Supersymmetric vacua in N = 2 supergravity

Jan Louis,Hagen Triendl,Paul Smyth

doi:10.1007/jhep08(2012)039

Abstract

Abstract We use the embedding tensor formalism to analyse maximally symmetric backgrounds of N = 2 gauged supergravities which have the full N = 2 supersymmetry. We state the condition for N = 2 vacua and discuss some of their general properties. We show that if the gauged isometries leave the SU(2) R-symmetry invariant, then the N = 2 vacuum must be Minkowski. This implies that there are no AdS backgrounds with eight unbroken supercharges in the effective N = 2 supergravity of six-dimensional SU(3) × SU(3) structure compactifications of type II string theory and M-theory. Combined with previous results on N = 1 vacua, we show that there exist N = 2 supergravities with a given set of gauged Abelian isometries that have both N = 2 and N = 1 vacua. We also argue that an analogue of our analysis holds in five and six spacetime dimensions.

Highlights

This means in particular that SU(3) × SU(3)-structure backgrounds with four- and fivedimensional N = 2 AdS vacua as found in [7,8,9] do not have any description in terms of N = 2 gauged supergravity
We show that if the gauged isometries leave the SU(2) R-symmetry invariant, the N = 2 vacuum must be Minkowski
The conditions for N = 2 Minkowski vacua are linear in the fibre coordinates and holomorphic in the coordinates on the special-Kahler manifold suggesting that generically a solution exists

Summary

Gauged supergravity with eight supercharges

The second component of the field space Mh, spanned by the scalars qu in the hypermultiplets, is quaternionic-Kahler and of real dimension 4nh These manifolds admit a triplet of almost complex structures Ix, x = 1, 2, 3 satisfying IxIy = −δxy1 + ǫxyzIz, with the metric being Hermitian with respect to all three Ix. The associated two-forms Kx are the field strength of the SU(2) connection ωx, i.e. In gauged supergravities the multiplets can be charged under a set of electric and magnetic gauge fields. From the Killing vectors kλ that are non-zero at the vacuum locus nc = rk(ΘλΛkλu) holomorphic conditions arise for the vector multiplet scalars ti which in turn imply that there are nc further massive gauge boson..

Λ ΘλΛ Pλx

Gauging the isometries of the c-map