Spectral correspondence for cyclic Higgs bundles

TL;DR

This paper develops a spectral correspondence framework for cyclic Higgs bundles using quiver bundles, linking them to sheaves on noncommutative surfaces and generalizing existing spectral correspondences.

Contribution

It introduces a novel spectral correspondence for cyclic Higgs bundles via quiver bundles, connecting to noncommutative geometry and Clifford algebra modules.

Findings

Establishes a one-to-one correspondence between cyclic Higgs bundles and sheaves on noncommutative surfaces.

Generalizes spectral correspondence for $U(p,p)$-Higgs bundles.

Connects spectral data for $U(p,q)$-Higgs bundles with modules over even Clifford algebra sheaves.

Abstract

In this paper, we describe the spectral correspondence for cyclic Higgs bundles from the viewpoint of quiver bundles. Under this framework, we establish a one-to-one correspondence between cyclic Higgs bundles on a curve and sheaves on a noncommutative surface whose noncommutative structure originates from the path algebra associated to the cyclic quiver. As applications, this correspondence generalizes the known spectral correspondence for $U(p,p)$-Higgs bundles and establish a connection between the spectral data for $U(p,q)$-Higgs bundles and modules over the sheaf of even Clifford algebras of a conic fibration.