Compactifications with S-Duality Twists(9)

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

From(2.35)and(2.37)itfollowsthatMabisasymmetricmatrixifnisevenandantisymmetricifnisodd:

Mab=( 1)nMba.

(2.38)Letthedimensionofker(M)bel.NowthematrixMabcanbebroughtintothecanonicalform

Mab=

′′00′′0mαβ(2.39)wheremαβisaninvertible(r l)×(r l)matrixwhichisdiagonalifnisevenandskew-diagonalifnisodd.Herewehavesplittheindicesa→(α,α′)whereαrunsfrom1tolandα′runsfroml+1tor.Similarlythegauge eldsAcanbewrittenintheblockform

A=

Performingthegaugetransformation

′¯(n 1)α′+( 1)n 1(m 1)α′β′DAβ¯(n 1)α′→AAn 2 A′Aαα (2.40)(2.41)

¯(n 1)α′becomemassive,havingeatenther l eldsAα′,whileoneseesthatther l eldsAn 2¯Aαn 2andA(n 1)αbothremaininthetheoryasmasslessgauge elds,withlofeach.The eld

strengthsforthe(n 2)-form eldsin(2.36)become

α′nα′β′¯Hn=( 1)mA(n 1)β′, 1ααHn 1=DAn 2(2.42)

andhencetheterm(2.30)canbewrittenas

Lb=

112′′¯(n)β′¯(n)α′∧ He 2(n 1)α δαβH(2.43)

2′′¯(n 1)β′.¯(n 1)α′∧ Ae2(D n)α (mTm)αβA

Wehavechosenthenormalisationof sothat ac bdδcd=δab.

Thegaugegroup,thecouplingsandthescalarpotentialoftheD-dimensionaltheoryfoundabovearegivenexplicitlyintermsofthemassmatrixM,andtwotheoriesaredistinctifthemonodromiesareindistinctG-conjugacyclasses.ForthecaseG=SL(2,R)therearethreeconjugacyclasses,thehyperbolic,ellipticandparabolicconjugacyclassesandsotherearethreedistinctreductions[6].Thehyperbolic,ellipticandparabolicmonodromymatricesandmassmatricescanbetakentobe:

Mh= e

0m0e m

,Me=

cosmsinm

sinmcosm ,Mp=

Mp= 1m01 .(2.44)(2.45)Mh=m0

0 m,Me=0m

m0

,0m00.