Generalized Quivers, Orthogonal and Symplectic Representations, and Hitchin-Kobayashi Correspondences

TL;DR

This paper extends the theory of quiver bundles to generalized quivers associated with any reductive group, exploring orthogonal, symplectic, and supermixed representations, and establishing Hitchin-Kobayashi correspondences for these structures.

Contribution

It introduces generalized quiver bundles for arbitrary reductive groups and studies their orthogonal, symplectic, and supermixed representations, linking them with Hitchin-Kobayashi correspondences.

Findings

Interpretation of orthogonal and symplectic bundle representations as symmetric quivers.

Introduction of supermixed quivers with combined orthogonal and symplectic symmetries.

Discussion of Hitchin-Kobayashi correspondences for these generalized quiver bundles.

Abstract

We review the theory of quiver bundles over a K\"ahler manifold, and then introduce the concept of generalized quiver bundles for an arbitrary reductive group G. We first study the case when G=O(V) or Sp(V), interpreting them as orthogonal (resp. symplectic) bundle representations of the symmetric quivers introduced by Derksen-Weyman. We also study supermixed quivers, which simultaneously involve both orthogonal and symplectic symmetries. Finally, we discuss Hitchin-Kobayashi correspondences for these objects.