BPS black holes, quantum attractor flows and automorphic forms

TL;DR

The work addresses the problem of exactly counting BPS black hole microstates in ${\mathcal N}\ge 2$ supergravities by exploiting timelike reduction to three dimensions, which turns 4D attractor flows into geodesic motion on $M_3^*=G_3/H_3$ and allows radial quantization. It identifies the three-dimensional U-duality group $G_3$ as a spectrum-generating symmetry, with BPS states arising from quantizing small nilpotent coadjoint orbits ${\mathcal O}_i$ into unipotent representations ${\mathcal H}_i$ of $G_3$. Automorphic forms for $G_3$ built from these representations have Fourier coefficients along the unipotent radical that are proposed to count 4D black hole degeneracies, linking to the topological string via a distinguished automorphic vector whose near-horizon limit mirrors a tree-level amplitude $\Psi_{\mathrm{top}}$. The framework offers an OSV-like perspective where degeneracies are encoded in products of real and adelic data rather than the squared topological-string amplitude, and it suggests deep connections between black hole microphysics, automorphic forms, and Calabi–Yau geometry with potential extensions to higher dimensions and more general black hole solutions.

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.

Figures (1)

• Figure 1: Number of supercharges, vector fields in $d=4$, moduli spaces before and after $d=4 \to 3$ reduction along a timelike Killing vector, and associated Jordan algebra. Real forms of exceptional groups are indicated by the notation $E_{n(m)}$ where $m$ is the difference between the number of non-compact and compact generators.