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Quantum field theory and the Jones polynomial. (English) Zbl 0667.57005
One of the problems recently proposed by M. 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
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