Compactifications with S-Duality Twists(16)

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

¯(n 1)α′,halvingthenumberImposingtheconstraintimposesself-dualityonthemassive eldsA

¯(n 1)αtoAα,sothathalfofthemcanbeeliminated(e.g.ofdegreesoffreedom,andrelatesAn 2¯(n 1)α,orA¯(n 1)αcanbeeliminatedleavingAα).ThusoneAαcanbeeliminatedleavingAn 2n 2

2n 2isleftwithkcn 1degreesoffreedom,asrequired.

The eldequationsfromtheD-dimensionallagrangian(2.28)with(2.29),(4.68)aresup-plementedbytheDdimensionalconstraint(4.69).Thisimpliesthatthe eldstrengthsHnandHn 1in(2.29)arenotindependentbutarerelatedviathedualitycondition(4.69).Notethatifthisconstraintwereappliedtotheaction,itwouldmakethegauge eldkineticterm(4.68)vanish.Thiswastobeexpectedasthetwistedself-dualitycondition(3.64),fromwhichthedualitycondition(4.69)isobtained,impliesthevanishingofthegaugekinetictermintheD+1-dimensionaldoubledlagrangian(3.63).Thusitisimportantthatone rstvariestheactionandthenimposestheconstraint(4.69).

Itisstraightforwardtoverifythatthe eldequationsderivedfromLbforthepotentialsAn 1areconsistentwiththeDdimensionalconstraint(4.69).Aftersomecomputationone ndsthattheconditionforconsistencyisthatthemassmatrixMshouldsatisfytheequation(2.37).

4.2LagrangianforReducedTheory

Theodddimensionalmassiveself-dualitycondition(4.71)canbeobtainedfromaChern-Simonsactionoftheform(1.6),aswenowshow.InthecaseinwhichMisinvertible,theD-dimensionalconstraint(4.71)followsfromthefollowinglagrangian:

L′b=1