Computing Non-Archimedean Cylinder Counts in Blowups of Toric Surfaces

TL;DR

This paper develops a formula linking counts of certain analytic cylinders on affine log Calabi-Yau surfaces to counts on a single blowup of a toric surface, offering a new approach to non-archimedean Gromov-Witten theory in mirror symmetry.

Contribution

It provides a closed-form formula relating cylinder counts on log Calabi-Yau surfaces to counts on toric surface blowups, contrasting with scattering diagram methods.

Findings

Derived a formula connecting cylinder counts on different surfaces.

Used non-archimedean Gromov-Witten theory and tropical degeneration.

Simplified computation of analytic cylinder counts in mirror symmetry.

Abstract

Counts of holomorphic disks are at the heart of the SYZ approach to mirror symmetry. In the non archimedean framework, these counts are expressed as counts of analytic cylinders. In simple cases, such as cluster varieties, these counts can be extracted from a combinatorial algorithm encoded by a scattering diagram. The relevant scattering diagram encodes information about infinitesimal analytic cylinders. In this paper, we give a formula relating counts of a restricted class of cylinders on affine log Calabi Yau surfaces to counts on a single blowup of a toric surface, using non archimedean Gromov Witten theory and a degeneration procedure parametrized by tropical data. This closed form formula constrasts with the scattering diagram approach.