Classifying Spaces and Dirac Operators Coupled to Instantons

Abstract

Let $M(k,SU(l))$ denote the moduli space of based gauge equivalence classes of $SU(l)$ instantons on principal bundles over ${S^4}$ with second Chern class equal to $k$. In this paper we use Dirac operators coupled to such connections to study the topology of these moduli spaces as $l$ increases relative to $k$. This "coupling" procedure produces maps ${\partial _u}:M(k,SU(l)) \to BU(k)$, and we prove that in the limit over $l$ such maps recover Kirwan’s $[\text {K}]$ homotopy equivalence $M(k,SU) \simeq BU(k)$. We also compute, for any $k$ and $l$, the image of the homology map ${({\partial _u})_ * }:{H_ * }(M(k,SU(l));Z) \to {H_ * }(BU(k);Z)$. Finally, we prove all the analogous results for $Sp(l)$ instantons.