BPS black holes, quantum attractor flows and automorphic forms

TL;DR

This paper develops a framework for counting microstates of 4D BPS black holes in supergravity by linking attractor flows, automorphic forms, and quantum geodesic quantization, revealing deep symmetry structures.

Contribution

It introduces a novel approach connecting black hole microstate counting with automorphic forms and quantum geodesic flows in supergravity theories.

Findings

Black hole degeneracies linked to Fourier coefficients of modular forms.

Quantum attractor flows correspond to automorphic representations.

Symmetries of supergravity theories inform microstate counting methods.

Abstract

We propose a program for counting microstates of four-dimensional BPS black holes in N >= 2 supergravities with symmetric-space valued scalars by exploiting the symmetries of timelike reduction to three dimensions. Inspired by the equivalence between the four dimensional attractor flow and geodesic flow on the three-dimensional scalar manifold, we radially quantize stationary, spherically symmetric BPS geometries. Connections between the topological string amplitude, attractor wave function, the Ooguri-Strominger-Vafa conjecture and the theory of automorphic forms suggest that black hole degeneracies are counted by Fourier coefficients of modular forms for the three-dimensional U-duality group, associated to special "unipotent" representations which appear in the supersymmetric Hilbert space of the quantum attractor flow.