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

a=M iVa(V 1)j.NotethatonehaswherePab=Pij(V 1)ia(V 1)jb=( 1)acMcbandMbjib

Pab=Pcd ca db=( 1)n 1Mab.WhenMisnotinvertible,Aαn 1dropsoutfromthislagrangian,

¯(n 1)α′:whichisnowjustalagrangianforA

L′b1=1

4¯(n)α∧ H¯(n)β 1e γδαβH

2

Thenthetotallagrangianis¯(n 1)α∧ DA¯(n 1)β.eγδαβDA(4.84)

L′D=Lg+Ls+L′b1+L′b2(4.85)

whereLgandLsareasin

(2.29).Itisstraightforwardtoshowthatthesegivetheright eldequations,byanargumentsimilartothatintheinvertiblecaseabove.

4.3G=SL(2,IR)Case

InthissubsectionwewillconsiderthecaseG=SL(2,IR).InthiscasethematricesKand areasin(2.15)and(3.56).TherearethreedistinctreductionscorrespondingtothethreeconjugacyclassesofSL(2,IR)asdiscussedinsection2.Themassmatricesrepresentingthethreeconjugacyclassesaregivenin(2.45).NowwewillgivethereducedlagrangiansforeachmassmatrixMe,MhandMp.

Me:

Therearetwomassive,(n 1)-formsinthetheorywhichwewillcallA1andA2.ThisisanSO(2)-gaugedtheorysinceMegeneratestheSO(2)subgroupofSL(2,IR).(Ifn=2,thereareadditionalgauge eldsandthegaugegroupisISO(2).)Thisistheonlycasethetheoryhasastableminimumofthepotential[14].Theglobalminimumofthepotentialisatχ=φ=0.Thelagrangianis: