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Quantum field theory and the Jones polynomial.(English)Zbl 0667.57005

One of the problems recently proposed byM. Atiyah [Proc. Symp. Pure Math. 48, 285-299 (1988)] for experts in quantum field theory was to find an intrinsically three dimensional definition of the Jones polynomial [V. F. R. Jones, Bull. Am. Math. Soc., New Ser. 12, 103- 111 (1985;Zbl 0564.57006); Ann. Math., II. Ser. 126, 335-388 (1987;Zbl 0631.57005)] of knot theory. In this work the author considers the quantum field theory defined by the nonabelian Chern-Simons action and shows that it is exactly soluble and has important implications for three dimensional geometry and two dimensional conformal field theory. Among the geometrical implications of the \(2+1\) dimensional quantum Yang-Mills theory with a pure Chern-Simons action is a natural framework for the understanding of the Jones polynomial. In this description, the Jones polynomial can be generalized from \(S^ 3\) to arbitrary three manifolds, giving invariants of three manifolds that are computable for a surgery presentation. The work provides a penetrating insight into the conformal field theory in \(1+1\) dimensions.
Reviewer: Ch.Sharma

MSC:

57N10 Topology of general \(3\)-manifolds (MSC2010)
57M25 Knots and links in the \(3\)-sphere (MSC2010)
81T17 Renormalization group methods applied to problems in quantum field theory
57R65 Surgery and handlebodies

Cite

References:

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This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
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