwomassive,(n 1)-formsinthetheorywhichwewillcallA1andA2,asbefore.ThegaugegroupisSO(1,1)inthiscase(forn>2).Thelagrangianis:

1

2LD=R 1

+1e 2(D 1)α F2∧ F2(4.87)

4tr(DK∧ DK 1) 2e2(D 1)α m2[1+χ2e2φ] 1.

Mp:

¯1,onemassless(n 1)-form eldA¯2andoneThereisonemassive(n 1)-form eldA

massless(n 2)-form eldB2.HoweveronecaneliminateB2byusingthereducedconstraint(4.74),aswasdiscussedintheprevioussubsection.ThegaugegroupisSO(1,1)inthiscase(forn>2).

LD=R 1

+

+11

212e 2(D 1)α F2∧ F22¯2∧ DA¯2eγDA(4.88)e2(D 1)α m2(e φ+eφχ2)2 1.

5SupergravityApplications

Inthissection,wewillapplyourresultstothetwistedreductionofsupergravitytheoriesind=D+1=4,6,8dimensionstoD=3,5,7.Wewilldiscussgeneralfeatureshere,andgivedetailsofthefulllagrangiansandoftheclassi cationoftheorieselsewhere.

We consider generalised Scherk Schwarz reductions of supergravity and superstring theories with twists by electromagnetic dualities that are symmetries of the equations of motion but not of the action, such as the S-duality of D=4, N=4 super-Yang-Mills cou

5.1Reductionofd=8MaximalSupergravity

TheN=2d=8maximalsupergravity[2]canbeobtainedfrom11-dimensionalsuper-gravitybytoroidalcompacti cationandhas eldequationsinvariantunderthedualitygroupSL(2,IR)×SL(3,IR).Thebosonic eldsconsistofametric,a3-formgauge eldA3,6vec-tor eldsinthe(2,3)representationofSL(2,IR)×SL(3,IR),32-formgauge eldsinthe(1,3)representationofSL(2,IR)×SL(3,IR),andscalarstakingvaluesinthecosetspace

3SL(3,IR)/SO(3)×SL(2,IR)/SO(2).Thegauge eldA3combineswiththedualgauge eldA

toformadoubletunderSL(2,IR)andSL(3,IR)isasymmetryoftheactionwhereasSL(2,IR)isasymmetryofthe eldequationsonly,asitactsthroughelectro-magneticdualityonthe3-formgauge elds.

ThereisaconsistenttruncationofthistheorywhereonlytheSL(3,IR)singletsarekeptandalltheother eldsaresettozero[29].Thenthetruncatedtheoryconsistsofametric,a3-formgauge eldandscalarstakingvaluesinSL(2,IR)/SO(2),withanSL(2,IR)S-dualitysymmetry.Thistruncatedtheoryispreciselyoftheform(3.46)withn=4andthetwistedreductionwithanSL(2,R)twistgivesthreedistinctreducedtheoriescorrespondingtothethreeconjugacyclasses,withlagrangians(4.86),(4.87)or(4.88).

Thiscanbeextendedtothefulltheory,asthereductionofthe eldsthatarenotSL(3,IR)singletsisastandardScherk-Schwarzreduction.TherearesomecomplicationsresultingfromtheChern-Simonsinteractionsofthed=8theory,andwewillnotpresentthefullresultshere.Therearethreedistinctclassicaltheories,whilethedistinctquantumtheoriescorrespondtothedistinctSL(2,Z)conjugacyclasses.

5.2Reductionofd=4,N=4Supergravity

N=4supergravitycoupledtopvectormultipletshasanO(6,p)symmetryoftheactionandanSL(2,IR)S-dualitysymmetryoftheequationsofmotion.Thevector eldsAI1(I=1,2,...,6+p)

Iareinthefundamental6+prepresentationofO(6,p)andcombinewithdualpotentialsA1

toform6+pdoubletsAmI(m=1,2)transforminginthe(2,6+p)ofSL(2,IR)×O(6,p).1

ThescalarstakevaluesinthecosetSL(2,IR)/SO(2)×O(6,22)/O(6)×O(22).ThescalarsinO(6,22)/O(6)×O(22)canberepresentedbyacosetspacemetricNIJwhilethe2scalarsφ,χinSL(2,IR)/SO(2)canberepresentedbyacosetspacemetricKmnwhichisofthesameformas(2.15).

Thelagrangianforthebosonicsectorcanbewrittenas[20,30,31]:

L=R 1+

214tr(dN∧ dN 1) 1(5.89)

IJχF2LIJ∧F2

We consider generalised Scherk Schwarz reductions of supergravity and superstring theories with twists by electromagnetic dualities that are symmetries of the equations of motion but not of the action, such as the S-duality of D=4, N=4 super-Yang-Mills cou

whereListheO(6,p)invariantmetricandthematricesNandLsatisfy

NT=N,NTLN=L.(5.90)

Nowthevector eldequationcanbewrittenasdGI2=0where

1IJGI2=(L)δL

4

2tr(dK∧ dK 1)+1IF2LIJ∧GJ2

I IAsbefore,the eldequationsdGI2=0implytheexistenceofdualpotentialsA1,withG2=

I.Thenthefullsetofvector eldsAiinthedoubledformalismisAmI=(AI,A I)wheredA11111

i=1,...,2(6+p)becomesthecompositeindexmI.The eldstrengthsarethe6+pSL(2,IR)-doublets: IdAI1.(5.94)H2=I dA1

Wealsoimposethetwistedself-dualityconstraint

mInJH2=JmnRIJ H2.(5.95)

whereJmnisasin(3.55),Jmn= mpKpn.SothematrixQin(3.64)isnowthe(12+2p)×(12+2p)matrix

Q=J R

whichsatis esQ2= 1sinceJ2= 1andR2=+1.Thedoubledlagrangian

L=R 1+

114tr(dN∧ dN 1)(5.97)(5.96)

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