BPS Black Hole Degeneracies and Minimal Automorphic Representations

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 $SO(8,24,Z)$.

36 pages, uses JHEP3.cls; v5: the "extra charge", which was, mistakenly identified as angular momentum, is now correctly identified as NUT charge (see historical note on p29), a few other cosmetic changes too

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• Exact Black Hole Degeneracies and the Topological String
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• A Unified Approach to the Minimal Unitary Realizations of Noncompact Groups and Supergroups
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• Observations on Arithmetic Invariants and U-Duality Orbits in N =8 Supergravity
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• The magic three-qubit Veldkamp line: A finite geometric underpinning for form theories of gravity and black hole entropy
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