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Mock Modularity Of Twisted Index In CHL Models

Nabamita Banerjee, Vedant Bhutra, Ranveer Kumar Singh

Abstract

We study the twisted partition function of quarter BPS states in CHL models and show that for a large class of single-centered black holes, the degeneracy of microstates is given by the Fourier coefficients of mock Jacobi forms. Our analysis is a continuation of the programme initiated by Dabholkar, Murthy and Zagier (DMZ) for $1/4$-BPS dyons in $\mathcal{N} = 4$ string theory and further extended by Bhand, Sen and Singh (BSS) to quarter BPS states in CHL models. We also present the multiplicative lift construction of the partition function and comment on the additive lift of the same.

Mock Modularity Of Twisted Index In CHL Models

Abstract

We study the twisted partition function of quarter BPS states in CHL models and show that for a large class of single-centered black holes, the degeneracy of microstates is given by the Fourier coefficients of mock Jacobi forms. Our analysis is a continuation of the programme initiated by Dabholkar, Murthy and Zagier (DMZ) for -BPS dyons in string theory and further extended by Bhand, Sen and Singh (BSS) to quarter BPS states in CHL models. We also present the multiplicative lift construction of the partition function and comment on the additive lift of the same.

Paper Structure

This paper contains 26 sections, 8 theorems, 164 equations, 5 tables.

Table of Contents

  1. Introduction
  2. Mathematical preliminaries
  3. Conventions.
  4. Jacobi and Siegel modular forms
  5. Mock modular forms
  6. Finite and polar parts
  7. Mock Jacobi forms from twisted partition function in CHL models
  8. The system.
  9. Twisted-twined partition function
  10. Invariance groups of $\Phi^{(M,N)}_k$
  11. Fourier-Jacobi coefficients of the partition function
  12. Single-centered degeneracies
  13. Dyon partition function through lifts
  14. Multiplicative and additive lifts
  15. Twisted-twined elliptic genus and Mathieu Moonshine
  16. ...and 11 more sections

Key Result

Theorem 2.6

Let $\varphi(\tau, z)$ be a meromorphic Jacobi form with simple poles at $z_s$ for all $s \in S(\varphi)$, some discrete subset of $\mathds{Q}^2$, then $\varphi$ has the decomposition where $\varphi^F$ and $\varphi^P$ are as defined above.

Theorems & Definitions (21)

  • Definition 2.1: Jacobi form on $\Gamma \ltimes (M\mathds{Z} \times \mathds{Z})$ of weight $k$ and index $m/M$
  • Definition 2.2: Siegel modular form on $G < \mathrm{Sp}(4,\mathds{Z})$ of weight $k$
  • Definition 2.3: Pure mock modular form
  • Definition 2.4: Pure mock Jacobi form
  • Definition 2.5: Mixed mock modular forms
  • Theorem 2.6: dabholkar2012quantumbhand2023mock
  • Theorem 2.7: dabholkar2012quantumbhand2023mock
  • proof
  • Theorem 3.1
  • proof
  • ...and 11 more