Third Group Cohomology and Gerbes over Lie Groups
Jouko Mickelsson, Stefan Wagner
TL;DR
The paper develops a framework tying gerbes over a Lie group $H$ to locally smooth degree-3 cohomology via transgression from abelian extensions in a group extension $1\to N\to G\to H\to 1$, under the key vanishing condition $H_s^1(N,A)=0$. It constructs a transgression map $\tau: H_s^2(N,A)\to H_s^3(H,A^N)$ using smooth crossed modules and identifies $\tau([f])$ with the characteristic class $\chi(\alpha,\widehat S)$, with vanishing of $\tau([f])$ equivalent to prolonging the extension to $1\to \widehat N\to \widehat G\to H\to 1$. The work also provides alternative lifting constructions when $A=Map(G,S^1)$, analyzes the case of non-simply connected bases (notably tori), and discusses gauge theory applications where gerbes model anomalies and central extensions of loop groups. Together, these results clarify how local smooth data on $N$ and its extension governs global gerbe structures on $H$, and they supply concrete tools for gauge-theoretic and representation-theoretic contexts. The framework is ultimately applicable to the study of loop group extensions, crossed modules, and the geometry of gauge-theory moduli spaces.
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.