Quantum Attractor Flows

TL;DR

This paper explores the quantum structure of BPS black holes in N=2 supergravity by relating radial evolution to geodesic motion on moduli space, and quantizing the BPS phase space via twistor space sheaf cohomology.

Contribution

It establishes a geometric and quantum framework for BPS black holes using twistor space and geodesic correspondence, extending previous classical and holographic analyses.

Findings

Identifies BPS black holes with holomorphic geodesics on twistor space.

Provides a quantization scheme for the BPS phase space using sheaf cohomology.

Calculates the exact wave function of BPS black holes with fixed charges.

Abstract

Motivated by the interpretation of the Ooguri-Strominger-Vafa conjecture as a holographic correspondence in the mini-superspace approximation, we study the radial quantization of stationary, spherically symmetric black holes in four dimensions. A key ingredient is the classical equivalence between the radial evolution equation and geodesic motion of a fiducial particle on the moduli space M^*_3 of the three-dimensional theory after reduction along the time direction. In the case of N=2 supergravity, M^*_3 is a para-quaternionic-Kahler manifold; in this case, we show that BPS black holes correspond to a particular class of geodesics which lift holomorphically to the twistor space Z of M^*_3, and identify Z as the BPS phase space. We give a natural quantization of the BPS phase space in terms of the sheaf cohomology of Z, and compute the exact wave function of a BPS black hole with fixed electric and magnetic charges in this framework. We comment on the relation to the topological string amplitude, extensions to N>2 supergravity theories, and applications to automorphic black hole partition functions.