Quivers, Flow Trees, and 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.

66 pages, 4 figures. Final version accepted for publication in Algebraic Geometry