Enumerative Geometry of Quantum Periods

TL;DR

This paper explores the quantum geometry of log Calabi-Yau surfaces, connecting quantum periods, mirror symmetry, and Gromov-Witten invariants, revealing new relations and discrepancies at higher genus.

Contribution

It introduces a $q$-refined theta function as a quantum period, extends the relation between open and closed invariants, and generalizes existing correspondences to arbitrary genus and winding.

Findings

Explicit relation between quantum periods and all-genus invariants.

Discrepancy at higher genus in open Gromov-Witten invariants.

Correspondence between open invariants and closed Gopakumar-Vafa invariants.

Abstract

We interpret the $q$-refined theta function $\vartheta_1$ of a log Calabi-Yau surface $(\mathbb{P},E)$ as a natural $q$-refinement of the open mirror map, defined by quantum periods of mirror curves for outer Aganagic-Vafa branes on the local Calabi-Yau $K_{\mathbb{P}}$. The series coefficients are all-genus logarithmic two-point invariants, directly extending the relation found in [GRZ]. Yet we find an explicit discrepancy at higher genus in the relation to open Gromov-Witten invariants of the Aganagic-Vafa brane. Using a degeneration argument, we express the difference in terms of relative invariants of an elliptic curve. With $\pi: \widehat{\mathbb{P}} \rightarrow \mathbb{P}$ the toric blow up of a point, we use the Topological Vertex [AKMV] to show a correspondence between open invariants of $K_{\mathbb{P}}$ and closed invariants of $K_{\widehat{\mathbb{P}}}$ generalizing a variant of [CLLT][LLW] to arbitrary genus and winding. We also equate winding-1, open-BPS invariants with closed Gopakumar-Vafa invariants.