Universal covers and the GW/Kronecker correspondence

TL;DR

This paper explores the geometric relationship between tropical geometry, quiver representations, and Gromov-Witten invariants, providing a new perspective on the GW/Kronecker correspondence through tropical curves.

Contribution

It constructs rational tropical curves from subquivers of the universal cover of m-Kronecker quivers, offering a geometric interpretation of the GW/Kronecker correspondence.

Findings

Constructed rational tropical curves from subquivers.

Provided a geometric picture behind the GW/Kronecker correspondence.

Linked tropical geometry with quiver moduli and Gromov-Witten invariants.

Abstract

The tropical vertex is an incarnation of mirror symmetry found by Gross, Pandharipande and Siebert. It can be applied to m-Kronecker quivers K(m) (together with a result of Reineke) to compute the Euler characteristics of the moduli spaces of their (framed) representations in terms of Gromov-Witten invariants (as shown by Gross and Pandharipande). In this paper, we study a possible geometric picture behind this correspondence, in particular constructing rational tropical curves from subquivers of the universal covering quiver of K(m). Additional motivation comes from the physical interpretation of m-Kronecker quivers in the context of quiver quantum mechanics (especially work of F. Denef).