Compactifications with S-Duality Twists(7)

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

Here isthescalar eldcomingfromthereductionofthemetric,F=dAandAisthegraviphoton.TheconstantsαandβdependonDandare:

α2=1

4

scalarpotential

[1]: ∧ dK 1)fromD+1dimensionstoDdimensionsgivesascalarkinetictermplusatr(dK

Ls=1

2

Theansatz(2.21)impliese2(D 1)α tr(M2+MK 1MTK) 1(2.24)

n(x,y)=e MyHn(x)+e MyHn 1(x)∧(dy+A)H

n=dA n 1.HeretheD-dimensional eldstrengthsareforthen-form eldstrengthsH

Hn 1(x)=dAn 2 ( 1)n 1MAn 1,

Reductionofthekinetictermgives

TK ∧ n→[e 2(n 1)α HTK∧ Hn+e2(D n)α HTK∧ Hn 1]∧dy HHnnn 1

CollectingtheresultswecannowwritedowntheD-dimensionallagrangianas:

LD=Lg+Lb+LsHn(x)=dAn 1 Hn 1∧A.(2.25)(2.26)(2.27)(2.28)