The Rank Stable Topology of Instantons on $\cpbar$

Abstract

Let $\M_{k}^{n}$ be the moduli space of based (anti-self-dual) instantons on $\cpbar$ of charge $k$ and rank $n$. There is a natural inclusion of rank $n$ instantons into rank $n+1$. We show that the direct limit space is homotopy equivalent to $BU(k)\times BU(k)$. The moduli spaces also have the following algebro-geometric interpretation: Let $\linf$ be a line in the complex projective plane and consider the blow-up at a point away from $\linf$. $\M _{k}^{n}$ can be described as the moduli space of rank $n$ holomorphic bundles on the blownup projective plane with $c_1=0$ and $c_2=k$ and with a fixed holomorphic trivialization on $\linf$.

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