Scattering diagrams, theta functions, and refined tropical curve counts

TL;DR

This paper links scattering diagrams, theta functions, and tropical curve counts across various algebraic settings, providing new geometric insights and proofs for conjectures in quantum cluster algebras and mirror symmetry.

Contribution

It introduces a tropical geometric framework for understanding scattering diagrams and theta functions, proving several key conjectures and establishing new connections in algebraic geometry.

Findings

Tropical curve counts correspond to descendant log Gromov-Witten invariants.

Proved the quantum Frobenius conjecture of Fock and Goncharov.

Established the non-degeneracy of the trace-pairing in Frobenius structures.

Abstract

Working over various graded Lie algebras and in arbitrary dimension, we express scattering diagrams and theta functions in terms of counts of tropical curves/disks, weighted by multiplicities given in terms of iterated Lie brackets. Over the tropical vertex group, our tropical curve counts are known to give certain descendant log Gromov-Witten invariants. Working over the quantum torus algebra yields theta functions for quantum cluster varieties, and our tropical description sets up for geometric interpretations of these. As an immediate application, we prove the quantum Frobenius conjecture of Fock and Goncharov. We also prove a refined version of the Carl-Pumperla-Siebert result on consistency of theta functions, and we prove the non-degeneracy of the trace-pairing for the Gross-Hacking-Keel Frobenius structure conjecture.