Compactifications with S-Duality Twists(14)

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

dimensionalreductionof(3.64).The eldequationsin2n 1dimensionsarethenthosefromthereducedactiontogetherwiththereducedconstraints,andwegoontoseekanactionin2n 1dimensionsthatgivesboththeconstraintsandthereduced eldequations.

4.1DimensionalReductionintheDoubledFormalism

Thelagrangian(3.63)inthedoubledformalismisofthesameformas(2.11),butwithanextrafactorof1/2inthenormalisationofthegauge eldkineticterm.TheScherk-Schwarzreductionofthelagrangian(2.11)wasalreadydiscussedinsection2,whereweshowedthatityieldsthelagrangian(2.28)inDdimensions.Itfollowsthatthereductionof(3.63)shouldgive(2.28)butnowwith(2.30)dividedbytwotogive:

Lb= 1

4Te2(D n)α Hn 1K∧ Hn 1(4.68)

Justasthelagrangian(3.63)shouldbesupplementedbytheD+1dimensionalconstraint(3.64)inordertogivethecorrectD+1dimensional eldequations,theDdimensionallagrangian(2.28)with(2.29),(4.68)shouldbesupplementedbytheconstraintwhichisobtainedbythedimensionalreductionof(3.64).InthissectionwewilldescribethereductionoftheD+1-dimensionalconstraint(3.64).NotethatitisG-covariant,sotheydependenceofthe eldsintheansatz(1.1)cancelsoutinthereduction.

Usingtheansatz(2.20),(2.21)theD+1dimensionalconstraint(3.64)reducestotheD-dimensionalconstraint:

Hn=eγQ Hn 1(4.69)

whereQisasin(3.65),Kisgivenby(3.66)andwehavede nedγ≡2(D n)α .Asaresult,then-form eldstrengthsaredualtothen 1-form eldstrengths.Theconstraint(4.69)canberewrittenusing atindicesas

nbc¯¯(n)a=eγδab (DAbH(n 2)+( 1)MA(n 1)c).(4.70)

Foranuntwistedreduction(i.e.onewithM=0,sothatitisastandardreduction)thisconstraintcanbeusedtoeliminatethe2kpotentialsAn 1sothatthetheorycanbewrittenintermsofthe2kpotentialsAn 2(oralternativelythepotentialsAn 2canbeeliminatedandthetheorywrittenintermsoftheAn 1,ormoregenerallyintermsofspotentialsAn 2and2k spotentialsAn 1).InthetwistedcasewithinvertibleM,onecangotothegaugeinwhichthe eldsAin 2aresetzero,aswasdiscussedinsection2.Inthisgaugethe eldstrengthsHnandHn 1aregivenin(2.33)and(2.34)sothatthedualitycondition(4.69)is:

An 1DAn 1=( 1)neγM

(4.71)