  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 $X$.
