BPS black holes in N=2 D=4 gauged supergravities

TL;DR

This work advances the study of BPS black holes in gauged $N=2$, $D=4$ supergravity with charged hypermultiplets by showing that spontaneous symmetry breaking in maximally supersymmetric vacua provides a robust method to generate gauged BHs from ungauged examples while keeping scalar hair at bay. It also analyzes half-BPS configurations via Killing spinor identities, revealing two branches of solutions: one leading to maximally symmetric spacetimes without horizons, and another reducing to a Behrndt–Lüst–Sabra-type system with modified Maxwell equations when $P^x_oldsymbol{ u} L^oldsymbol{ u}=0$. The paper then investigates scalar hair and finds that, within the explored models and consistent positive-definite kinetic terms, static hair is obstructed unless ghost modes are allowed in the vector multiplet sector, with explicit quadratic and cubic prepotential examples illustrating ripple-like metric behavior and negative-norm scalars. It concludes with a discussion of fermionic hair as a potential workaround for the ghost problem and outlines future directions involving AdS$_4$ BHs, higher-derivative effects, and holographic applications. Overall, the results clarify constraints on scalar hair in gauged $N=2$ supergravity and provide a concrete framework for embedding ungauged BPS black holes into gauged theories.

Abstract

We construct and analyze BPS black hole solutions in gauged N=2, D=4 supergravity with charged hypermultiplets. A class of solutions can be found through spontaneous symmetry breaking in vacua that preserve maximal supersymmetry. The resulting black holes do not carry any hair for the scalars. We demonstrate this with explicit examples of both asymptotically flat and anti-de Sitter black holes. Next, we analyze the BPS conditions for asymptotically flat black holes with scalar hair and spherical or axial symmetry. We find solutions only in cases when the metric contains ripples and the vector multiplet scalars become ghost-like. We give explicit examples that can be analyzed numerically. Finally, we comment on a way to circumvent the ghost-problem by introducing also fermionic hair.

Figures (1)

• Figure 1: Plots of the solution to the differential equation \ref{['eq:diff-h1']} for $q_0 =1$, using boundary conditions $H_1(1) = 10$ and $H_1'(1) = 1$. The scalar $z$ approaches zero at the horizon at $r=0$, and the Kähler potential ${\rm e}^{-\mathcal{K}}$ approaches $1$ as $r \to \infty$.