Cohomological twisting of 2-linearization and extended TQFT
type: Item
instance of: scholarly article
title: Cohomological twisting of 2-linearization and extended TQFT
zbMATH Open document ID: 1333.18012
published in: Journal of Homotopy and Related Structures
publication date: 2015-06-22T00:00:00Z
full work available at URL: https://arxiv.org/abs/1003.5603
review text: An \(n\)-dimensional topological quantum field theory (TQFT) is a symmetric monoidal functor from a category, \(\mathbf{nCob}\), of \((n-1)\)-dimensional manifolds and cobordisms between them and the category of vector spaces (or Hilbert spaces) and linear maps. One important example of such is the Dijkgraaf-Witten (DW) model, associated to a finite group, \(G\), or more generally to such a \(G\) together with a `twisting' 3-cocycle \(\omega\in Z^3(BG,U(1))\). Fully extended TQFTs are based on a higher order notion of cobordism so that one has, in general, an \(n\)-category whose objects are points and whose \(n\)-morphisms are cobordisms \textit{with corners}, and then with a codomain \(n\)-category being an \(n\)-category of some form of \(n\)-vector space. It is useful, however, also to consider a more specific form in which we have a cobordism bicategory, \(\mathbf{nCob}_2\), whose objects are \((n-2)\)-categories without boundaries, whose 1-morphisms are \((n-1)\)-dimensional (collared) cobordisms between such and whose 2-morphisms are (approximately) \(n\)-dimensional collared cobordisms with corners. The codomain 2-category will be \(\mathbf{2Vect}\), the 2-category of (Kapranov-Voevodsky) 2-vector spaces, and, for this paper, an extended TQFT is then a symmetric weak 2-functor from \(\mathbf{nCob}_2\) to \(\mathbf{2Vect}\). Following Freed, Hopkins, Lurie and Teleman, [\textit{S. S. Freed} et al., ``Topological quantum field theories from compact Lie groups'', \url{arXiv:0905.0731}], one expects an ETGQFT to consist of two ingredients: (i) a classical field theory, in which the values of the fields live in an \(n\)-groupoid and (ii) a `quantization' functor which takes \(n\)-groupoids and spans between them to \((n+1)\)-algebras and morphisms between them. The main result of this paper is that given a finite group, \(G\), and \(\omega\) as before, there is a symmetric monoidal functor \(Z^\omega_G:\mathbf{nCob}_2\to \mathbf{2Vect}\) that reproduces, for \(n=…
Mathematics Subject Classification ID: 18E10
Mathematics Subject Classification ID: 20L05
Mathematics Subject Classification ID: 57R56
zbMATH DE Number: 6448495
zbMATH Keywords: TQFT
zbMATH Keywords: groupoids
zbMATH Keywords: 2-vector spaces
zbMATH Keywords: gauge theory
author: Jeffrey Colin Morton
reviewed by: Timothy Porter
MaRDI profile type: Publication
OpenAlex ID: W1988228253
arXiv ID: 1003.5603
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DOI: 10.1007/S40062-013-0047-2
description: scientific article