Flat connections and cohomology invariants

Abstract

The main goal of this article is to construct some geometric invariants for the topology of the set F of flat connections on a principal G-bundle P→M. Although the characteristic classes of principal bundles are trivial when F≠∅, their classical Chern–Weil construction can still be exploited to define a homomorphism from the set of homology classes of maps S→F to the cohomology group H2r−k(M,R), where S is null-cobordant (k−1)-manifold, once a G-invariant polynomial p of degree r on Lie(G) is fixed. For S=Sk−1, this gives a homomorphism πk−1(F)→H2r−k(M,R). The map is shown to be globally gauge invariant and furthermore it descends to the moduli space of flat connections F/ Gau P, modulo cohomology with integer coefficients. The construction is also adapted to complex manifolds. In this case, one works with the set F0,2 of connections with vanishing (0, 2)-part of the curvature, and the Dolbeault cohomology. Some examples and applications are presented.