Differential characters and cohomology of the moduli of flat Connections

Let $\pi \text{\hspace{0.17em}⁣}:P\to M$ be a principal bundle and $p$ an invariant polynomial of degree r on the Lie algebra of the structure group. The theory of Chern-Simons differential characters is exploited to define an homology map $\chi^{k} : H_{2r-k-1}(M)\times H_{k}($\mathcal{F}$/$\mathcal{G}$)\to $\mathbb{R}$/$\mathbb{Z}$$,for$k<r-1$,where$$\mathcal{F}$ /$\mathcal{G}$$ is the moduli space of flat connections of $\pi$ under the action of a subgroup $\mathcal{G}$ of the gauge group. The differential characters of first order are related to the Dijkgraaf-Witten action for Chern-Simons Theory. The second order characters are interpreted geometrically as the holonomy of a connection in a line bundle over $\mathcal{F}\mathrm{/}\mathcal{G})$. The relationship with other constructions in the literature is also analyzed.