Real instantons, dirac operators and quaternionic classifying spaces

Abstract

Let M(k, SO(n)) be the moduli space of based gauge equivalence classes of SO(n) instantons on principal SO(n) bundles over S4 with first Pontryagin class p1 = 2k. In this paper, we use a monad description (Y. Tian, The Atiyah-Jones conjecture for classical groups, preprint, S. K. Donaldson, Comm. Math. Phys. 93 (1984), 453-460) of these moduli spaces to show that in the limit over n, the moduli space is homotopy equivalent to the classifying space BSp(k). Finally, we use Dirac operators coupled to such connections to exhibit a particular and quite natural homotopy equivalence. © 1996 American Mathematical Society.