All timelike supersymmetric solutions of N=2, D=4 gauged supergravity coupled to abelian vector multiplets

TL;DR

This work provides a complete timelike-classification of supersymmetric solutions in four-dimensional ${\cal N}=2$ gauged supergravity with abelian vector multiplets using spinorial geometry. It reveals a generalized holonomy constraint ${\rm GL}((8-N)/2,\mathbb{C})\ltimes (N/2)\mathbb{C}^{(8-N)/2}$ and shows the spacetime is a U$(1)$-holonomy fibration over a three-dimensional base with torsion, guiding the construction of novel AdS$_4$ BPS black holes with nontrivial scalar profiles. In the timelike sector with U$(1)$ FI gauging and no hypermultiplets, the Killing spinor equations yield a linear system that fixes scalars and fluxes, resulting in a metric of the form $ds^2 = -4|b|^2 (dt+\sigma)^2 + |b|^{-2}(dz^2 + e^{2\Phi} dw d\bar{w})$ and a 3D base with reduced holonomy. The analysis sets the stage for exploring attractor phenomena and extensions to more general gaugings, hypermultiplets, and non-Abelian vector sectors in AdS backgrounds.

Abstract

The timelike supersymmetric solutions of N=2, D=4 gauged supergravity coupled to an arbitrary number of abelian vector multiplets are classified using spinorial geometry techniques. We show that the generalized holonomy group for vacua preserving N supersymmetries is GL((8-N)/2,C) $\ltimes$ N/2 C^((8-N)/2) $\subseteq$ GL(8,R), where N=0,2,4,6,8. The spacetime turns out to be a fibration over a three-dimensional base manifold with U(1) holonomy and nontrivial torsion. Our results can be used to construct new supersymmetric AdS black holes with nontrivial scalar fields turned on.