Theta Function Basis of the Cox ring of Postive 2d Looijenga pairs

TL;DR

This paper constructs a canonical basis of theta functions for the Cox ring of certain 2D Looijenga pairs, linking algebraic structures to counts of analytic disks in mirror symmetry.

Contribution

It introduces a canonical theta basis for the Cox ring of 2D Looijenga pairs, connecting algebraic geometry with disk counting in mirror symmetry.

Findings

Established a canonical theta basis for the Cox ring

Connected structure constants to counts of k-analytic disks

Applied to log Calabi-Yau surfaces

Abstract

We give a canonical basis of theta functions for the Cox ring of two dimensional Looijenga pairs with affine interior, with structure constants naive counts of k-analytic disks in the total space of the universal deformation of the mirror (which, as this is dimension two, is isomorphic to the log Calabi-Yau surface itself)