Equivariant characteristic classes in the bundle of connections

TL;DR

This paper explores equivariant characteristic classes in the bundle of connections, linking geometric forms on principal bundles to cohomology classes in the space of connections modulo gauge transformations, extending previous work by Atiyah and Singer.

Contribution

It introduces a framework for equivariant characteristic classes that extend characteristic forms to the space of connections and relates them to known classes by Atiyah and Singer.

Findings

Equivariant characteristic classes provide canonical extensions of characteristic forms.

These classes correspond to cohomology classes in the quotient space of connections.

The framework unifies geometric and topological aspects of gauge theories.

Abstract

The characteristic forms in the bundle of connections of a principal bundle P over M determine the characteristic classes of P for degree less or equal to the dimension of M, and differential forms on the space of connections for higher degree. The equivariant characteristic classes provide canonical equivariant extensions of this forms, and so cohomology classes in the quotient space of connections modulo gauge transformations. More generally, given a closed differential form in M and a characteristic form, we obtain a cohomology class in the space of connections modulo gauge transformations, and we show that these classes coincide with some classes previously defined by Atiyah and Singer.