Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quantum Black Holes, Wall Crossing, and Mock Modular Forms

Atish Dabholkar, Sameer Murthy, Don Zagier

TL;DR

This work establishes a canonical decomposition of meromorphic Jacobi forms counting quarter-BPS dyons into a finite part, which is a mock Jacobi form, and a polar Appell–Lerch part capturing wall-crossing; the finite part admits a modular completion whose shadow encodes the noncompact microscopic CFT and holographic structure. It develops a comprehensive theory of mock Jacobi forms, including precise structure theorems, optimal-growth constructions, and Hecke-like operators, and identifies two key families: weight-2 $oldsymbol ext{Phi}_{2,m}$ and weight-1 $oldsymbol ext{Phi}_{1,m}$, whose special instances recover Ramanujan mock theta functions, Hurwitz–Kronecker class numbers, and modules linked to Mathieu and Umbral moonshine. The results connect black-hole degeneracies to mock modular phenomena, providing both exact decompositions and asymptotic growth via Hardy-Ramanujan-type formulas, with implications for ${ m AdS}_3/{ m CFT}_2$ holography, wall-crossing, and noncompact CFTs. Collectively, the paper builds a rich mathematical framework tying mock modular forms, Jacobi forms, and Siegel modular forms to the microstate counting of immortal black holes and to deep structures in number theory and moonshine.

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.

Quantum Black Holes, Wall Crossing, and Mock Modular Forms

TL;DR

This work establishes a canonical decomposition of meromorphic Jacobi forms counting quarter-BPS dyons into a finite part, which is a mock Jacobi form, and a polar Appell–Lerch part capturing wall-crossing; the finite part admits a modular completion whose shadow encodes the noncompact microscopic CFT and holographic structure. It develops a comprehensive theory of mock Jacobi forms, including precise structure theorems, optimal-growth constructions, and Hecke-like operators, and identifies two key families: weight-2 and weight-1 , whose special instances recover Ramanujan mock theta functions, Hurwitz–Kronecker class numbers, and modules linked to Mathieu and Umbral moonshine. The results connect black-hole degeneracies to mock modular phenomena, providing both exact decompositions and asymptotic growth via Hardy-Ramanujan-type formulas, with implications for holography, wall-crossing, and noncompact CFTs. Collectively, the paper builds a rich mathematical framework tying mock modular forms, Jacobi forms, and Siegel modular forms to the microstate counting of immortal black holes and to deep structures in number theory and moonshine.

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 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.

Paper Structure

This paper contains 49 sections, 26 theorems, 432 equations, 1 figure, 5 tables.

Table of Contents

  1. Introduction
  2. Introduction for mathematicians
  3. Organization of the paper
  4. Review of Type-II superstring theory on $K3 \times T^2$
  5. Modular forms in one variable
  6. Basic definitions and properties
  7. Quantum black holes and modular forms
  8. Jacobi forms
  9. Basic definitions
  10. Theta expansion and Taylor expansion
  11. Example: Jacobi forms of index $1$
  12. Hecke-like operators
  13. Quantum black holes and Jacobi forms
  14. Siegel modular forms
  15. Definitions and examples of Siegel modular forms
  16. ...and 34 more sections

Key Result

Proposition 8.1

For $c\in{\mathbb R}$ and $z\in{\mathbb C}\smallsetminus{\mathbb Z}$ we have

Figures (1)

  • Figure 1: The diagram on the left shows the moduli space projected onto the $\Sigma$ upper-half plane divided into chambers separated by walls where a quarter-BPS state decays into two half-BPS states. The M-theory limit, shown on the right, corresponds to taking $\hbox{Im}(\Sigma)$ to be very large. In this limit several walls are no longer visible. There are still an infinite number of walls which can be crossed by varying $\hbox{Re}(\Sigma)$. Spectral flow transformation maps one chamber to another chamber.

Theorems & Definitions (31)

  • Proposition 8.1
  • Theorem 8.1
  • Theorem 8.2
  • Proposition 8.2
  • Proposition 8.3
  • Proposition 8.4
  • Theorem 8.3
  • Theorem 9.1
  • Theorem 9.2
  • Corollary
  • ...and 21 more