Simplicial and categorical methods in TQFTs and HQFTs

Methods for constructing topological quantum field theories from finite groups were well known in the 1990s. Given a finite group, $G$, one constructed a ‘state sum’, which measured isomorphism classes of principal $G$-bundles on a manifold, $X$, and which could trace the evolution of this state space through a cobordism from $X$ to another manifold $Y$.

In 1992-93, Yetter adapted these models first to give a slightly different presentation of the same basic idea using colourings of triangulations of the manifolds by elements of $G$, and then to replace the role of $G$ by a finite 2-type, represented by a finite crossed module. The interpretation of his invariants and the corresponding TQFT was far from clear. He also posed several questions, the third one of which concerned the possibility of adapting his method to taking a finite $n$-type as ‘coefficients’.

By reworking Yetter’s definition of colouring, Porter showed in 1995 that what Yetter’s construction took as the basic ‘colourings’ were just simplicial maps from an ordered triangulation of the manifold to the classifying space of the group or crossed module. They thus corresponded to principal simplicial bundles with a particularly simple simplicial automorphism group related to the group or crossed module. (The case of a group was just that of a particular class of covering space as had been already noted by Yetter.) This also linked the resulting TQFT with non-Abelian cohomology theory.

In 1998, continuing this work, Porter answered Yetter’s third problem asking how one might construct TQFTs of arbitrary dimension from algebraic models of finite n-types by `colouring’ ordered triangulations of both the manifolds and the cobordisms. The bundle theoretic aspect was still there, and linked up well with Grothendieck’s pursuit of stacks, as a generalisation of the covering space interpretation given by Yetter.

In collaboration with João Faria Martins, the corresponding Yetter invariant (value of the TQFT on a closed n+1 dimensional manifold) was interpreted as an alternating product, related to the homotopy groups of certain mapping spaces, and this was used to derive an extension of a well known construction of the Dijkgraaf-Witten invariant.

In 1999-2000, Vladimir Turaev introduced *Homotopy Quantum Field Theory*, i.e., HQFTs. (A similar idea had been considered by Porter (unpublished, see here, in research proposals at about the same time.) Turaev’s work considered manifolds with a characteristic map to a fixed ‘background’ space, with cobordisms between such allowing the characteristic maps to ‘evolve’ from one such to another. This was very amenable to analysis using methods adapted from the analysis of Yetter’s TQFTs, with the role of the finite group or crossed module which played the role of ‘coefficients’ in Yetter’s theory, being here replaced by their classifying space as background.

Although Turaev defined HQFTs with arbitrary base, he considered in detail only the case of the base being a 1-type, even though Rodrigues had showed that if looking, for instance, at 2-dimensional HQFTs, it was the 2-type of the base that mattered, so there was potential for extra structure to be available.

In 2007-8, Porter and Turaev solved the problem of what, in general, the extra structure available in a 2-type gave to the possible HQFTs. Turaev’s algebraic models for 2D HQFTs with base $BG$ were $G$-graded Frobenius algebras with some extra twisting structure. If $G$ was replaced by a crossed module, $C$, so that the background, $BC$, was now a 2-type, then the extra part of the crossed module structure acted by conjugation be some related units, on the graded Frobenius algebra structure.

Porter gave an interpretation of these HQFTs in terms of stacks and related structures and continues to research these links. In an effort to clarify the connections between this theory and the methods of non-Abelian cohomology, stacks, gerbes etc., Porter has written and made available on the internet a set of notes, The Crossed Menagerie:an introduction to crossed gadgetry and cohomology in algebra and topology, and has prepared some related material here on the nLab, so as to try to take a relatively complete and consistent view of the area.

Revised on April 17, 2012 09:18:10
by Tim Porter
(131.251.253.155)