Hitchin-Kobayashi correspondence, quivers and vortices
by
Oscar García-Prada
Universidad Autónoma de Madrid, Spain
Coauthors: Luis Álvarez-Cónsul

Oral Communication

The so-called Hitchin-Kobayashi correspondence, proved by Donaldson, Uhlenbeck and Yau, establishes that a holomorphic vector bundle over a compact Kähler manifold admits a Hermitian-Einstein metric if and only if the bundle satisifies the Mumford-Takemoto stability condition. In this talk I will consider a ``deformed'' version of this correspondence for G-equivariant bundles on the product of a compact Kähler manifold X by a flag manifold G/P, where G is a complex semisimple Lie group and P is a parabolic subgroup. The deformation is naturally defined by the equivariant condition on the bundle.

The study of invariant solutions to the ``deformed'' Hermitian-Einstein equation over X×G/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 a ``quiver'' whose structure is entirely determined by the parabolic subgroup P. In the simplest case, when the flag manifold is the complex projective line, one recovers the theory of vortices, as studied by S. Bradlow, the author and others.

Date received: July 3, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cafh-39.