Log Calabi-Yau surfaces and Jeffrey-Kirwan residues

TL;DR

This paper establishes a connection between Jeffrey-Kirwan residues, Donaldson-Thomas invariants, and mirror symmetry for log Calabi-Yau surfaces, confirming predictions from physics and providing new computational tools.

Contribution

It proves an equality between Jeffrey-Kirwan residues and Donaldson-Thomas invariants for certain quivers, and links these residues to mirror symmetry via scattering diagrams and theta functions.

Findings

Jeffrey-Kirwan residues match Donaldson-Thomas invariants for specific quivers.

Residues are determined by the Gross-Hacking-Keel mirror family in special cases.

The results confirm physical predictions and connect algebraic and geometric invariants.

Abstract

We prove an equality, predicted in the physical literature, between the Jeffrey-Kirwan residues of certain explicit meromorphic forms attached to a quiver without loops or oriented cycles and its Donaldson-Thomas type invariants. In the special case of complete bipartite quivers we also show independently, using scattering diagrams and theta functions, that the same Jeffrey-Kirwan residues are determined by the the Gross-Hacking-Keel mirror family to a log Calabi-Yau surface.