# Flat connections and cohomology invariants

## TL;DR

This paper develops geometric invariants for the topology of flat connection spaces on principal bundles, using a generalized Chern-Weil approach that applies to both real and complex manifolds, revealing new gauge-invariant cohomological structures.

## Contribution

It introduces a novel method to define cohomological invariants for flat connections, extending classical characteristic class theory to the moduli space of flat connections and complex manifolds.

## Key findings

- Constructed gauge-invariant homomorphisms to cohomology groups
- Extended invariants to complex manifolds using Dolbeault cohomology
- Provided examples demonstrating applications of the invariants

## Abstract

The main goal of this article is to construct some geometric invariants for the topology of the set $\mathcal{F}$ of flat connections on a principal $G$-bundle $P\,\longrightarrow\, M$. Although the characteristic classes of principal bundles are trivial when $\mathcal{F}\neq \emptyset$, their classical Chern-Weil construction can still be exploited to define a homomorphism from the set of homology classes of maps $S\longrightarrow \mathcal{F}$ to the cohomology group $H^{2r-k}(M,\mathbb{R})$, where $S$ is null-cobordant $(k-1)$-manifold, once a $G$-invariant polynomial $p$ of degree $r$ on $\text{Lie}(G)$ is fixed. For $S=S^{k-1}$, this gives a homomorphism $\pi_{k-1}(\mathcal{F})\longrightarrow H^{2r-k}(M,\mathbb{R})$. The map is shown to be globally gauge invariant and furthermore it descends to the moduli space of flat connections $\mathcal{F}/\mathrm{Gau}P$, modulo cohomology with integer coefficients. The construction is also adapted to complex manifolds. In this case, one works with the set $\mathcal{F}^{0,2}$ of connections with vanishing $(0,2)$-part of the curvature, and the Dolbeault cohomology. Some examples and applications are presented.