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

thelagrangian(3.46)canbewrittenas

L=

strengthFnare

dFn=0

dGn=d( e φ Fn χFn)=0

whichcanbecombinedas

dHn=0

whereHnistheSL(2,IR)doublet

Hn= 12F∧G(3.48)whereKisasin(2.15).TheBianchiidentityandtheequationofmotionforthen-form eld(3.49)(3.50) Fn

Gn.(3.51)

The eldequationsaremanifestlySL(2,R)invariant,buttheF∧Gterminthelagrangian(3.48)isnotinvariant.However,aninvariantlagrangiancanbeconstructedasin[21]ifthe

nsothatGn=dA n,which eldequationdGn=0issolvedbyintroducingadualpotentialA

icanbecombinedwithAntoformanSL(2,R)doublet,with eldstrengthsHngivenby

Hn= dAn ndA .(3.52)

ThenthenaturalSL(2,IR)invariantlagrangianis

L′=1

4ijHnKij∧ Hn.(3.53)

whichisoftheformconsideredintheprevioussection.

n 1areindependent elds,sothatthenumberofn 1Forthisaction,bothAn 1andA

formdegreesoffreedomhasbeendoubled.Tohalvethemagain,forevennthisactioncanbesupplementedbytheSL(2,R)covariantconstraint[21]

ijHn=Jij Hn(3.54)

whereJistheSL(2,IR)matrix

Jij= ikKkj.

Here istheSL(2,IR)invariantmatrix

= (3.55)01

10

.(3.56)