Quivers, curves, and the tropical vertex

TL;DR

This paper explores the tropical vertex group, revealing connections between symplectomorphisms, quiver moduli spaces, and Gromov-Witten invariants, and proves new results on scattering diagram symmetries.

Contribution

It provides new insights into the rays and symmetries of scattering diagrams, confirming conjectures by Gross-Siebert and Kontsevich, with perspectives from quiver theory and Gromov-Witten invariants.

Findings

Proved new results on scattering diagram symmetries.

Confirmed conjectures by Gross-Siebert and Kontsevich.

Established links between tropical vertex group, quiver moduli, and Gromov-Witten theory.

Abstract

Elements of the tropical vertex group are formal families of symplectomorphisms of the 2-dimensional algebraic torus. Commutators in the group are related to Euler characteristics of the moduli spaces of quiver representations and the Gromov-Witten theory of toric surfaces. After a short survey of the subject (based on lectures of Pandharipande at the 2009 Geometry summer school in Lisbon), we prove new results about the rays and symmetries of scattering diagrams of commutators (including previous conjectures by Gross-Siebert and Kontsevich). Where possible, we present both the quiver and Gromov-Witten perspectives.