Compactifications with S-Duality Twists(4)

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].