On the Moduli Space of Anti-Self-Dual Yang–Mills Connections on Kähler Surfaces

Abstract

The purpose of this paper is to give a complete proof of the result announced in [7]. In fact in this paper we discuss the dimension of moduli space consisting of anti-self-dual solutions of Yang-Mills equation in the case where the base space is Kahler and we obtain the dimension formula which is similar to the case of moduli space of self-dual solutions over a self-dual base space, given by Atiyah, Hitchin and Singer [2]. Further we get on a compact Kahler surface a suggestive observation that the moduli space of anti-self-dual Yang-Mills connections may have a close relation to moduli space of holomorphic vector bundles. Yang-Mills connections, namely, solutions to Yang-Mills equation have originated from field theory in physics ([12]). Yang-Mills equation is considered as a generalization of Maxwell equation from a viewpoint of non-abelian gauge group. Mathematically, Yang-Mills connections are formulated by the aid of notions of connections on a principal fibre bundle. Let P be a principal bundle over a compact oriented Riemannian 4-manifold M with a compact semi-simple Lie group G. Let E be an associated complex vector bundle. A functional ^M is defined over the space CE,G consisting of all G-connections on E by ^5^(7)=(1/2) H^!! for the curvature form R^ of 7. The Euler-Lagrangian equation of the functional is written by d(^)—0. A G-connection which gives a solution to this equation is called a Yang-Mills Gconnection. From Bianchi's identity an (anti-)self-dual G-connection which satisfies that *R^=±R^ is a special Yang-Mills G-connection. If we denote by J>E,G the set of all (anti-)self-dual G-connections, then the space M%tGt which is a quotient space of J.E, G, modulo the group of gauge transformations SP represents moduli space of essentially distinct (anti-)self-dual G-connections. Now suppose that M is a Kahler surface. Then M admits the canonical orientation induced from the complex structure of M. With respect to the space of infinitesimal deformations of anti-self-dual G-connections, that is, the tangent