Enumerative Geometry of Quantum Periods

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 $\ddot{I}€:\hat{\mathbb{P}}\to \mathbb{P}$ 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 $K_{\hat{\mathbb{P}}}$ generalizing a variant of [CLLT][LLW] to arbitrary genus and winding. We also equate winding-1, open-BPS invariants with closed Gopakumar-Vafa invariants.

50 pages, comments welcome; v2: extended Proposition 4.1, updated Section 5, added an Appendix, other small changes