Lectures on Superconformal Quantum Mechanics and Multi-Black Hole Moduli Spaces

TL;DR

The paper addresses the low-energy quantum mechanics of multiple supersymmetric black holes by formulating dynamics on their moduli space $\mathcal{M}_N$ and uncovering conformal and superconformal symmetries, notably $SL(2,\mathbb{R})$ and $D(2,1;\alpha)$. It develops the conformal and superconformal structure for $N$-particle systems, examines a test particle in a Reissner-Nordström background as a warm-up, and analyzes the five-dimensional black-hole moduli space with a detailed near-horizon limit revealing an $AdS_2\times S^3$ geometry and an extended ${\cal N}=4$ superconformal symmetry. A key technical insight is the DFF trick, which replaces the Hamiltonian with a well-behaved $L_0=\tfrac{1}{2}(H+K)$ to produce a discrete spectrum and resolve infrared issues associated with noncompact moduli spaces. The work proposes that near-horizon multi-black-hole dynamics decouples from the asymptotic sector and may be dual to aspects of the Higgs branch of multiply wrapped M2-branes, offering a potential route toward understanding black-hole microstates via $AdS_2/CFT_1$ ideas.

Abstract

This contribution to the proceedings of the 1999 NATO ASI on Quantum Geometry at Akureyri, Iceland, is based on notes of lectures given by A. Strominger. Topics include N-particle conformal quantum mechanics, extended superconformal quantum mechanics and multi-black hole moduli spaces.

Figures (4)

• Figure 1: A comparison between the potentials for $H$ and $L_0$. The dashed line is the potential energy part of $H$ and the solid line is that for $L_0$. Note that the former has no minimum while the latter is a well.
• Figure 2: The geometry of $AdS_2.$ The time conjugate to $h+k$ is a good global coordinate.
• Figure 3: (a) Widely separated black holes. (b) Near-coincident black holes. (c) The near-horizon limit.
• Figure 4: (a) Regions of the two-black hole moduli space. (b) The near-horizon limit.