Quiver Bundles and Wall Crossing for Chains
P. B. Gothen, A. Nozad

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.
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