BPS Black Hole Degeneracies and Minimal Automorphic Representations

TL;DR

This paper explores the microscopic degeneracies of 4D and 5D BPS black holes in string theory, linking them to automorphic representations of exceptional groups and proposing a novel automorphic framework for their exact counting.

Contribution

It introduces a new automorphic representation-based approach to compute exact black hole degeneracies, connecting them to minimal unitary representations of exceptional groups like E8.

Findings

Exact degeneracies for certain 4D 1/8-BPS black holes computed.

Degeneracies related to the cubic invariant of E6 and automorphic representations.

Proposes automorphic forms as the underlying structure for black hole microstates.

Abstract

We discuss the degeneracies of 4D and 5D BPS black holes in toroidal compactifications of M-theory or type II string theory, using U-duality as a tool. We generalize the 4D/5D lift to include all charges in N=8 supergravity, and compute the exact indexed degeneracies of certain 4D 1/8-BPS black holes. Using the attractor formalism, we obtain the leading micro-canonical entropy for arbitrary Legendre invariant prepotentials and non-vanishing D6-brane charge. In particular, we find that the N=8 prepotential is given to leading order by the cubic invariant of $E_6$. This suggests that the minimal unitary representation of $E_8$, based on the same cubic prepotential, underlies the microscopic degeneracies of N=8 black holes. We propose that the exact degeneracies are given by the Wigner function of the $E_8(Z)$ invariant vector in this automorphic representation. A similar conjecture relates the degeneracies of N=4 black holes to the minimal unipotent representation of $SO(8,24,Z)$.