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

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 $X$.