Special Vinberg Cones and the Entropy of BPS Extremal Black Holes

TL;DR

This paper derives a general formula for the entropy of BPS extremal black holes in N=2 supergravity with homogeneous scalar manifolds, extending previous results to non-symmetric cases using Vinberg's theory.

Contribution

It introduces a novel application of Vinberg's theory of homogeneous cones to explicitly invert quadratic maps related to black hole entropy formulas.

Findings

Explicit entropy formula for BPS black holes with homogeneous scalar manifolds

Extension of known results from symmetric to non-symmetric scalar manifolds

Application of Vinberg's theory to supergravity black hole entropy

Abstract

We consider the static, spherically symmetric and asymptotically flat BPS extremal black holes in ungauged N = 2 D = 4 supergravity theories, in which the scalar manifold of the vector multiplets is homogeneous. By a result of Shmakova on the BPS attractor equations, the entropy of this kind of black holes can be expressed only in terms of their electric and magnetic charges, provided that the inverse of a certain quadratic map (uniquely determined by the prepotential of the theory) is given. This inverse was previously known just for the cases in which the scalar manifold of the theory is a homogeneous symmetric space. In this paper we use Vinberg's theory of homogeneous cones to determine an explicit expression for such an inverse, under the assumption that the scalar manifold is homogeneous, but not necessarily symmetric. As immediate consequence, we get a formula for the entropy of BPS black holes that holds in any model of N = 2 supergravity with homogeneous scalar manifold.