Third group cohomology and gerbes over Lie groups

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• 作者：Jouko Mickelsson ^{jouko.mickelsson@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; Stefan Wagner ; ^{stefan.wagner@uni-hamburg.de" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 22E65 ; 22E67 ; secondary ; 20J06 ; 57T10
• 刊名：Journal of Geometry and Physics
• 出版年：2016
• 出版时间：October 2016
• 年：2016
• 卷：108
• 期：Complete
• 页码：49-70
• 全文大小：568 K

文摘

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$H$ is given by the third cohomology 20fdbdb692ae44b855223cadcfa2b8">$H^3(H,\mathbb{Z})$. When H$H$ 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$H$. We shall study in more detail this relation in the case of a group extension 1→N→G→H→1$1\to N\to G\to H\to 1$ when the gerbe is defined by an abelian extension $1\to A\to \hat{N}\to N\to 1$ of N$N$. In particular, when $H_s^1(N,A)$ vanishes we shall construct a transgression map 2005e48791dd1e0828e0da1cc69da40">$H_s^2(N,A)\to H_s^3(H,A^N)$, where A^N$A^N$ is the subgroup of N$N$-invariants in A$A$ and the subscript 2030685131" title="Click to view the MathML source">s$s$ denotes the locally smooth cohomology. Examples of this relation appear in gauge theory which are discussed in the paper.