# Theta bases and log Gromov-Witten invariants of cluster varieties

## TL;DR

This paper establishes a connection between theta bases in cluster varieties and descendant log Gromov-Witten invariants, providing a geometric interpretation and new technical tools for their computation.

## Contribution

It proves that theta bases are determined by specific Gromov-Witten invariants and introduces the concept of contractible tropical curves for analyzing log curves.

## Key findings

- Theta bases are determined by descendant log Gromov-Witten invariants.
- Gromov-Witten counts often correspond to naive rational curve counts.
- Introduction of 'contractible' tropical curves as a new technical tool.

## Abstract

Using heuristics from mirror symmetry, combinations of Gross, Hacking, Keel, Kontsevich, and Siebert have given combinatorial constructions of canonical bases of "theta functions" on the coordinate rings of various log Calabi-Yau spaces, including cluster varieties. We prove that the theta bases for cluster varieties are determined by certain descendant log Gromov-Witten invariants of the symplectic leaves of the mirror/Langlands dual cluster variety, as predicted in the Frobenius structure conjecture of Gross-Hacking-Keel. We further show that these Gromov-Witten counts are often given by naive counts of rational curves satisfying certain geometric conditions. As a key new technical tool, we introduce the notion of "contractible" tropical curves when showing that the relevant log curves are torically transverse.