Extremal non-BPS black holes and entropy extremization

TL;DR

The paper analyzes extremal static black holes in four-dimensional $N=2$ supergravity with a one-modulus conifold prepotential, focusing on attractor behavior derived from the black hole potential $V_{ m BH}$ and the entropy function $\mathcal{E}$. It solves the attractor equations for a two-charge configuration and compares the black hole potential and entropy-function formalisms, providing exact and small-$T$ (conifold) solutions for both BPS and non-BPS branches. It finds that near the conifold, the BPS branch has a local entropy maximum at $T=0$, while the non-BPS branch has a local minimum at $T=0$ but a nearby maximum, with the latter sometimes yielding higher entropy than the BPS case; stability analysis shows the BPS solution is robust, whereas the non-BPS attractor exists only in a restricted region of the modulus. Overall, the work highlights a rich structure of attractor loci in moduli space and demonstrates that entropy extremization can favor non-supersymmetric vacua in the conifold setting.

Abstract

At the horizon, a static extremal black hole solution in N=2 supergravity in four dimensions is determined by a set of so-called attractor equations which, in the absence of higher-curvature interactions, can be derived as extremization conditions for the black hole potential or, equivalently, for the entropy function. We contrast both methods by explicitly solving the attractor equations for a one-modulus prepotential associated with the conifold. We find that near the conifold point, the non-supersymmetric solution has a substantially different behavior than the supersymmetric solution. We analyze the stability of the solutions and the extrema of the resulting entropy as a function of the modulus. For the non-BPS solution the region of attractivity and the maximum of the entropy do not coincide with the conifold point.

Figures (4)

• Figure 1: Eigenvalue(s) $\lambda$ of the Hessian of the black hole potential as a function of $T$ for $Y^0 =1, \beta=-1/2$, $a=1$ in the BPS case.
• Figure 2: Eigenvalues $\lambda_{1,2}$ of the Hessian of the black hole potential as functions of $T$ for $Y^0=1$, $\beta=-1/2$, $a=1$ in the non-BPS case.
• Figure 3: $\mathcal{S}/\pi$ as a function of $T$ for $Y^0=1$, $\beta=-1/2$, $a=1$ in the BPS case. The left graph is a cross section along the positive $T$ semi-axis through the surface in the right graph.
• Figure 4: $\mathcal{S}/\pi$ as a function of $T$ for $Y^0=1$, $\beta=-1/2$, $a=1$ in the non-BPS case, similarly to Fig. \ref{['fig:susyS']}.