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