Mirror symmetry for the Tate curve via tropical and log corals

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.

72 pages, 13 figures, the introduction is rewritten

References in corpus (7)

• Functors and Computations in Floer homology with Applications Part II
• Enumerative tropical algebraic geometry in R2
• The canonical wall structure and intrinsic mirror symmetry
• Intrinsic Mirror Symmetry
• Punctured logarithmic maps
• Intrinsic mirror symmetry and punctured Gromov-Witten invariants
• The Quantum $A_{\infty }$-Relations on the Elliptic Curve