Third Group Cohomology and Gerbes over Lie Groups

TL;DR

This paper explores the relationship between third group cohomology and gerbes over Lie groups, providing a detailed analysis of how gerbes classified by third cohomology relate to group extensions and transgression maps, with applications in gauge theory.

Contribution

It establishes a detailed connection between topological gerbes, third group cohomology, and group extensions for Lie groups, including the construction of a transgression map under specific conditions.

Findings

Constructed a transgression map from second to third cohomology groups.

Analyzed the relation between gerbes and third cohomology in the context of Lie group extensions.

Provided examples relevant to gauge theory applications.

Abstract

The topological classification of gerbes, as principal bundles with the structure group the projective unitary group of a complex Hilbert space, over a topological space $H$ is given by the third cohomology $\text{H}^3(H, \Bbb Z)$. When $H$ is a topological group the integral cohomology is often related to a locally continuous (or in the case of a Lie group, locally smooth) third group cohomology of $H$. We shall study in more detail this relation in the case of a group extension $1\to N \to G \to H \to 1$ when the gerbe is defined by an abelian extension $1\to A \to \hat N \to N \to 1$ of $N$. In particular, when $\text{H}_s^1(N,A)$ vanishes we shall construct a transgression map $\text{H}^2_s(N, A) \to \text{H}^3_s(H, A^N)$, where $A^N$ is the subgroup of $N$-invariants in $A$ and the subscript $s$ denotes the locally smooth cohomology. Examples of this relation appear in gauge theory which are discussed in the paper.