A Farey tale for N=4 dyons

TL;DR

This paper explores the contributions of semi-classical AdS2 geometries to black hole degeneracies, connecting Farey tail expansions with modular forms in string theory, and providing new insights into dyon partition functions.

Contribution

It introduces a Farey tale expansion for dyon partition functions, linking semi-classical geometries to modular forms and extending the understanding of black hole microstates.

Findings

Identifies infinite families of semi-classical AdS2 geometries contributing to black hole entropy.

Relates Farey tail expansion to the sum over poles in Siegel modular forms.

Provides a formal mathematical lift from Hilbert to Siegel modular forms with poles.

Abstract

We study exponentially suppressed contributions to the degeneracies of extremal black holes. Within Sen's quantum entropy function framework and focusing on extremal black holes with an intermediate AdS3 region, we identify an infinite family of semi-classical AdS2 geometries which can contribute effects of order exp(S_0/c), where S_0 is the Bekenstein-Hawking-Wald entropy and c is an integer greater than one. These solutions lift to the extremal limit of the SL(2,Z) family of BTZ black holes familiar from the "black hole Farey tail". We test this understanding in N=4 string vacua, where exact dyon degeneracies are known to be given by Fourier coefficients of Siegel modular forms. We relate the sum over poles in the Siegel upper half plane to the Farey tail expansion, and derive a "Farey tale" expansion for the dyon partition function. Mathematically, this provides a (formal) lift from Hilbert modular forms to Siegel modular forms with a pole at the diagonal divisor.