All supersymmetric solutions ofN= 2,D= 4 gauged supergravity

Abstract

We review the classification of supersymmetric solutions to minimal gauged supergravity in four dimensions. After a short introduction to the main features of the theory, we explain how to obtain all its solutions admitting a Killing spinor. Then, we analyze the rich mathematical structure behind them and present the supersymmetric field configurations. Among them, we find supersymmetric black holes, quarter and half BPS traveling waves, kink solutions, and supersymmetric Kundt and Robinson-Trautman solutions. Finally, we generalize the classification to include external sources, and show a particular solution describing a supersymmetric Gödel-type universe.

All solutions of the localization equations for $ \mathcal{N}=2 $ quantum black hole entropy

“…Using Fierz identities to rearrange four fermion terms [41,42], we find the following algebraic relation between these quantities:…”

“…Some explicit relations are written e.g. in [41]. We present one such example in Appendix B, which will be useful to us later.…”

Supersymmetric solutions to gauged N=2 d=4 SUGRA: the full timelike shebang

“…The generic form of the solutions is the one already known from earlier investigations, but there are some fine differences; for example in ungauged case the base space is just R 3 and in (Abelian) gauged SUGRA the base space becomes torsionful [15], or said differently it must have a non-trivial SO(2) holonomy in order to be able to kill off the effective U(1) gauge field induced on the base space. In our case, see eq.…”

“…[13,14]; in the latter theories one can, depending on the model, construct analytic, globally regular monopole solutions and non-Abelian black holes. Caldarelli & Klemm [15] extended Tod's results to the case of minimal gauged N = 2, d = 4 supergravity and the resulting solutions were studied further by Cacciatori et al in refs. [16]; some examples of supersymmetric black holes had been already been obtained in refs.…”

All the timelike supersymmetric solutions of all ungauged d = 4 supergravities

“…These results were immediately extended to the Abelian gauged case [8] and later on to general matter contents and couplings [9] (always in the minimal N = 2 supergravity). The spinorbilinear method was subsequently applied to other 4-dimensional [10]- [21], 6-dimensional [22], 7-dimensional [23], 11-dimensional [24] and, recently, to 3-dimensional [25] supergravities.…”