The following is a slightly edited version of an (unsuccessful) research proposal that I submitted in 2002. An earlier version had been refused funding a few years earlier. I would be interested in reviewing this project in the light of the recent results on the cobordism conjecture and the classification of TQFTs.
Topological Quantum Field Theories (TQFTs) provide a relatively new tool for the study of geometric and topological properties of manifolds (and other spaces). Methods for generating TQFTs have been described that relate to knot theory, quantum groups and their representations, or to homotopical and cohomological invariants. Yetter’s constructions suggest a much wider range of possibilities and their higher dimensional analogues extend that even further, shedding light on the extended TQFTs (ETQFTs) of Freed, Kapranov and Voevodsky. Porter’s own results, interpret Yetter’s construction in a simplicial manner and indicated how this link with simplicial sets and simplicial group theory enables a far ranging generalisation of Yetter’s construction to be given. This is closely linked to ideas of Turaev on HQFTs?.
The aims of the project are:
(i) to test the properties of the simplicial methods known to work for the construction of TQFTs and ETQFTs including some indication of how general such simplicially generated TQFTs may be;
(ii) to use relative methods to encode more geometric information in such simplicially generated TQFTs;
(iii) to compare simplicially with categorically generated theroies;
(iv) to study and interpret such TQFTs in terms of stacks, non-abelian cohomology, etc.;
(v) to extend from crossed module / cat-1-group coefficients to their lax analogues, categorical groups, and to investigate such a laxification process in other simplical and categorical settings.
The overall aim of the research is to find a classification theorem for a large class of TQFTs and ETQFTs. The above aims relate, in part, to data gathering leading to the construction of a large class of non-trivial examples of TQFTs. This seems to be a necessary first step towards such a classification theorem as the number of known TQFTs is still relatively small in higher dimensions.
The initial measurable objectives are the successful completion of various specific families of examples, or more explicitly:
(i) development of TQFTs and ETQFTs with simplicial Hopf algebra?s and/or simplicial Lie algebras as coefficients (considered as test cases);
(ii) development of TQFTs and ETQFTs with relative coefficients, encoding G-structures and linking with Turaev’s HQFTs;
(iii) development of TQFTs and ETQFTs adapting simplicial techniques but with coefficients in models for weak $n$-categories. Followed by the study and interpretation of the ‘stack-like’ gadgets to be found in each case;
(iv) comparison of the constructions found in (iii) with known structures from other areas e.g. Hischowitz-Simpson’s higher stacks.
Several of the best known constructions of TQFTs have used triangulations and / or the dual decomposition of the manifolds involved. Typically the edges of the triangulation are labelled with elements or objects from an algebraic structure such as a finite group or a category of representations of some quantum group. The labelling is to satisfy equations specified by the 2-simplices of the triangulation, corresponding to ‘integrating’ the labels around a 1-boundary to get a trivial element. The innovation of Yetter, amongst others, was to replace the simple single layer of ‘coefficients’ by a graded object so that labels at one higher level than those for the edges replaced the equations. These higher-level labels give the ‘value’ of the result of integrating the field (i.e. the lower labelling) around the 1-boundary and thus measure the ‘holonomy’ of the field locally. The algebraic gadgets used by Yetter were cat-1-groups and symmetric monoidal categories, the latter coefficients also being used by many others following Reshetikhin and Turaev.
The Principal Investigator noted that this labelling could be separated into two parts: (i) a universal construction of a simplicial groupoid, $G(T)$, from the triangulation, $T$, of the manifold, and (ii) a simplicial homomorphism from $G(T)$ to the coefficients. (The construction of this simplicial groupoid, $G(T)$, was developed initially by Dwyer and Kan.) As $G(-)$ is adjoint to a functor $\overline{W}$, this identified the labellings as simplicial maps from $T$ to $\overline{W}(G)$, for coefficients in $G$, a finite simplicial group. Attempts to generalise this to the case of symmetric monoidal categories encounter a larger element of coherence theory and this will form part of the project (see below).
There are a series of categories from simplicial groups through simplicial Hopf algebras, simplicial Lie algebras to differential graded algebras, that encode homotopical invariants of spaces, cf. [homalg,rat]. The simplicial construction of TQFTs takes the set of labellings as a basis for a vector space over the complex numbers. Thus the elements of that vector space are labellings with values in the simplicial Hopf algebra (group algebra) of the simplicial group of coefficients. Noting that particular simplical Hopf algebras (simplical Quantum groups) seem to arise in connection with the Drinfeld-Yetter modules (also called crossed modules over a quantum group) considered, e.g. by Majid, in relation to representations of the quantum doubleconstruction, this suggests an attack on more general coefficients such as Lie groups, quantum groups etc. These will be part of the first phase of the programme. To some extent they are routine and will thus be very appropriate for trainig a RA in the techniques used in the area.
