On the moduli spaces of commuting elements in the projective unitary groups

Abstract

We provide descriptions for the moduli spaces $\text{Rep}(\Gamma ,PU(m))$, where $\Gamma$ is any finitely generated abelian group and $PU(m)$ is the group of $m\times m$ projective unitary matrices. As an application we show that for any connected CW-complex $X$ with ${\pi }_1(X)\cong {\mathbf{Z}}^n$, the natural map ${\pi }_0(\text{Rep}({\pi }_1(X),PU(m)))\to [X,BPU(m)]$ is injective, hence providing a complete enumeration of the isomorphism classes of flat principal $PU(m)$-bundles over $X$.

10 pages. Minor typos fixed. To appear in the Journal of Mathematical Physics