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