Quiver Bundles and Wall Crossing for Chains

TL;DR

This paper introduces a new quiver bundle construction for chains on Riemann surfaces, linking chain stability to quiver bundle stability and providing a novel proof of a key lemma in Higgs bundle theory.

Contribution

It associates a new quiver bundle to chains and proves that chain stability implies quiver bundle stability, offering a new proof of a fundamental lemma without using vortex equations.

Findings

Established a stability correspondence between chains and quiver bundles.

Provided a new proof of a key lemma in Higgs bundle moduli.

Enhanced understanding of fixed points of C*-actions on Higgs moduli.

Abstract

Holomorphic chains on a Riemann surface arise naturally as fixed points of the natural C*-action on the moduli space of Higgs bundles. In this paper we associate a new quiver bundle to the Hom-complex of two chains, and prove that stability of the chains implies stability of this new quiver bundle. Our approach uses the Hitchin-Kobayashi correspondence for quiver bundles. Moreover, we use our result to give a new proof of a key lemma on chains (due to \'Alvarez-C\'onsul, Garc\'ia-Prada and Schmitt), which has been important in the study of Higgs bundle moduli; this proof relies on stability and thus avoids the direct use of the chain vortex equations.