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

ij= ab(V 1)ia(V 1)jb,wherei,j=1,...,2kareindicesforthe2krepresentationofG,satisfying ij=( 1)n 1 ji.

IwithGI=dA ItoformAsbeforeweintroducepotential eldsAn 1nn 1

Hn= dAInI dAn (3.62)

transforminginthe2krepresentationofG.ThenthesystemcanbedescribedbytheG-invariantlagrangian1L′=

.(3.67)

sothattheconstraint(3.64)isconsistentasfor2n-dimensionalLorentzianspace-time Hn=( 1)n 1Hn.Itwasshownin[21]thatthe eldequationsfrom(3.63)areequivalenttothosefrom(3.57)togetherwiththeconstraint(3.64).

4ReductionwithDualityTwist

Thetheorywithlagrangian(3.57)hasaglobalsymmetryGoftheequationsofmotionwhichactsviadualitytransformations.InthissectionwewilldimensionallyreduceonacirclefromD+1=2ntoDdimensionswithatwistthathasmonodromyMinG.ForsomechoicesofmonodromyMinG,thisisinfactasymmetryoftheactionandthisisastandardScherk-Schwarzreduction,asinsection2.Ifitisonlyasymmetryoftheequationsofmotion,thenweusethedoubledformalismofsection3withlagrangian(3.63)supplementedbytheconstraint(3.64).Thelagrangian(3.63)isofthesameformas(2.11),sotheScherk-Schwarzreductionoftheactionproceedsasinsection2.Thisissupplementedbytheconstraintsarisingfromthe