The Attractor Mechanism in Five Dimensions

TL;DR

This work provides a pedagogical account of the attractor mechanism in five-dimensional $N=2$ supergravity arising from M-theory on Calabi–Yau three-folds, emphasizing real special geometry and its role in fixing moduli at black hole horizons. It derives both central-charge–driven gradient flows and explicit attractor formulas for spherically symmetric BPS black holes, relating horizon values of the moduli to electric charges and presenting the entropy in terms of a cubic invariant of the charges. The text then extends the mechanism to non-spherical, supersymmetric configurations including black rings, black strings, and multi-center solutions, highlighting the appearance of magnetic charges via dipoles and the associated near-horizon geometries ($AdS_2\times S^3$ for holes, $AdS_3\times S^2$ for rings) and supersymmetry enhancement. An alternative extremization viewpoint is developed, showing how the attractor values can follow from minimizing an effective potential or entropy-like functionals, and discussing the connections to thermodynamics and split attractor flows. Collectively, these results illuminate how horizon data in diverse extremal configurations are governed by extremization principles tied to the central charges and geometric invariants, with implications for microphysical interpretations and black object thermodynamics.

Abstract

We give a pedagogical introduction to the attractor mechanism. We begin by developing the formalism for the simplest example of spherically symmetric black holes in five dimensions which preserve supersymmetry. We then discuss the refinements needed when spherical symmetry is relaxed. This is motivated by rotating black holes and, especially, black rings. An introduction to non-BPS attractors is included, as is a discussion of thermodynamic interpretations of the attractor mechanism.