Cohomological invariants of a variation of flat connection

In this paper, we apply the theory of Chern-Cheeger-Simons to construct canonical invariants associated to a \(r\)-simplex whose points parametrize flat connections on a smooth manifold \(X\). These invariants lie in degrees \((2p-r-1)\)-cohomology with \(C/Z\)-coefficients, for \(p> r\geq 1\). In turn, this corresponds to a homomorphism on the higher homology groups of the moduli space of flat connections, and taking values in \(C/Z\)-cohomology of the underlying smooth manifold \(X\).