Differential characters and cohomology of the moduli of flat connections

Let \(\pi {:}\, P\rightarrow M\) 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 a homology map \(\chi ^{k} {:}\, H_{2r-k-1}(M)\times H_{k}({\mathcal {F}}/{\mathcal {G}})\rightarrow {\mathbb {R}}/{\mathbb {Z}}\), 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 \({\mathcal {F}}/{\mathcal {G}}\). The relationship with other constructions in the literature is also analyzed.