DIMENSIONAL REDUCTION, ${\rm SL} (2, {\mathbb C})$-EQUIVARIANT BUNDLES AND STABLE HOLOMORPHIC CHAINS

Abstract

In this paper we study gauge theory on SL(2, C)-equivariant bundles over X × P 1 , where X is a compact Kähler manifold, P 1 is the complex projective line, and the action of SL(2, C) is trivial on X and standard on P 1 . We first classify these bundles, showing that they are in correspondence with objects on X -that we call holomorphic chainsconsisting of a finite number of holomorphic bundles E i and morphisms E i → E i−1 . We then prove a Hitchin-Kobayashi correspondence relating the existence of solutions to certain natural gauge-theoretic equations and an appropriate notion of stability for an equivariant bundle and the corresponding chain. A central tool in this paper is a dimensional reduction procedure which allow us to go from X × P 1 to X.

Quiver gauge theory of non-Abelian vortices and noncommutative instantons in higher dimensions

“…The CP 1 dependence in this case is uniquely determined by the above SU(2)-invariant Dirac monopole configurations [33,34]. The u(k)-valued gauge potential A thus splits into k i × k j blocks A ij , 10) where the indices i, j, .…”

“…All fields (A i , φ i+1 ) depend only on the coordinates x ′ ∈ M q . Every SU(2)-invariant unitary connection A on M q × CP 1 is of the form given in (3.10)-(3.14) [34], which follow from the fact that the complexified cotangent bundle of CP 1 is L 2 ⊕ L −2 . This ansatz amounts to an equivariant decomposition of the original rank k SU(2)-equivariant bundle E → M q × CP 1 in the form 16) where E k i → M q is a hermitean vector bundle of rank k i with typical fibre the module V k i , and…”

Hitchin–Kobayashi Correspondence, Quivers, and Vortices

“…First, the symplectic form v∈Q 0 σ v ω v + ω R on A × Ω 0 (cf. §2.2) has been deformed by the parameters σ whenever σ v = σ v ′ for some v, v ′ ∈ Q 0 ; as a matter of fact, the vortex equations (2.2) depend on new parameters even for holomorphic triples or chains [AG1,BG], hence generalizing their Hitchin-Kobayashi correspondences (in the case of a holomorphic pair (E, φ), consisting of a holomorphic vector bundle E and a holomorphic section φ ∈ H 0 (X, E), as considered in [B], which can be understood as a holomorphic triple φ : O X → E, the new parameter can actually be absorbed in φ, so no new parameters are really present). Second, the twisting bundles M a , for a ∈ Q 1 , are not considered in [Ba, M].…”

Homological Algebra of Twisted Quiver Bundles

“…In this case a Q-bundle is called a holomorphic chain, these where studied bý Alvarez-Cónsul and García-Prada [1]. There is a generalization of the notion of Q-bundle to that of twisted Q-bundle, where the morphisms φ a are twisted by a vector bundle M a (see Section 3 for precise definitions).…”