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Mock Modularity In CHL Models

Ajit Bhand, Ashoke Sen, Ranveer Kumar Singh

TL;DR

This work extends the Dabholkar-Murthy-Zagier (DMZ) framework from integral to rational indices by formulating a robust theory of meromorphic Jacobi forms of index m/N for congruence subgroups Γ, including a fractional-index Appell-Lerch apparatus. The authors establish that any such form φ decomposes canonically as φ^P+φ^F, where φ^F is a mixed mock Jacobi form, with the polar part governed by universal Appell-Lerch sums adapted to rational indices. They apply this to CHL model partition functions, proving that a wide class of single-centered black hole degeneracies are Fourier coefficients of φ^F or its companion ψ^F, and provide precise charge-sets A and B capturing when mock modularity describes degeneracies. The paper also demonstrates limitations of the DMZ proposal by constructing explicit charges outside A,B, analyzes wall-crossing, and extends the mock-modularity framework to Z_M×Z_N CHL orbifolds, offering a uniform, modularly robust description of dyon degeneracies across a broad family of CHL models.

Abstract

Dabholkar, Murthy and Zagier (DMZ) proved that there is a canonical decomposition of a meromorphic Jacobi form of integral index for $\mathrm{SL}(2, \mathbb{Z})$ with poles on torsion points into polar and finite parts, and showed that the finite part is a mock Jacobi form. In this paper we generalize the results of DMZ to meromorphic Jacobi forms of rational index for congruence subgroups of $\mathrm{SL}(2, \mathbb{Z})$. As an application, we establish that a large class of single-centered black hole degeneracies in CHL models are given by the Fourier coefficients of mock Jacobi forms. In this process we refine the result of DMZ regarding the set of charges for which the single-centered black hole degeneracies are given by a mock modular form. In particular, in the case studied by DMZ, we present examples of charges for which the single-centered degeneracies are not captured by the mock modular form of the expected index.

Mock Modularity In CHL Models

TL;DR

This work extends the Dabholkar-Murthy-Zagier (DMZ) framework from integral to rational indices by formulating a robust theory of meromorphic Jacobi forms of index m/N for congruence subgroups Γ, including a fractional-index Appell-Lerch apparatus. The authors establish that any such form φ decomposes canonically as φ^P+φ^F, where φ^F is a mixed mock Jacobi form, with the polar part governed by universal Appell-Lerch sums adapted to rational indices. They apply this to CHL model partition functions, proving that a wide class of single-centered black hole degeneracies are Fourier coefficients of φ^F or its companion ψ^F, and provide precise charge-sets A and B capturing when mock modularity describes degeneracies. The paper also demonstrates limitations of the DMZ proposal by constructing explicit charges outside A,B, analyzes wall-crossing, and extends the mock-modularity framework to Z_M×Z_N CHL orbifolds, offering a uniform, modularly robust description of dyon degeneracies across a broad family of CHL models.

Abstract

Dabholkar, Murthy and Zagier (DMZ) proved that there is a canonical decomposition of a meromorphic Jacobi form of integral index for with poles on torsion points into polar and finite parts, and showed that the finite part is a mock Jacobi form. In this paper we generalize the results of DMZ to meromorphic Jacobi forms of rational index for congruence subgroups of . As an application, we establish that a large class of single-centered black hole degeneracies in CHL models are given by the Fourier coefficients of mock Jacobi forms. In this process we refine the result of DMZ regarding the set of charges for which the single-centered black hole degeneracies are given by a mock modular form. In particular, in the case studied by DMZ, we present examples of charges for which the single-centered degeneracies are not captured by the mock modular form of the expected index.
Paper Structure (22 sections, 20 theorems, 339 equations, 1 figure)

This paper contains 22 sections, 20 theorems, 339 equations, 1 figure.

Table of Contents

  1. Introduction and Summary
  2. Introduction
  3. Summary of results
  4. Mathematical preliminaries
  5. Mock modular forms
  6. Weak Jacobi forms
  7. Siegel modular forms
  8. Results of Dabholkar-Murthy-Zagier
  9. Generalisation to certain rational index meromorphic Jacobi forms
  10. The Space $J_{k,\frac{m}{N}}^N(\Gamma)$
  11. Universal Appell-Lerch sum of index $m/N$
  12. Mock Jacobi forms from CHL model partition functions
  13. The CHL Partition Function
  14. Single-centered degeneracy and mock Jacobi forms
  15. Comparison with Dabholkar-Murthy-Zagier
  16. ...and 7 more sections

Key Result

Proposition 3.2

Let $\varphi$ be a meromorphic Jacobi forms with simple poles at $z=z_s=\alpha\tau+\beta$ for $s=(\alpha,\beta)\in S(\varphi)\subset\mathbb{Q}^2$. Then the residues $D_s(\tau)$ defined in eq:1resn=1 satisfies the following properties: where $s\gamma=(a\alpha+c\beta,b\alpha+d\beta).$

Figures (1)

  • Figure 1: Contour $C(P,\tau,\ell)$ for the integral

Theorems & Definitions (46)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 3.1
  • Proposition 3.2
  • proof
  • ...and 36 more