Compactifications with S-Duality Twists(10)

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