Real instantons, Dirac operators and quaternionic classifying spaces

Abstract

Let M ( k , S O ( n ) ) M(k,SO(n)) be the moduli space of based gauge equivalence classes of S O ( n ) SO(n) instantons on principal S O ( n ) SO(n) bundles over S 4 S^4 with first Pontryagin class p 1 = 2 k p_1=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 n , the moduli space is homotopy equivalent to the classifying space B S p ( k ) BSp(k) . Finally, we use Dirac operators coupled to such connections to exhibit a particular and quite natural homotopy equivalence.