U-duality Invariant Quantum Entropy from Sums of Kloosterman Sums

Abstract

U-duality plays a special role in the study of the microscopic degrees of freedom of supersymmetric black holes. To be consistent with duality, the black hole quantum degeneracy must obey special arithmetic properties, which are non-perturbative in nature. In this work, we study these properties from a holographic point of view, establishing a connection between arithmetic properties of Kloosterman sums and quantum gravity in ${\text{AdS}}_2$ space. To this end, we consider the entropy of black holes that carry non-primitive charges, in both $\mathcal{N}=8$ and $\mathcal{N}=4$ four dimensional compactifications; our analysis includes all the perturbative and non-perturbative bulk quantum corrections. The key result relies on special arithmetic properties of generalized Kloosterman sums that we develop. These are a generalization of the known Selberg identity of classical Kloosterman sums. In both the $\mathcal{N}=8$ and $\mathcal{N}=4$ examples, we recover, from the bulk quantum gravity, the non-primitive answer which is a sum over the primitive degeneracies, depending non-trivially on the discrete duality invariants. In particular, for the $\mathcal{N}=4$ case we show that the quantum gravity answer reproduces the dependence on the torsion invariant $I=\text{gcd}(Q\wedge P)$, in agreement with the microscopic formulas. For the $\mathcal{N}=8$ case, we solve a puzzle related to U-duality invariance of the supergravity answer and the corresponding one-eighth BPS degeneracy.

arXiv comment44 pages

References in corpus (21)

• Strong-Weak Coupling Duality in Four Dimensional String Theory
• Quantum Entropy Function from AdS(2)/CFT(1) Correspondence
• Microscopic Black Hole Entropy in Theories with Higher Derivatives
• Quantum Foam and Topological Strings
• Quantum Black Holes, Wall Crossing, and Mock Modular Forms
• Precision Counting of Small Black Holes
• A Black Hole Farey Tail
• CHL Dyons and Statistical Entropy Function from D1-D5 System
• Counting BPS Blackholes in Toroidal Type II String Theory
• Nonperturbative black hole entropy and Kloosterman sums
• Comments on the Spectrum of CHL Dyons
• Supersymmetric Index from Black Hole Entropy
• Black hole bound states in AdS_3 x S^2
• A Farey tale for N=4 dyons
• All solutions of the localization equations for N=2 quantum black hole entropy
• Generalities of Quarter BPS Dyon Partition Function and Dyons of Torsion Two
• Arithmetic of N=8 Black Holes
• U-duality Invariant Dyon Spectrum in type II on T^6
• Single-centered black hole microstate degeneracies from instantons in supergravity
• Quantum Black Hole Entropy, Localization and the Stringy Exclusion Principle
• Generalized Kloosterman Sums from M2-branes

Cited by in corpus (1)

• On the exact entropy of $\mathcal{N}=2$ black holes