Third Group Cohomology and Gerbes over Lie Groups
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arXiv
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| Natura: | Preprint |
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2016
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| _version_ | 1866914215320616960 |
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| author | Mickelsson, Jouko Wagner, Stefan |
| author_facet | Mickelsson, Jouko Wagner, Stefan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1602_02565 |
| institution | arXiv |
| publishDate | 2016 |
| record_format | arxiv |
| spellingShingle | Third Group Cohomology and Gerbes over Lie Groups Mickelsson, Jouko Wagner, Stefan Mathematical Physics Differential Geometry Representation Theory 22E65, 22E67 (primary), 20J06, 57T10, 81R10 (secondary) 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. |
| title | Third Group Cohomology and Gerbes over Lie Groups |
| topic | Mathematical Physics Differential Geometry Representation Theory 22E65, 22E67 (primary), 20J06, 57T10, 81R10 (secondary) |
| url | https://arxiv.org/abs/1602.02565 |