BPS Black Hole Entropy and Attractors in Very Special Geometry. Cubic Forms, Gradient Maps and their Inversion

TL;DR

This paper derives explicit formulas for BPS black hole entropy and attractors in certain supergravity models by analyzing the invertibility of gradient maps of cubic forms, enabling direct solutions to complex BPS equations.

Contribution

It establishes invertibility conditions for gradient maps of cubic forms in non-symmetric scalar manifolds, leading to explicit solutions for BPS black hole entropy and attractors.

Findings

Derived explicit BPS black hole entropy formulas.

Proved invertibility conditions for gradient maps of cubic forms.

Provided solutions for infinite classes of models, including non-symmetric cases.

Abstract

We consider Bekenstein-Hawking entropy and attractors in extremal BPS black holes of $\mathcal{N}=2$, $D=4$ ungauged supergravity obtained as reduction of minimal, matter-coupled $D=5$ supergravity. They are generally expressed in terms of solutions to an inhomogeneous system of coupled quadratic equations, named BPS system, depending on the cubic prepotential as well as on the electric-magnetic fluxes in the extremal black hole background. Focussing on homogeneous non-symmetric scalar manifolds (whose classification is known in terms of $L(q,P,\dot{P})$ models), under certain assumptions on the Clifford matrices pertaining to the related cubic prepotential, we formulate and prove an invertibility condition for the gradient map of the corresponding cubic form (to have a birational inverse map which is an homogeneous polynomial of degree four), and therefore for the solutions to the BPS system to be explicitly determined, in turn providing novel, explicit expressions for the BPS black hole entropy and the related attractors as solution of the BPS attractor equations. After a general treatment, we present a number of explicit examples with $\dot{P}=0$, such as $L(q,P)$, $1\leqslant q\leqslant 3$ and $P\geqslant 1$,or $L(q,1)$, $4\leqslant q\leqslant 9$, and one model with $\dot{P}=1$, namely $L(4,1,1)$. We also briefly comment on Kleinian signatures and split algebras. In particular, we provide, for the first time, the explicit form of the BPS black hole entropy and of the related BPS attractors for the infinite class of $L(1,P)$ $P\geqslant 2$ non-symmetric models of $\mathcal{N}=2$, $D=4$ supergravity.