Nonperturbative black hole entropy and Kloosterman sums

TL;DR

The paper shows that nonperturbative corrections to supersymmetric black hole entropy, encoded by Kloosterman sums, arise naturally from a bulk $AdS_{2}$ localization path integral supplemented by topological boundary data. By organizing subleading saddle points as $SL(2,Z)$ Dehn-filled orbifolds and evaluating boundary Chern-Simons terms and Wilson lines, the authors reproduce the full Kloosterman phase structure and derive an explicit multiplier system governing the generalized sums. This work connects topology, number theory, and quantum gravity, offering a concrete holographic realization of nonperturbative black-hole entropy and suggesting deep links with knot theory via Chern-Simons surgery. The results mark a significant step toward computable quantum holography in which both bulk and boundary sectors admit exact quantum corrections and integrality constraints consistent with microscopic degeneracies.

Abstract

Non-perturbative quantum corrections to supersymmetric black hole entropy often involve nontrivial number-theoretic phases called Kloosterman sums. We show how these sums can be obtained naturally from the functional integral of supergravity in asymptotically AdS_2 space for a class of black holes. They are essentially topological in origin and correspond to charge-dependent phases arising from the various gauge and gravitational Chern-Simons terms and boundary Wilson lines evaluated on Dehn-filled solid 2-torus. These corrections are essential to obtain an integer from supergravity in agreement with the quantum degeneracies, and reveal an intriguing connection between topology, number theory, and quantum gravity. We give an assessment of the current understanding of quantum entropy of black holes.