Categorification of Dijkgraaf-Witten theory (Q1691344)

By using cohomology with coefficients in the Picard groupoid of Hermitian lines, categorification of Dijkgraaf-Witten (DW) theory [\textit{R. Dijkgraaf} and \textit{E. Witten}, Commun. Math. Phys. 129, No. 2, 393--429 (1990; Zbl 0703.58011)]. which provides foundations for a direct construction of DW theory as an Extended Topological Quantum Field Theory (ETQFd), is given. DW theory is a gauge theory with a finite gauge group \(G\). It is a toy model of gauge theory with a compact gauge group. In [loc. cit.], DW-theory descibed as a Toplogucal Quantum Field Theory (TQFT) \(\Phi\), a functor on the category of 3-dimensional (3d) cobordism to a category of vector spaces. Objects of this category are cohomology groups with coefficients in certain line bundle over \(Y\) regarded as \(\partial_\pm X\), where \(X\) is a 3d manifold. Construction in [loc. cit.]. and later improvements of this constructios [\textit{D. S. Freed} and \textit{F. Quinn}, Commun. Math. Phys. 156, No. 3, 435--472 (1993; Zbl 0788.58013)], [\textit{D. S. Freed} et al., in: A celebration of the mathematical legacy of Raoul Bott. Based on the conference, CRM, Montreal, Canada, June 9--13, 2008. Providence, RI: American Mathematical Society (AMS). 367--403 (2010; Zbl 1232.57022)], defined relative to some ``fundamental class'' That is \(\Phi\) is not defined absolutely. In this paper, using cohomology with coefficients in the Picard groupoid, \(\mathcal{L}\), absolute definition of \(\Phi\) is given. The authors say this is obtained during to search how to extend \(\Phi\) as ETQFT, which is defined on cobordisms with cornor [\textit{J. Lurie}, in: Current developments in mathematics, 2008. Somerville, MA: International Press. 129--280 (2009; Zbl 1180.81122)]. Definitions of Picard groupoid \(\mathcal{L}\) and cohomology with coefficients in \(\mathcal{L}\), together with construction of the long cohomology exact sequences, and cap product which already appeared in [Freed and Quinn, loc. cit.] incomplete form, in \S2. Related topics to this subject; ``The Hom 2-chain comples,'' and ``Pic categpories'', are explained in Appendix A and B. \S3 deals with (flat) Hermitian line \(n\)-gerbes (Def. 3.1). A flat hermitian line 0-gerbe can be represented by a functor from the first fundamental groupoid of \(X\) into the Picard groupoid of hermitian lines \(\mathcal{L}\) (Example 3.2. Def.3.5). Then, the relation between flat hermitian 1-gerbe (def.3.9) and the 0-gerbe is studied (Porp. 3.11). It shows the flat hermitian line 1-gerbe describes certain geometric objects over a topological space \(X\) which are classified by the Čech cohomology of \(X\) with coefficients in \(U(1)\). After these preparations, categorification of DW theory (an absolute definition of \(\Phi\)) is given in \S4. For this purpose, first the functor \[ F:C_Y\to \mathcal{L} \] where the fundamental cycle groupoid \(C_Y\) is the full subgroupoid formed by all possibly cycles representing the fundamental cycle and connected by equivalence classes of morphisms given in \(n\)-boundaries modulo \((n+1)\)-boundaries, and \(\mathcal{L}\) is the category of hermitian lines, is defined by using (restriction of) cap product.and show this functor has a limit \(\lim_{C_Y}F\) (Prop.4.1). Hence, we need not select ``fundamental cycle'' in the construction of \(\Phi\) [Freed and Quinn, loc. cit.]. This Prop. is used to study a linear isometry corresponding to an \((n+1)\) cobordism. By using them, pullback map \(p_-^\ast\) , intermediate map \(H^0(\mathcal{L}_X)\), and pushforward map \((p_+)_\ast\) associated to the diagram of spaces \[ \mathrm{Map}(\partial_-X,BG)\leftarrow{p_-}\mathrm{Map}(X,BG)\rightarrow{p_+}\mathrm{Map}(\partial_+X,BG), \] are constructed. The TQFT functor \[ Z^\alpha(X):H^0(\mathrm{Map}(\partial_-X,BG);\mathcal{L}_{\partial_-X})\to H^0(\mathrm{Map}(\partial_+X,BG);\mathcal{L}_{\partial_+X}), \] is defined as the compositions of \((p_+)_\ast,H^0(\mathcal{L}_X)\) and \(p_-^\ast\).