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

where

Lg=R 1

Ls=

and

Lb= 11221e 2(D 1)α F2∧ F2e2(D 1)α tr(M2+MK 1MTK) 1Te2(D n)α Hn 1K∧ Hn 1(2.29)2

The eldstrengths(2.26)areinvariantunderthefollowinggaugetransformations:

δAn 1=dΛ,δAn 2=( 1)n 1MΛ.(2.30)(2.31)

IfMisinvertible,thesecanbeusedtogaugeAn 2tozerobyperformingthegaugetransfor-mation:

An 1→An 1+( 1)n 1M 1dAn 2.

InthisgaugetheD-dimensional eldstrengthsbecome

Hn=DAn 1=dAn 1 ( 1)nMAn 1∧A

Hn 1=( 1)nMAn 1.(2.33)(2.34)(2.32)

ThenAn 2disappearsfromthetheory,andthetermHn 1∧ Hn 1isamasstermforAn 1.Thedegreesoffreedomrepresentedbyther eldsAn 2havebeenabsorbedbyther(n 1)-form eldsAn 1whichhavebecomemassive.NowHn=DAn 1isagaugecovariantderivativewherethegaugegroupisthesubgroupofGgeneratedbyMandthecorrespondinggauge eldisthegraviphotonA.

NowwewillanalyzethecaseMisnotinvertible.Itisusefultoworkwith atindicesHa=VaiHi,Aa=VaiAi.ThenHa=DAa=dAa+ωabAbwhereωistheconnection1-formωab=Vai(dV 1)ib.ThegroupsGarisinginthesupergravitytheoriesofinteresthereallhaveaG-invariantmatrix whichissymmetricifnisoddandanti-symmetricifniseven

ab=( 1)n 1 ba.

¯a=( 1)abHbandMab=Mac cb.NowonehasUsingthis,weintroduceH

aan 1¯(n 1)b,HnMabA 1=DAn 2 ( 1)(2.35)¯(n)a=DA¯(n 1)a H¯(n 1)a∧A=D A¯(n 1)a(2.36)H

isthecovariantderivativewithconnectionsωabandA.whereDNotethatM=eMandMT 1M= 1sinceM∈Gand isG-invariant.(Forcomplexrepresentations,theconditionisM 1M= 1.)AsaresultthemassmatrixMabsatis es:

MT 1+ 1M=0.

(2.37)