Differential characters and cohomology of the moduli of flat connections

Abstract

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.