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

TL;DR

This paper investigates the modularity properties of generating series of logarithmic Gromov-Witten invariants in elliptic fibrations, conjecturing they are mock modular forms and proving this for certain cases related to the mirror of P^2.

Contribution

It establishes the mock modularity of these generating series for a class of invariants of the rational elliptic surface mirror to P^2, connecting Gromov-Witten and Vafa-Witten invariants.

Findings

Proves the mock modularity conjecture for specific invariants.

Establishes a correspondence between log Gromov-Witten and Vafa-Witten invariants.

Demonstrates the modularity properties in the context of elliptic fibrations.

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.