Quantum Black Holes, Wall Crossing, and Mock Modular Forms

TL;DR

This paper explores the decomposition of meromorphic Jacobi forms related to BPS black hole states in string theory, revealing connections to mock modular forms, wall-crossing phenomena, and holography.

Contribution

It introduces a canonical decomposition of meromorphic Jacobi forms into mock Jacobi forms and Appell-Lerch sums, linking black hole degeneracies to mock modular forms and wall-crossing.

Findings

Decomposition of Jacobi forms into mock forms and Appell-Lerch sums.

Identification of special mock Jacobi forms for each magnetic charge.

Connection of these forms to known mock modular forms and moonshine phenomena.

Abstract

We show that the meromorphic Jacobi form that counts the quarter-BPS states in N=4 string theories can be canonically decomposed as a sum of a mock Jacobi form and an Appell-Lerch sum. The quantum degeneracies of single-centered black holes are Fourier coefficients of this mock Jacobi form, while the Appell-Lerch sum captures the degeneracies of multi-centered black holes which decay upon wall-crossing. The completion of the mock Jacobi form restores the modular symmetries expected from $AdS_3/CFT_2$ holography but has a holomorphic anomaly reflecting the non-compactness of the microscopic CFT. For every positive integral value m of the magnetic charge invariant of the black hole, our analysis leads to a special mock Jacobi form of weight two and index m, which we characterize uniquely up to a Jacobi cusp form. This family of special forms and another closely related family of weight-one forms contain almost all the known mock modular forms including the mock theta functions of Ramanujan, the generating function of Hurwitz-Kronecker class numbers, the mock modular forms appearing in the Mathieu and Umbral moonshine, as well as an infinite number of new examples.