Cohomology of flat bundles and a Chern-Simons functional

TL;DR

This paper demonstrates that flat principal bundles with compact connected structure groups share cohomology with trivial bundles, and uses this to define invariants related to gauge transformations and Chern-Simons functionals.

Contribution

It establishes cohomological equivalences for flat bundles and constructs a new invariant of gauge transformations affecting Chern-Simons functionals.

Findings

Flat principal bundles have the same cohomology as trivial bundles.

Constructed a cohomology class related to the Maurer-Cartan 3-form.

Defined an invariant of gauge transformations impacting Chern-Simons functionals.

Abstract

We show that a flat principal bundle with compact connected structure group and its adjoint bundles of Lie groups have the same cohomology as the trivial bundle, which is done by proving they satisfy the condition for the Leray-Hirsch theorem. This information has been used to construct a cohomology class of the adjoint bundle of a flat bundle whose restriction to each fiber is the class of the Maurer-Cartan 3-form. Then we use this result to define an invariant of a gauge transformation of a flat bundle which describes the effect of the gauge transformation on a Chern-Simons functional.