Wall-crossings for Twisted Quiver Bundles

TL;DR

This paper develops a wall-crossing formula for generalized Donaldson-Thomas invariants of twisted quiver sheaves on curves, extending ADHM invariants through homological algebra and stability condition analysis.

Contribution

It introduces a new wall-crossing formula for twisted quiver sheaves, generalizing ADHM invariants and utilizing Joyce and Song's theory in a novel setting.

Findings

Derived a wall-crossing formula for invariants of twisted quiver sheaves.

Extended ADHM invariants to a broader class of quiver sheaves.

Connected homological algebra with stability conditions in the context of quiver moduli.

Abstract

Given a double quiver, we study homological algebra of twisted quiver sheaves with the moment map relation using the short exact sequence of Crawley-Boevey, Holland, Gothen, and King. Then in a certain one-parameter space of the stability conditions, we obtain a wall-crossing formula for the generalized Donaldson-Thomas invariants of the abelian category of framed twisted quiver sheaves on a smooth projective curve. To do so, we closely follow the approach of Chuang, Diaconescu, and Pan in the ADHM quiver case, which makes use of the theory of Joyce and Song. The invariants virtually count framed twisted quiver sheaves with the moment map relation and directly generalize the ADHM invariants of Diaconescu.