BPS Black Hole Degeneracies and Minimal Automorphic Representations

TL;DR

The paper investigates subleading microstate counts for 4D/5D BPS black holes in maximal string/M-theory compactifications by leveraging U-duality and a 4D/5D lift. It shows the leading entropy is governed by the cubic invariant $I_3$ of $E_6$ and proposes that exact degeneracies are captured by the Wigner function of an $E_8(\mathbb{Z})$-invariant distribution in the minimal unipotent representation of $E_8$, i.e., an automorphic $E_8$ theta series. The work extends the lift to all charges, derives Legendre-invariant prepotentials, and analyzes how subleading corrections might be encoded by automorphic data, with parallel constructions for ${\cal N}=4$ via $D_{16}$. It moreover connects these black-hole partition functions to conformal quantum mechanics and suggests testable predictions and avenues for generalization to ${\cal N}=2$ scenarios, highlighting a potential unifying role for minimal representations in black-hole microphysics.

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 $SO(8,24,Z)$.