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)