Third Group Cohomology and Gerbes over Lie Groups

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 ${\text{H}}^3(H,\mathbb{Z})$. 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 ${\text{H}}_s^2(N,A)\to {\text{H}}_s^3(H,A^N)$, 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.

in J. Geom. Phys. (2016)

References in corpus (3)

• Introduction to the language of stacks and gerbes
• Abelian extensions of infinite-dimensional Lie groups
• From gauge anomalies to gerbes and gerbal actions

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