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

exceptfortheKaluza-Klein eld .Then

δκLb=

=

= 1eγeγδκ∧ Hn δκ

δκ1δκeγ M∧ An 1 ∧ Hn 1δκM∧ An 1(4.77)M∧ An 1=δκL′b.

establishesthatthetwolagrangiansLandL′havethesame eldequationsforthescalar elds.Inthesecondlinewehaveimposedtheconstraint(4.71).In

the

third

lineweusedthesymmetrypropertiesofthematrices andK,thefactthat isG-invariantandalsothatδK = MT 1 KM= MTKM.This.Thelastequalityin(4.77)holdsbecausePMδκ

Inordertochecktheequivalenceofthe eldequationsforthemetric,itisusefultonotethefollowingrelation:

δ(i)αµ1···µn 1(j)β g( nHnHnµ1···µn 1+1

δgαβ=

2eγeγδ1¯kl(n 1)A(k)ασ1···σn 2A(l)β gM (n 1)(n 1)σ1···σn 2

δgαβ(4.79)

¯kl=KijMiMj=(MTKM)kl.Theequivalenceofthe eldequationswherewehavede nedMkl

fortheKaluza-Klein eld arealsoeasilychecked.

AsaresultwehaveanewD-dimensionallagrangianwhichyieldstheD-dimensional eld

1

2equationsandalsotheconstraint:LD=R 1

+1e 2(D 1)α F2∧ F2

1(4.80)

4tr(DK∧ DK 1)

2γ bac bPab[( 1)n 1Aan 1∧DAn 1+eMcAn 1∧ An 1],(4.81)