Black Holes and Critical Points in Moduli Space

TL;DR

The paper investigates how scalars stabilize near supersymmetric black-hole horizons within 4D theories governed by special geometry. By recasting radial dynamics as geodesic motion on an enlarged moduli space and identifying the black-hole potential with the first symplectic invariant, it shows that critical points of the central charge coincide with those of the potential, and that a positive moduli-space metric yields a unique BPS-mass minimum. It further connects these results to thermodynamic geometry, demonstrating that the moduli-space metric essentially mirrors the Weinhold metric in the extremal limit. The framework extends to extended supergravities and discusses implications for attractor behavior and possible phase transitions in the moduli space.

Abstract

We study the stabilization of scalars near a supersymmetric black hole horizon using the equation of motion of a particle moving in a potential and background metric. When the relevant 4-dimensional theory is described by special geometry, the generic properties of the critical points of this potential can be studied. We find that the extremal value of the central charge provides the minimal value of the BPS mass and of the potential under the condition that the moduli space metric is positive at the critical point. We relate these ideas to the Weinhold and Ruppeiner metrics introduced in the geometric approach to thermodynamics and used for study of critical phenomena.