The Non-BPS Black Hole Attractor Equation

TL;DR

The paper extends the black hole attractor paradigm to extremal non-BPS black holes in ${\cal N}=2$ supergravity, deriving a general non-supersymmetric attractor equation via the Hodge decomposition of $H_3$ and the minimization of $V_{BH}$. It argues that extremality—with its infinite throat—is essential for moduli to flow to charges-determined horizon values, whereas non-extremal black holes lack a true attractor due to finite horizon distance. The authors solve the attractor equation in explicit models (large-volume Calabi–Yau without D6-branes, a non-BPS double-extremal example, and the mirror quintic near the Gepner point), obtaining SUSY and non-SUSY fixed points that match the minima of the effective potential and illustrating the role of charge-dependent moduli. They discuss the stability and physical significance of non-BPS attractors, their relation to SUSY-breaking scenarios, and potential connections to flux vacua, outlining directions for further exploration of stable non-BPS attractors in broader contexts.

Abstract

We study the attractor mechanism for extremal non-BPS black holes with an infinite throat near horizon geometry, developing, as we do so, a physical argument as to why such a mechanism does not exist in non-extremal cases. We present a detailed derivation of the non-supersymmetric attractor equation. This equation defines the stabilization of moduli near the black hole horizon: the fixed moduli take values specified by electric and magnetic charges corresponding to the fluxes in a Calabi Yau compactification of string theory. They also define the so-called double-extremal solutions. In some examples, studied previously by Tripathy and Trivedi, we solve the equation and show that the moduli are fixed at values which may also be derived from the critical points of the black hole potential.

Figures (2)

• Figure 1: On the left there is an Euclidean section of the near horizon "cigar" geometry of the non-extremal black holes with non-vanishing temperature. On the right there is an Euclidean section of an infinite throat near horizon geometry of the extremal black holes with vanishing temperature.
• Figure 2: Evolution of the dilaton field $e^{-2z}$ for various initial conditions at $\rho = 0$ ($r = \infty$) to a common fixed point at $\rho =-\infty$ ($r = 0)$ where $\rho= -{1\over r}$. The special case of the horizontal line represents a double-extremal black hole.