Nakajima quiver bundles

TL;DR

This paper introduces Nakajima bundle representations, generalizing quiver representations and bundles, with gauge-theoretic characterizations, deformation theory, stability conditions, and examples on algebraic curves.

Contribution

It develops a new framework for Nakajima bundle representations, connecting quiver theory, gauge theory, and algebraic geometry, with a Hitchin-Kobayashi correspondence and moduli space analysis.

Findings

Established a gauge-theoretic characterization of Nakajima bundle representations.

Proved a Hitchin-Kobayashi correspondence for stability of these representations.

Constructed examples recovering known and new moduli spaces.

Abstract

We introduce the notion of a Nakajima bundle representation. Given a labelled quiver and a variety or manifold $X$, such a representation involves an assignment of a complex vector bundle on $X$ to each node of the doubled quiver; to the edges, we assign sections of, and connections on, associated twisted bundles. We for the most part restrict attention in our development to algebraic curves or Riemann surfaces. Our construction simultaneously generalizes ordinary Nakajima quiver representations on the one hand and quiver bundles on the other hand. These representations admit gauge-theoretic characterizations, analogous to the ADHM equations in the original work of Nakajima, allowing for the construction of these generalized quiver varieties using a reduction procedure with moment maps. We study the deformation theory of Nakajima bundle representations, prove a Hitchin-Kobayashi correspondence between such representations and stable quiver bundles, examine the natural torus action on the resulting moduli varieties, and comment on scenarios where the variety is hyperk\"ahler. Finally, we produce concrete examples that recover known and new moduli spaces.