The rank stable topology of instantons on $\overline {\mathbf {CP}}^2$

Abstract:

Let $\mathcal {M} _{k}^{n}$ be the moduli space of based (anti-self-dual) instantons on $\overline {\mathbf {CP}}^2$ of charge $k$ and rank $n$. There is a natural inclusion $\mathcal {M} _{k}^{n}\hookrightarrow \mathcal {M}_{k}^{n+1}$. We show that the direct limit space $\mathcal {M}_k^\infty$ is homotopy equivalent to $BU(k)\times BU(k)$. Let $\ell _{\infty }$ be a line in the complex projective plane and let $\widetilde { {\mathbf C} {\mathbf {P}}}^{2}$ be the blow-up at a point away from $\ell _{\infty }$. $\mathcal {M} _{k}^{n}$ can be alternatively described as the moduli space of rank $n$ holomorphic bundles on $\widetilde { \mathbf {C} \mathbf {P}}^{2}$ with $c_{1}=0$ and $c_{2}=k$ and with a fixed holomorphic trivialization on $\ell _{\infty }$.