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

LD=R 1

+112e 2(D 1)α F2∧ F2(4.86)

4tr(DK∧ DK 1) 2e2(D 1)α m2[sinh2φ+χ2(2+e2φ(2+χ2))] 1.

Mh:

Therearetwomassive,(n 1)-formsinthetheorywhichwewillcallA1andA2,asbefore.ThegaugegroupisSO(1,1)inthiscase(forn>2).Thelagrangianis:

1

2LD=R 1

+1e 2(D 1)α F2∧ F2(4.87)

4tr(DK∧ DK 1) 2e2(D 1)α m2[1+χ2e2φ] 1.

Mp:

¯1,onemassless(n 1)-form eldA¯2andoneThereisonemassive(n 1)-form eldA

massless(n 2)-form eldB2.HoweveronecaneliminateB2byusingthereducedconstraint(4.74),aswasdiscussedintheprevioussubsection.ThegaugegroupisSO(1,1)inthiscase(forn>2).

LD=R 1

+

+11

212e 2(D 1)α F2∧ F22¯2∧ DA¯2eγDA(4.88)e2(D 1)α m2(e φ+eφχ2)2 1.

5SupergravityApplications

Inthissection,wewillapplyourresultstothetwistedreductionofsupergravitytheoriesind=D+1=4,6,8dimensionstoD=3,5,7.Wewilldiscussgeneralfeatureshere,andgivedetailsofthefulllagrangiansandoftheclassi cationoftheorieselsewhere.