Quantum field theory and the Jones polynomial
One of the problems recently proposed by \textit{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 [\textit{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.
scientific article; zbMATH DE number 4092352
scientific article