Real instantons, Dirac operators and quaternionic classifying spaces

Let $M(k,SO(n))$ be the moduli space of based gauge equivalence classes of $SO(n)$ instantons on principal $SO(n)$ bundles over $S^4$ with first Pontryagin class $p_1=2k$. 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 $n$, 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.