Cohomological invariants of a variation of flat connection

TL;DR

This paper constructs new cohomological invariants for flat connections on manifolds using Chern-Cheeger-Simons theory, revealing homomorphisms on higher homology groups of the moduli space.

Contribution

It introduces a novel method to associate canonical cohomological invariants to simplices parametrizing flat connections, expanding the understanding of moduli space topology.

Findings

Invariants lie in specific cohomology groups with $C/Z$-coefficients.

Establishes homomorphisms on higher homology groups of the moduli space.

Connects invariants to the $C/Z$-cohomology of the manifold.

Abstract

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 $X$.