BPS Black Hole Degeneracies and Minimal Automorphic Representations

Original · EN

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 E₆. This suggests that the minimal unitary representation of E₈, 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 E₈(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).