Instantons on S4 and CP2, rank stabilization, and Bott periodicity

We study the large n limit of the moduli spaces of Gn-instantons on S4 and CP2 where Gn is SU(n), Sp(n/2), or SO(n). We show that in the direct limit topology, the moduli space is homotopic to a classifying space. For example, the moduli space of Sp(∞) or SO(∞) instantons on CP2 has the homotopy type of BU(k) where k is the charge of the instantons. We use our results along with Taubes’ result concerning the k→∞ limit to obtain a novel proof of the homotopy equivalences in the eight-fold Bott periodicity spectrum. We work with the algebro-geometric realization of the instanton spaces as moduli spaces of framed holomorphic bundles on CP2 and CP2 blown-up at a point. We give explicit constructions for these moduli spaces (see Table 1).