The maps from $T$ to $\overline{W}(G)$ are well known to correspond to simplicial fibre bundles on $T$ with fibres isomorphic to $G$ as simplicial sets. Thus any conjectured labellings in, say, a simplicial quantum group, should correspond, likewise, to some sort of fibre bundle with the simplicial quantum group as fibre. Such objects would seem to be as yet illknown (although possibly related to ideas of Majid). This is only one of three or four structures available for study. The situation is very rich and its interpretation, both in terms of fibre bundles and more generally of stacks (i.e. fibre bundles up to homotopy coherence) and non-abelian cohomology, is technically exacting. The limits of this simplicial ‘technology’ and of its interpretation are unknown. (Indications that it may have considerable geometric impact come from work linking topological quantum field theory with stacks of groupoids and non-abelian cohomology and extensive work by Mac Kaaij?, Picken and others on the related notion of gerbes in this context.)
One secondary aim of the project is to test these simplicial methods to find their main properties. For each of the possible classes of simple coefficient systems, (simplicial Hopf algebras, simplicial Lie algebras, simplicial quantum groups), the necessary finiteness conditions to ensure that the conjectured TQFT is finite dimensional will be sought. In addition, the case of \emph{homotopy finite} simplicial groups will be analysed and if possible extended. The framework of the construction should be the same in many cases, but the technical details are in most cases as yet unknown, e.g. for the detailed structure of the various $\overline{W }$-constructions within the finiteness constraints assumed.
In each case, a routine investigation will be made of any universal bundle-like construction corresponding to $WG \rightarrow \overline{W} G$ in the group case. Alternative descriptions of such structures will be sought where possible. Analogues of non-abelian cohomology will be investigated and interpreted. Examples of properties satisfied by simplicially generated TQFTs but which may fail for more general examples will be sought. (The extendibility to a simplicially enriched ETQFT is expected to be one such property.)
These initial steps for the first class of coefficients should be completed quite rapidly as the project will at this stage be building on existing simplicial technology. A key question will be: under what circumstances do different types of simplicial coefficients give isomorphic TQFTs (or equivalent ETQFTs)? This should take approximately 6 to 9 months allowing for the initial ‘training period’ (3 months).
The second class of ‘coefficients’ to be considered poses new problems. The nerves of strict monoidal categories have structure similar to that of a cat-1-group. This latter structure, that of a T-complex, involves a type of ‘canonical’ or ‘thin’ filler for all boxes, thus these nerves are Kan complexes. The class of ‘weak’ T-complexes, in which certain boxes need not have these ‘thin’ fillers, gives weak Kan complexes and corresponds approximately to the nerves of strict monoidal categories. Infinity categories have similar structure in their nerves as shown by Street and Verity. (The ‘best’ way to laxify infinity category theory is unknown, but the work of Baez, Kapranov, Voevodsky, Tamsamani, Simpson, Leinster and others indicates how important this may be for several areas of mathematics, whilst the homotopy coherent category theory of Cordier and Porter develops a possible candidate in a locally weakly Kan version.) The potential of left adjoints to any nerve functor will be investigated and exploited. If present, say, in the weak $\infty$-category case, it would allow a treatment using $\infty$-category techniques somewhat nearer tithe existing Reshetikhin and Turaev approach using monoidal categories.
This second class of coefficients should by itself take about nine months of work, but concurrently with this group, the question of relative theories will be examined. Many geometric structures correspond to $G$-structure maps to a classifying space $\overline{W} (G)$ for $G$ a simplicial group, however it is rare that the simplicial group involved is finite. Key topological problems (e.g. the Hauptvermutung ) correspond to lifting a structure map back up a fibration or ‘reducing’ it to a subgroup. The interesting case where the homotopy fibre of the comparison map is homotopy finite will be examined. Again interpretation in terms of bundles, stacks and non-abelian cohomology will be sought and algebraic models (generalising Frobenius algebras) wil be examined. This relative TQFT is a potential way of studying constructions of HQFTs (Turaev). The place of his ‘base space’ $X$ is taken by $\overline{W}(G)$, the classifying space of the simplicial group. Allowing higher homotopy information, i.e. not restricting $X$ to be a $K(G,1)$, has been noted as important by Turner and Willerton who view this emerging theory from a different perspective, closer to conformal field theory. Relative cases of the Hopf and Lie algebra cases will also be looked at. The completion of this ‘relative’ phase is only expected towards the 24 to 30 month mark as the interpretation of it should also look at any obvious geometric consequences.
