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Mock modularity of Calabi-Yau threefolds

Sergei Alexandrov, Khalil Bendriss

TL;DR

The paper addresses the modularity of D4-D2-D0 BPS indices on compact Calabi–Yau threefolds by showing that the generating functions $h_{r,\mu}(\tau)$ are depth $r-1$ mock modular forms obeying a modular anomaly. A two-pronged strategy is developed: (i) for simple charge configurations (two charges or all unit charges) to relate to mock modular forms of optimal growth and Vafa–Witten invariants; (ii) for general charges to construct solutions via indefinite theta series on an extended lattice with refinement, and to fix holomorphic modular ambiguities using Jacobi-like forms. This leads to explicit constructions of anomalous coefficients $g^{({\boldsymbol r})}_{\mu,{\boldsymbol \mu}}(\tau)$ for two and three charges and a general refined framework for arbitrary charges, with an unrefined limit conjectured to exist under a zero-mode condition. The work connects D4-D2-D0 BPS counting to VW invariants and mock modularity, providing a method to determine $h_{r,\mu}$ up to a finite holomorphic modular ambiguity and offering a path toward full determination via polar data. The analysis has potential applications to CYs with multiple moduli and builds bridges between BPS state counting, wall-crossing, and the theory of higher-depth mock modular forms.

Abstract

Generating functions $h_r(τ)$ of D4-D2-D0 BPS indices, appearing in Calabi-Yau compactifications of type IIA string theory and identical to rank 0 Donaldson-Thomas invariants, are known to be higher depth mock modular forms satisfying a specific modular anomaly equation, with depth determined by the D4-brane charge $r$. We develop a method to solve the anomaly equation for arbitrary charges, in terms of indefinite theta series. This allows us to find the generating functions up to modular forms that can be fixed by computing just a finite number of Fourier coefficients of $h_r$.

Mock modularity of Calabi-Yau threefolds

TL;DR

The paper addresses the modularity of D4-D2-D0 BPS indices on compact Calabi–Yau threefolds by showing that the generating functions are depth mock modular forms obeying a modular anomaly. A two-pronged strategy is developed: (i) for simple charge configurations (two charges or all unit charges) to relate to mock modular forms of optimal growth and Vafa–Witten invariants; (ii) for general charges to construct solutions via indefinite theta series on an extended lattice with refinement, and to fix holomorphic modular ambiguities using Jacobi-like forms. This leads to explicit constructions of anomalous coefficients for two and three charges and a general refined framework for arbitrary charges, with an unrefined limit conjectured to exist under a zero-mode condition. The work connects D4-D2-D0 BPS counting to VW invariants and mock modularity, providing a method to determine up to a finite holomorphic modular ambiguity and offering a path toward full determination via polar data. The analysis has potential applications to CYs with multiple moduli and builds bridges between BPS state counting, wall-crossing, and the theory of higher-depth mock modular forms.

Abstract

Generating functions of D4-D2-D0 BPS indices, appearing in Calabi-Yau compactifications of type IIA string theory and identical to rank 0 Donaldson-Thomas invariants, are known to be higher depth mock modular forms satisfying a specific modular anomaly equation, with depth determined by the D4-brane charge . We develop a method to solve the anomaly equation for arbitrary charges, in terms of indefinite theta series. This allows us to find the generating functions up to modular forms that can be fixed by computing just a finite number of Fourier coefficients of .

Paper Structure

This paper contains 58 sections, 9 theorems, 284 equations, 2 figures.

Table of Contents

  1. Introduction
  2. BPS indices and their modular anomaly
  3. D4-D2-D0 BPS indices and their generating series
  4. Modular symmetry
  5. Redefinition
  6. Anomalous coefficients
  7. Partial solutions
  8. Two charges and Hecke-like operators
  9. Unit charges and Vafa-Witten
  10. Solution via indefinite theta series
  11. Motivation and strategy
  12. Refinement
  13. Lattice extension
  14. Lattice factorization
  15. Two charges
  16. ...and 43 more sections

Key Result

Theorem 3.1

Let $g^{(r)}_{\mu,\mu'}=\delta_{\mu,\mu'}$ and $\tilde{h}^{(0)}_{r,\mu}$ be a set of holomorphic modular forms. Then is a depth $r-1$ modular form with completion of the form exp-twhh provided $g^{({\boldsymbol r})}_{\mu,{\boldsymbol \mu}}$ are depth $n-1$ mock modular forms (where $n$ is the number of charges $r_i$) with completions satisfying whereNote that while the sets ${\boldsymbol r}$ and

Figures (2)

  • Figure 1: A representation of contributions to the r.h.s. of \ref{['compl-gi']} in terms of rooted trees of depth 2.
  • Figure 2: Construction of the refined anomalous coefficients through the refinement and lattice extension and their relation to the generating functions of BPS indices.

Theorems & Definitions (15)

  • Theorem 3.1
  • proof
  • Theorem 5.1
  • Conjecture 5.1
  • Theorem 5.2
  • proof
  • Proposition A.1
  • Proposition A.2
  • proof
  • Theorem C.1
  • ...and 5 more