Mirror symmetry for the Tate curve via tropical and log corals

TL;DR

This paper establishes a correspondence between tropical corals and holomorphic polygons, linking tropical geometry with log Gromov--Witten invariants to confirm a mirror symmetry prediction for the Tate curve.

Contribution

It introduces tropical corals and proves a correspondence theorem connecting tropical counts to log Gromov--Witten invariants, advancing understanding of mirror symmetry for elliptic curves.

Findings

Tropical corals correspond to holomorphic polygons in Floer theory

Counts of tropical corals match punctured log Gromov--Witten invariants

Mirror symmetry prediction for the Tate curve is confirmed

Abstract

We introduce tropical corals, balanced trees in a half-space, and show that they correspond to holomorphic polygons capturing the product rule in Lagrangian Floer theory for the elliptic curve. We then prove a correspondence theorem equating counts of tropical corals to punctured log Gromov--Witten invariants of the Tate curve. This implies that the homogeneous coordinate ring of the mirror to the Tate curve is isomorphic to the degree-zero part of symplectic homology, confirming a prediction of homological mirror symmetry.