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

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.

18 pages, 4 figures,