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Links, quantum groups and TQFTs. (English) Zbl 0872.57002

The paper is an excellent introduction to and survey of link and 3-manifold invariants arising in the framework of tangle functors and quantum groups. “This paper is intended to introduce and invite a large mathematical audience to this exciting field. The focus is on giving the flavor and illustrating some of the power of a few simple ideas. This is attempted by proving an important result with a minimum of machinery”. The paper is written in a clear and organic way, with emphasis on the motivation and the main ideas of the constructions.
The first section discusses the Jones polynomial of a link, its Kauffman bracket form and their skein relations. In the second section, the monoidal category of framed tangles is introduced and the Kauffman bracket is constructed explicitly as a monoidal functor from this category to the category of vector spaces and linear maps. Then, generalizing this construction, a set of axioms on an algebra is given which guarantee that its representation category is naturally the range of such a tangle functor. This requires, in growing order of complexity, bialgebras, Hopf algebras, quasitriangular and ribbon Hopf algebras. At this point Lie algebras come into the play as a natural source of ribbon Hopf algebras which, however, give trivial link information because they are cocommutative. This suggests deforming them within the set of all ribbon Hopf algebras to obtain interesting link invariants thus arriving at the concept of quantum groups and modular Hopf algebras (depending on a complex parameter). The case of \(sl_2\) is discussed in detail. Next the notion of topological quantum field theory (TQFT) is discussed as a functor from the cobordism category of manifolds (which now replaces the tangle category) to the category of vector spaces. In dimension three, such a cobordism (between 2-manifolds) can be encoded purely combinatorially in terms of framed tangles, and now the above machinery of functors from the tangle category giving link invariants is extended to this more general situation leading to invariants of links in 3-manifolds and of 3-manifolds themselves (the case of the empty link). “Section 4 discusses the properties of the quantum groups at roots of unity, summarized as their being modular Hopf algebras, which allows one to construct TQFTs from them. Section 5 gives a categorical formulation of TQFTs. Section 6 sketches a purely combinatorial description of the relevant category, that of biframed 3-dimensional cobordisms, in terms of surgery on links. This allows us in section 7 to construct the TQFT out of the link invariants we have already defined”.
The field has gone a long and fascinating way in the 13 years after Jones’ 1984 discovery of his polynomial, occuring as the trace of a representation of the braid group invariant under the Markov moves which served as a model for all further developments. Up to this point, the emphasis has been almost exclusively on the development of the formalism and problems related to this. As noted in the introduction it is still not clear what impact this will have on the rest of mathematics, in particular on the theory of 3-manifolds itself (in knot theory, the new polynomial invariants have proven themselves extremely useful, for example in resolving the old Tait conjectures). “We have a wealth of invariants, all readily computable, but standing decidedly outside the traditions of knot and 3-manifold theory. They lack a geometric interpretation and consequently have been of almost no use in answering questions one might have asked before their creation. In effect, we are left looking for the branch of mathematics from which these should have come organically”. So the most interesting part of the story has probably still to come.

MSC:

57-02 Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes
57M25 Knots and links in the \(3\)-sphere (MSC2010)
81T99 Quantum field theory; related classical field theories
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