MPS degeneration formula for quiver moduli and refined GW/Kronecker correspondence

TL;DR

The paper generalizes a formula for the topology of quiver moduli spaces, linking it to Gromov-Witten invariants and tropical geometry, providing new computational tools and insights into quiver representations.

Contribution

It proves a motivic generalization of the MPS formula for arbitrary quivers and relates it to Gromov-Witten invariants and tropical curves.

Findings

Motivic generalization of MPS formula for all quivers.

Identification of MPS formula with Gromov-Witten degeneration formula.

New combinatorial formula for quiver Euler characteristics as sums over trees.

Abstract

Motivated by string-theoretic arguments Manschot, Pioline and Sen discovered a new remarkable formula for the Poincare polynomial of a smooth compact moduli space of stable quiver representations which effectively reduces to the abelian case (i.e. thin dimension vectors). We first prove a motivic generalization of this formula, valid for arbitrary quivers, dimension vectors and stabilities. In the case of complete bipartite quivers we use the refined GW/Kronecker correspondence between Euler characteristics of quiver moduli and Gromov-Witten invariants to identify the MPS formula for Euler characteristics with a standard degeneration formula in Gromov-Witten theory. Finally we combine the MPS formula with localization techniques, obtaining a new formula for quiver Euler characteristics as a sum over trees, and constructing many examples of explicit correspondences between quiver representations and tropical curves.