Arithmetic of Quantum Entropy Function

TL;DR

This paper explains how the quantum entropy function formalism accounts for differences in microscopic degeneracies of black hole states in N=4 string theories, highlighting the role of arithmetical properties and additional saddle points.

Contribution

It demonstrates that the quantum entropy function naturally incorporates subtle arithmetical differences in black hole microstates through orbifold saddle points.

Findings

Additional saddle points depend on black hole charge properties.

Quantum entropy function is insensitive to infrared cutoff details.

It can be used to compute both degeneracy and index.

Abstract

Quantum entropy function is a proposal for computing the entropy associated with the horizon of a black hole in the extremal limit, and is related via AdS/CFT correspondence to the dimension of the Hilbert space in a dual quantum mechanics. We show that in N=4 supersymmetric string theories, quantum entropy function formalism naturally explains the origin of the subtle differences between the microscopic degeneracies of quarter BPS dyons carrying different torsion, i.e. different arithmetical properties. These arise from additional saddle points in the path integral -- whose existence depends on the arithmetical properties of the black hole charges -- constructed as freely acting orbifolds of the original AdS_2\times S^2 near horizon geometry. During this analysis we demonstrate that the quantum entropy function is insensitive to the details of the infrared cutoff used in the computation, and the details of the boundary terms added to the action. We also discuss the role of the asymptotic symmetries of AdS_2 in carrying out the path integral in the definition of quantum entropy function. Finally we show that even though quantum entropy function is expected to compute the absolute degeneracy in a given charge and angular momentum sector, it can also be used to compute the index. This can then be compared with the microscopic computation of the index.