Hitchin–Kobayashi Correspondence, Quivers, and Vortices

Abstract

ABSTRACT. A twisted quiver bundle is a set of holomorphic vector bundles over a complex manifold, labelled by the vertices of a quiver, linked by a set of morphisms twisted by a fixed collection of holomorphic vector bundles, labelled by the arrows. When the manifold is Kähler, quiver bundles admit natural gauge-theoretic equations, which unify many known equations for bundles with extra structure. In this paper we prove a Hitchin-Kobayashi correspondence for twisted quiver bundles over a compact Kähler manifold, relating the existence of solutions to the gauge equations to a stability criterion, and consider its application to a number of situations related to Higgs bundles and dimensional reductions of the Hermitian-Einstein equations.

Moduli of ADHM sheaves and the local Donaldson–Thomas theory

“…2.9.2.42] proves that the stability condition (2.2) is in fact the asymptotic form of a more general GIT stability condition for ADHM sheaves. Analogous stability conditions for quiver sheaves were previously formulated and studied in [73,30,74], and in the context of HitchinKobayashi correspondence in [2,1]. This is a natural generalization of previous work on decorated sheaves, including [13,79,14,15,9,41,40,75].…”

“…In particular, I owe special thanks to Ionut Ciocan-Fontanine for pointing out references [46,19,29] [2,1,31]. Let X be an arbitrary scheme, M 1 , M 2 be fixed invertible sheaves on X and E ∞ be a coherent O X -module.…”

Homological Algebra of Twisted Quiver Bundles

“…In a similar way to Proposition 2.3, the category of twisted Q-sheaves is equivalent to the category of sheaves of B-modules on X (a detailed proof can be found in [3]). Thus we shall interchangeably use the expressions "twisted Q-sheaf" and "sheaf of B-modules", or, for short, "B-module".…”

Rank two quiver gauge theory, graded connections and noncommutative vortices

“…Then the original rank k hermitean vector bundle (2.8) 15) where E k iα → M is a hermitean vector bundle of rank k iα with typical fibre the module V k iα in (2.6), and 2) is the bundle with fibres…”

“…For Kähler manifolds X the stability of such bundles (BPS conditions) is controlled by the DonaldsonUhlenbeck-Yau (DUY) equations [11]. For G-equivariant bundles E → X one finds that Yang-Mills theory on X reduces to a quiver gauge theory on M [12]- [15].…”