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Precision Counting of Small Black Holes

Atish Dabholkar, Frederik Denef, Gregory W. Moore, Boris Pioline

TL;DR

The work probes the OSV conjecture by focusing on small black holes whose microstate degeneracies are exactly computable via heterotic Dabholkar–Harvey states and their Type II duals. By combining holomorphic topological-string amplitudes (notably $F_1$) with an attractor formalism, the authors demonstrate that the macroscopic Bekenstein–Hawking–Wald entropy, including infinite higher-derivative corrections, is reproduced by the microscopic degeneracies to all orders in an inverse-charge expansion in many ${\\cal N}=4$ models, and in several ${\\cal N}=2$ constructions after accounting for instantons and monodromy. However, the agreement is not universal: in certain ${\\cal N}=2$ asymmetric orbifolds and in cases where moduli drive instanton series strongly, nonperturbative contributions and the integration measure complicate the OSV matching, signaling the need for refined ensembles or measures. The paper also develops a regulator-based framework for the black hole partition function, revealing a finite, duality-consistent structure that aligns with topological-string expectations in several regimes while highlighting remaining ambiguities in nonperturbative sectors. Overall, the results provide strong evidence that, for a broad class of small black holes, macroscopic predictions from topological amplitudes encode detailed microscopic degeneracies, with important caveats tied to duality, moduli, and nonperturbative corrections.

Abstract

It has recently been proposed that a class of supersymmetric higher-derivative interactions in N=2 supergravity may encapsulate an infinite number of finite size corrections to the microscopic entropy of certain supersymmetric black holes. If this proposal is correct, it allows one to probe the string theory description of black-hole micro-states to far greater accuracy than has been possible before. We test this proposal for ``small'' black holes whose microscopic degeneracies can be computed exactly by counting the corresponding perturbative BPS states. We also study the ``black hole partition sum'' using general properties of of BPS degeneracies. This complements and extends our earlier work in hep-th/0502157

Precision Counting of Small Black Holes

TL;DR

The work probes the OSV conjecture by focusing on small black holes whose microstate degeneracies are exactly computable via heterotic Dabholkar–Harvey states and their Type II duals. By combining holomorphic topological-string amplitudes (notably ) with an attractor formalism, the authors demonstrate that the macroscopic Bekenstein–Hawking–Wald entropy, including infinite higher-derivative corrections, is reproduced by the microscopic degeneracies to all orders in an inverse-charge expansion in many models, and in several constructions after accounting for instantons and monodromy. However, the agreement is not universal: in certain asymmetric orbifolds and in cases where moduli drive instanton series strongly, nonperturbative contributions and the integration measure complicate the OSV matching, signaling the need for refined ensembles or measures. The paper also develops a regulator-based framework for the black hole partition function, revealing a finite, duality-consistent structure that aligns with topological-string expectations in several regimes while highlighting remaining ambiguities in nonperturbative sectors. Overall, the results provide strong evidence that, for a broad class of small black holes, macroscopic predictions from topological amplitudes encode detailed microscopic degeneracies, with important caveats tied to duality, moduli, and nonperturbative corrections.

Abstract

It has recently been proposed that a class of supersymmetric higher-derivative interactions in N=2 supergravity may encapsulate an infinite number of finite size corrections to the microscopic entropy of certain supersymmetric black holes. If this proposal is correct, it allows one to probe the string theory description of black-hole micro-states to far greater accuracy than has been possible before. We test this proposal for ``small'' black holes whose microscopic degeneracies can be computed exactly by counting the corresponding perturbative BPS states. We also study the ``black hole partition sum'' using general properties of of BPS degeneracies. This complements and extends our earlier work in hep-th/0502157

Paper Structure

This paper contains 39 sections, 377 equations, 1 table.

Table of Contents

  1. Introduction
  2. The Black Hole Attractor and the OSV Conjecture
  3. Small Black Holes
  4. Summary of Main Results
  5. Outline of the Paper
  6. A Benchmark ${\cal N}=4$ Example: Type $IIA/K3\times T^2$
  7. Review of Heterotic/Type II Duality in 4 Dimensions
  8. Small Black Holes and DH States in the $Het(4,24)$ Model
  9. Asymptotic Degeneracies and the Rademacher Formula
  10. Generalized Attractor Formalism for ${\cal N}=4$ and Leading Entropy
  11. Testing the OSV Formula
  12. Degeneracies vs. Helicity Supertrace
  13. DH states in type II/$K3\times T^2$
  14. Small Black Holes in ${\cal N}=4$ Models with Reduced Rank
  15. $F_1$ in Reduced Rank Type II Models
  16. ...and 24 more sections