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

Lorentzianspaceofdimension4m,thenQ2=

andQisaproductstructure.Inthe

itheorieswewillconsider,theHntransforminanr-dimensionalrepresentationofarigidduality

groupG.Ind=4,N=8supergravity,therearer=562-form eldstrengthstransformingasa56ofthedualitygroupG=E7[22,23].Thesesplitinto28 eldstrengthsF=dAand

= F+...,withQacomplexstructureonR56.Ind=6,N=828dual eldstrengthsF

supergravity,thereare53-form eldstrengthswhichsplitinto5self-dualonesand5anti-selfdualones,andthese10transformasa10ofG=SO(5,5)[24].The103-form eldstrengths iwithi=1,...,10,satisfy(anti)self-dualityconstraintsoftheform(1.4)withQrelatedtoHn

theSO(5,5)-invariantmetric.Ind=8maximalsupergravity,thereisa3-formpotential,andits eldstrengthanditsdualcombineintoanSL(2,R)doublet,satisfyingaconstraintoftheform(1.4)withQ=iσ2.

OurmaininteresthereisinreductionsinwhichthemonodromyM∈Gisasymmetryof

iviatransformationstheequationsofmotionbutnottheaction,actingonthe eldstrengthsHn

involvingHodgeorelectromagneticdualities,sothattheycannotberealisedlocallyonthefundamentaln 1formpotentials.We ndthat(inthecaseinwhichMisinvertible)the eld

isatisfyingtheconstraint(1.4)giverisetorn 1formpotentialsAiin2n 1strengthsHnn 1

dimensionssatisfyingmassiveself-dualityconstraintsoftheform

An 1DAn 1=M(1.5)

whereDisagauge-covariantexteriorderivative, isnowtheHodgedualinDdimensions

∝QM.Suchodd-dimensionalself-dualityconditionswere rstconsideredandthematrixM

in[26]andoftenoccurinodd-dimensionalgaugedsupergravitytheories,andfollowfromaChern-Simonsactionwithmasstermoftheform

ijAi∧ AjL=PijAi∧DAj+M(1.6)

=PM andPijisasuitablychosenconstantmatrix.InthegeneralcaseinwhichMwhereM

isnotinvertible,someofthegauge eldsremainmassless.

Indimensionallyreducingatheorywithatwistthatisasymmetryoftheequationsofmotionandnotoftheaction,oneneedstoreducetheequationsofmotion,nottheaction.However,forthecasesofinterestherethereisadoubledformalism[21]inwhichdualpotentials n 1areintroducedforeachn 1formpotentialAn 1,inwhichthedualitysymmetrybecomesA

],whichissupplementedbyaduality-invariantconstraintthatasymmetryoftheactionS[A,A

intermsofA.ThisdoubledactionandconstraintcanthenbecouldbeusedtoeliminateA

dimensionallyreducedinthestandardwaywithatwistbythedualitysymmetry.Thisgreatlysimpli esthecalculations.