Quivers, Flow Trees, and Log Curves

TL;DR

This paper establishes a deep connection between Donaldson-Thomas invariants of quivers, attractor flow trees, and genus 0 log Gromov-Witten invariants of toric varieties, revealing a new geometric interpretation.

Contribution

It proves that the coefficients in the universal formula for DT invariants are genus 0 log Gromov-Witten invariants, linking combinatorial and geometric invariants.

Findings

Coefficients are genus 0 log Gromov-Witten invariants.

Established a log-tropical correspondence theorem.

Connected attractor flow trees with rational log curves.

Abstract

Donaldson-Thomas (DT) invariants of a quiver with potential can be expressed in terms of simpler attractor DT invariants by a universal formula. The coefficients in this formula are calculated combinatorially using attractor flow trees. In this paper, we prove that these coefficients are genus 0 log Gromov--Witten invariants of $d$-dimensional toric varieties, where $d$ is the number of vertices of the quiver. This result follows from a log-tropical correspondence theorem which relates $(d-2)$-dimensional families of tropical curves obtained as universal deformations of attractor flow trees, and rational log curves in toric varieties.