Compactifications with S-Duality Twists(5)

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

Ourmaininterestwillbeinthereductionofsupergravityandsuperstringtheories.ExtendedsupergravitytheoriestypicallyhaveaglobalsymmetryGandthescalarstakevaluesinthecosetspaceG/HwhereGisanon-compactgroupandHisthemaximalcompactsubgroupofG.ThescalarsectorofthetheoryistheninvariantunderthegroupGandthissymmetrytypicallyextendstothefulltheoryforsupergravitiesinodddimensions.Insomeevendimensionaltheories,thesymmetryGextendstoasymmetryoftheequationsofmotiononly,actingthroughdualitytransformationsexchanging eldequationswithBianchiidentitites.

ThetheorycanbeformulatedwithalocalHsymmetryaswellasaglobalGsymmetry.ThescalarsinthecosetspaceG/HcanberepresentedbyavielbeinV(x)∈GwhichtransformsunderglobalGandlocalHtransformationsas

V→h(x)V,V→Vg,

L= 1Thelagrangianish(x)∈Hg∈G(2.7)

sothatδabisaninvariant,butthegeneralisationtoother

representationsisstraightforward.

AnalternativeformulationthatdoesnotinvolveextrascalarsistouseametricKonG/Hinsteadofavielbein,transformingas(forarealrepresentationofG)

K→gTKg(2.9)

SuchametriccanbeconstructedfromthevielbeinasKij=δabVaiVbj,whereiandaarethecurvedand atindicesrespectively.KisinvariantunderlocalHtransformationsashTh=

,wewouldusethehermitianmetric

K=V VtransformingasK→g Kg.)ThelagrangiancanbewrittenintermsofKas

L=1