Superconformal Multi-Black Hole Quantum Mechanics

TL;DR

The paper derives the moduli-space metric for $N$ slowly moving BPS black holes in five-dimensional $\mathcal{N}=1$ supergravity and shows the low-energy dynamics possess $\mathcal{N}=4B$ supersymmetry. In the near-horizon limit, the moduli space decouples into a strongly interacting region that exhibits an enhanced $D(2,1;0)$ superconformal symmetry; because the Hamiltonian $H$ has noncompact directions, the authors promote to $L_0=\tfrac{1}{2}(H+K)$, yielding a discrete, normalizable spectrum. The near-horizon quantum mechanics can be interpreted as internal states of a composite black hole with total charge, potentially encoding black hole entropy and offering insights into AdS$_2$/CFT$_1$ via an $M$-theory/CALABI–Yau framework. The work provides a concrete, supersymmetric quantum-mechanical model for multi-black-hole dynamics and highlights the delicate role of conformal symmetry in achieving a well-defined Hilbert space.

Abstract

The quantum mechanics of N slowly-moving charged BPS black holes in five-dimensional ${\cal N}=1$ supergravity is considered. The moduli space metric of the N black holes is derived and shown to admit 4 supersymmetries. A near-horizon limit is found in which the dynamics of widely separated black holes decouples from that of strongly-interacting, near-coincident black holes. This decoupling suggests that the quantum states supported in the near-horizon moduli space can be interpreted as internal states of a single composite black hole carrying all of the charge. The near-horizon theory is shown to have an enhanced D(2,1;0) superconformal symmetry. Eigenstates of the Hamiltonian H of the near-horizon theory are ill-defined due to noncompact regions of the moduli space corresponding to highly redshifted near-coincident black holes. It is argued that one should consider, instead of H eigenstates, eigenstates of $2 L_0 = H+K$, where K is the generator of special conformal transformations. The result is a well-defined Hilbert space with a discrete spectrum describing the N-black hole dynamics.