This will concentrate on examining extended TQFTs. The Yetter construction naturally leads not only to a TQFT with finite dimensional vector spaces as its values, but also to one with simplicial vector spaces. There is a well-developed theory of simplicial vector spaces dating from the late 1960s and well-structured invariants can be expected. This will be further strengthened by methods from simplicial Hopf algebra theory. Some investigation of the enriched / extended TQFT structures available will be made during the earlier phase, (for instance it is ‘clear’ that any simplicially enriched ETQFT should yield a TQFT, but this does need proving in detail.) Such results will be re-examined in this final phase and, where it is felt that they reveal relevant structure, they will be studied in detail and depth in this last phase of the project.
TQFTs are a means for a) studying topological and geometric invariants, that are often less accessible by other means, and b) studying the topological and geometric underpinning of the quantum field theories of theoretical physics. The project aims initially to contribute mainly to a), but several theoretical physicists, notably Baez and Freed, have shown a considerable interest in the relationship of these constructions to b).
One central problem in TQFT theory is the classification of TQFTs. Various researchers (Quinn, Turaev, etc.) have suggested algebraic structures generalising Frobenius algebras, that are quite successful in some dimensions, but it seems to be unknown if they generalise well to all dimensions or if they can be extended to ETQFTs. This project is founded on the assumption that a larger bank of examples and a deeper knowledge of how (E)TQFTs and HQFTs can be generated will help to crack this classification problem.
The research is timely. The key results of the Principal Investigator were proved in 1995 and published 1997-98. The referee for the first paper was of the opinion that the deep results of that paper broke significant new ground’ in our understanding of the links between TQFTs and homotopical methods. Turaev’s work on HQFTs followed shortly. Considerable work by Hirschwitz, Simpson, etc., on higher stacks has appeared in the last three years and relevant work by Turner, Willerton and others on HQFTs and non-abelian cohomology has appeared only very recently in preprint form.
Many problems in quantum field theory (as physics) are related to the mathematical problem of understanding path integrals. These are ‘understood’ at a physical level, but are not even known to exist at a mathematical level except for various relatively simple cases and ‘toy models’ where finiteness plays a large part. These problems relate to deep functional analytic questions about the existence of measures on certain spaces of connections. The study of TQFTs is one of the mainstays of the attack on this problem as, by ignoring the deep Riemannian geometry of the physical case and concentrating on the topology involved, it is hoped that the exact meaning of the path integrals will become clearer. This links with approaches that try to integrate geometric structures (work again of Baez, Freed, etc.) instead of merely integrating functions on spaces of connections.
[Cordier-Porter,97] J.-M. Cordier and T. Porter, Homotopy Coherent Category Theory, Trans. Amer. Math. Soc. 349 (1997) 1-54.
[Porter,96] T. Porter, Interpretations of Yetter’s notion of G-coloring : simplicial fibre bundles and non-abelian cohomology, J. Knot Theory and its Ramifications 5 (1996) 687-720.
[Porter,98] T. Porter, TQFTs from Homotopy n-types, J. London Math. Soc., 58 (1998) 723 – 732.
[Porter-Turaev] Formal Homotopy Quantum Field Theories I : Formal Maps and Crossed C -algebras, (arXiv, math.QA/0512032), Journal of Homotopy and Related Structures, 3(1), 2008, 113 – 159.
[Porter,07] Formal Homotopy Quantum Field Theories II: Simplicial Formal Maps, in Cont. Math. 431, (2007), 375-404 (Streetfest volume: Categories in Algebra, Geometry and Mathematical Physics - A. Davydov, M. Batanin, and M. Johnson, Macquarie University, S. Lack, University of Western Sydney, and A. Neeman, Australian National University,)(available also at arXiv, math.QA/0512034).
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[Quillen,69] D.A. Quillen, Rational homotopy theory, Ann. Math. 90 (1969) 205 - 295.
[Turaev,99] V. Turaev, Homotopy field theory in dimension 2 and group-algebras, preprint arXiv: math.QA/9910010
[Turaev,00] V. Turaev, Homotopy field theory in dimension 3 and crossed group-categories., preprint arXiv:math.GT/0005291 v1
[Yetter,93] D.N. Yetter, TQFTs from homotopy 2-types, J. Knot Theory and its Ramifications, 2 (1993) 113 - 123.