The homotopy types of $\mathrm{PU}(3)$– and $\mathrm{PSp}(2)$–gauge groups

Abstract

Let $G$ be a compact connected simple Lie group. Any principal $G$–bundle over $S^4$ is classified by an integer $k\in \mathbb{Z}\cong {\pi }_3\left(G\right)$, and we denote the corresponding gauge group by ${\mathcal{G}}_k\left(G\right)$. We prove that ${\mathcal{G}}_k\left(PU\left(3\right)\right)\simeq {\mathcal{G}}_{\ell }\left(PU\left(3\right)\right)$ if and only if $\left(24,k\right)=\left(24,\ell \right)$, and ${\mathcal{G}}_k\left(PSp\left(2\right)\right){\simeq }_{\left(p\right)}{\mathcal{G}}_{\ell }\left(PSp\left(2\right)\right)$ for any prime $p$ if and only if $\left(40,k\right)=\left(40,\ell \right)$, where $\left(m,n\right)$ is the gcd of integers $m,n$.