Dimensional reduction and quiver bundles

Abstract

The so-called Hitchin-Kobayashi correspondence, proved by Donaldson, Uhlenbeck and Yau, establishes that an indecomposable holomorphic vector bundle over a compact Kahler manifold admits a Hermitian-Einstein metric if and only if the bundle satisfies the Mumford-Takemoto stability condition. In this paper we consider a variant of this correspondence for G-equivariant vector bundles on the product of a compact Kahler manifold X by a flag manifold G/P, where G is a complex semisimple Lie group and P is a parabolic subgroup. The modification that we consider is determined by a filtration of the vector bundle which is naturally defined by the equivariance of the bundle. The study of invariant solutions to the modified Hermitian-Einstein equation over XxG/P leads, via dimensional reduction techniques, to gauge-theoretic equations on X. These are equations for hermitian metrics on a set of holomorphic bundles on X linked by morphisms, defining what we call a quiver bundle for a quiver with relations whose structure is entirely determined by the parabolic subgroup P. Similarly, the corresponding stability condition for the invariant filtration over XxG/P gives rise to a stability condition for the quiver bundle on X, and hence to a Hitchin-Kobayashi correspondence. In the simplest case, when the flag manifold is the complex projective line, one recovers the theory of vortices, stable triples and stable chains, as studied by Bradlow, the authors, and others.