We provide descriptions for the moduli spaces $\mbox{Rep}(\Gamma,PU(m))$, 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 $\pi_{1}(X)\cong\mathbb{Z}^{n}$, the natural map $\pi_{0}(\mbox{Rep}(\pi_{1}(X),PU(m)))\rightarrow [X,BPU(m)]$ is injective, hence providing a complete enumeration of the isomorphism classes of flat principal $PU(m)$-bundles over $X$. This is joint work with Man Chuen Cheng.