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Mock modularity of log Gromov--Witten Invariants: the mirror to $\mathbb{P}^2$

Hülya Argüz

TL;DR

The article investigates the emergence of mock modular forms in the generating series of logarithmic Gromov–Witten invariants for elliptic fibrations relative to singular fibers, proposing that these series are mock modular and proving this for a broad class in the rational elliptic surface mirror to $\mathbb{P}^2$. It builds a bridge between log GW theory (via the Gross–Siebert program) and Vafa–Witten invariants, establishing a precise correspondence on the mirror pair that transfers modularity properties from VW to log GW counts. Concretely, it proves that the log GW generating series for the mirror $(Y,D)$ to $\mathbb{P}^2$ coincides with VW generating series in a regime, implying their mock modularity with weight $-\tfrac{3}{2}$ and depth matching the rank. The results provide a robust mathematical realization of VW-inspired mock modularity in log GW theory, and they predict deeper mock modular structures for higher ranks, connecting enumerative geometry, mirror symmetry, and four-dimensional gauge theory.

Abstract

We study modularity properties of generating series of logarithmic Gromov-Witten invariants of elliptic fibrations relative to singular fibers. Motivated by predictions from Vafa-Witten theory, we conjecture that such generating series are mock modular forms. We prove this conjecture for a large class of invariants of the rational elliptic surface mirror to $\mathbb{P}^2$, relative to a cycle of nine rational curves. The proof uses a correspondence between log Gromov-Witten invariants of the mirror and Vafa-Witten invariants of $\mathbb{P}^2$ established in previous work joint with Bousseau, together with known mock modularity results on the Vafa-Witten side.

Mock modularity of log Gromov--Witten Invariants: the mirror to $\mathbb{P}^2$

TL;DR

The article investigates the emergence of mock modular forms in the generating series of logarithmic Gromov–Witten invariants for elliptic fibrations relative to singular fibers, proposing that these series are mock modular and proving this for a broad class in the rational elliptic surface mirror to . It builds a bridge between log GW theory (via the Gross–Siebert program) and Vafa–Witten invariants, establishing a precise correspondence on the mirror pair that transfers modularity properties from VW to log GW counts. Concretely, it proves that the log GW generating series for the mirror to coincides with VW generating series in a regime, implying their mock modularity with weight and depth matching the rank. The results provide a robust mathematical realization of VW-inspired mock modularity in log GW theory, and they predict deeper mock modular structures for higher ranks, connecting enumerative geometry, mirror symmetry, and four-dimensional gauge theory.

Abstract

We study modularity properties of generating series of logarithmic Gromov-Witten invariants of elliptic fibrations relative to singular fibers. Motivated by predictions from Vafa-Witten theory, we conjecture that such generating series are mock modular forms. We prove this conjecture for a large class of invariants of the rational elliptic surface mirror to , relative to a cycle of nine rational curves. The proof uses a correspondence between log Gromov-Witten invariants of the mirror and Vafa-Witten invariants of established in previous work joint with Bousseau, together with known mock modularity results on the Vafa-Witten side.
Paper Structure (14 sections, 3 theorems, 43 equations, 4 figures)

This paper contains 14 sections, 3 theorems, 43 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Background and context
  3. Main results
  4. Mirror symmetry and log Gromov--Witten theory
  5. The Gross--Siebert program
  6. Log Gromov--Witten invariants
  7. The rational elliptic surface $(Y,D)$ mirror to $\mathbb{P}^2$
  8. The log Gromov--Witten/Vafa--Witten correspondence
  9. Vafa--Witten invariants of $\mathbb{P}^2$
  10. The log Gromov--Witten/Vafa--Witten correspondence
  11. Mock modularity of log Gromov--Witten generating series
  12. Modularity and mock modularity
  13. Mock modularity and Vafa--Witten invariants of $\mathbb{P}^2$.
  14. Mock modularity of log Gromov--Witten invariants of the mirror $(Y,D)$ to $\mathbb{P}^2$

Key Result

Theorem 4.1

For any $\gamma = (r,c_1,c_2) \in \mathbb{Z}^3$ such that $r > 0$ and $-1 < \frac{c_1}{r} \leq 0$, we have where the contact order $v_{\gamma}\in \mathbb{Z}^2$, and the curve class $\beta_{\gamma} \in H_2(Y,\mathbb{Z})$ are given as in eq:vgamma2 and eq:betagamma, respectively.

Figures (4)

  • Figure 1.1: A rational elliptic surface $Y$ and a curve with tangencies relative to a smooth fiber $D$.
  • Figure 1.2: The rational elliptic surface $Y$, mirror to $\mathbb{P}^2$, together with a curve having prescribed tangency to the singular fiber $D$.
  • Figure 3.1: The fan $\Sigma \subset \mathbb{R}^2$ for $\overline{Y}$.
  • Figure 3.2: An anti-canonical curve $\overline{C}$ in $\overline{Y}$.

Theorems & Definitions (12)

  • Conjecture 1.1
  • Theorem 4.1
  • Definition 5.1
  • Example 5.2
  • Definition 5.3
  • Definition 5.4
  • Definition 5.5
  • Definition 5.6
  • Theorem 5.7
  • proof
  • ...and 2 more