Flat connections in three-manifolds and classical Chern-Simons invariant

TL;DR

This paper introduces a method for constructing smooth flat connections on 3-manifolds using Heegaard splittings and relates the classical Chern-Simons invariant to intersection data and representation volumes, with explicit examples.

Contribution

It provides a new systematic approach to build flat connections on 3-manifolds and expresses the Chern-Simons invariant in a canonical form based on topological and representation-theoretic data.

Findings

Explicit construction of flat connections from fundamental group representations

Canonical form of the Chern-Simons invariant involving intersections and volumes

Examples demonstrating the computation of invariants in nontrivial manifolds

Abstract

A general method for the construction of smooth flat connections on 3-manifolds is introduced. The procedure is strictly connected with the deduction of the fundamental group of a manifold M by means of a Heegaard splitting presentation of M. For any given matrix representation of the fundamental group of M, a corresponding flat connection A on M is specified. It is shown that the associated classical Chern-Simons invariant assumes then a canonical form which is given by the sum of two contributions: the first term is determined by the intersections of the curves in the Heegaard diagram, and the second term is the volume of a region in the representation group which is determined by the representation of pi_1(M) and by the Heegaard gluing homeomorphism. Examples of flat connections in topologically nontrivial manifolds are presented and the computations of the associated classical Chern-Simons invariants are illustrated.