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

whereListheO(6,p)invariantmetricandthematricesNandLsatisfy

NT=N,NTLN=L.(5.90)

Nowthevector eldequationcanbewrittenasdGI2=0where

1IJGI2=(L)δL

4

2tr(dK∧ dK 1)+1IF2LIJ∧GJ2

I IAsbefore,the eldequationsdGI2=0implytheexistenceofdualpotentialsA1,withG2=

I.Thenthefullsetofvector eldsAiinthedoubledformalismisAmI=(AI,A I)wheredA11111

i=1,...,2(6+p)becomesthecompositeindexmI.The eldstrengthsarethe6+pSL(2,IR)-doublets: IdAI1.(5.94)H2=I dA1

Wealsoimposethetwistedself-dualityconstraint

mInJH2=JmnRIJ H2.(5.95)

whereJmnisasin(3.55),Jmn= mpKpn.SothematrixQin(3.64)isnowthe(12+2p)×(12+2p)matrix

Q=J R

whichsatis esQ2= 1sinceJ2= 1andR2=+1.Thedoubledlagrangian

L=R 1+

114tr(dN∧ dN 1)(5.97)(5.96)