Differentiable Cohomology of Gauge Groups

TL;DR

This paper introduces a new framework for differentiable cohomology of Lie groups, connecting it to classical cohomology theories, secondary characteristic classes, and gauge group extensions, with implications for complex Lie groups and conjectures on differential forms.

Contribution

It defines differentiable cohomology for Lie groups, relates it to existing cohomologies, and explores its applications to gauge groups, secondary classes, and complex Lie groups, proposing new conjectures.

Findings

Differentiable cohomology maps to discrete and Lie algebra cohomology.

Secondary classes from Beilinson lead to differentiable cohomology classes.

Generalization of central extensions of loop groups via differentiable cohomology.

Abstract

We give a definition of differentiable cohomology of a Lie group G (possibly infinite-dimensional) with coefficients in any abelian Lie group. This differentiable cohomology maps both to the cohomology of the group made discrete and to Lie algebra cohomology. We show that the secondary characteristic classes of Beilinson lead to differentiable cohomology classes with coefficients in C*. These may be viewed as an enrichment of the Chern-Simons differential forms. By transgression, classes in differentiable cohomology of a Lie group G lead to differentiable cohomology classes for gauge groups Map(M,G). These classes generalize the central extensions of loop groups. We also discuss holomorphic cohomology of complex Lie groups as the natural place to construct secondary classes. We present several conjectures relating the above cohomology classes to the differential forms of Bott-Shulman-Stasheff.