Cohomological twisting of 2-linearization and extended TQFT
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 \(n=3\), the DW-model with twisting cocycle \(\omega\), and, moreover, is constructed as a composite \(\Lambda^{U(1)}\circ \mathcal{A}_0(-)^\omega_G\), where \(\mathcal{A}_0(-)^\omega_G\) is an \(\omega\)-twisted form of a functor from \(\mathbf{nCob}_2\) to \(Span(Gpd)\), induced by the groupoid of flat connections functor, and \(\Lambda^{U(1)}\) is a quantization functor as expected. The paper explores both the untwisted and the twisted cases in a lot of enlightening detail, showing the feasibility of doing detailed calculations in numerous example cases. There is extensive use made of both spans and cospans in various contexts and these are applied to provide a categorical and calculatory underpinning to the theory.
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