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Abelian Chern-Simons theory and linking numbers via oscillatory integrals. (English) Zbl 0869.46022

The work is dedicated to the creation of a mathematical model of Abelian Chern-Simons theory based on the theory of infinite-dimensional oscillatory integrals, developed in [the first author and R. Hoegh-Krohn, Mathematical theory of Feynman path integrals, Lect. Notes in Math. 523 (1976; Zbl 0337.28009)]. A Chern-Simons path integral is considered as a Fresnel integral in a certain Hilbert space.

Willson loop variables are determined as Fresnel integrable functions. It is mathematically shown that there exists a connection between expectation values of products of Willson loops with respect to the path integral and linking numbers, which is a topological invariant. The topological invariant can be calculated in terms of pairwise linking numbers of the loops as was conjectured by E. Witten [Commun. Math. Phys. 117, No. 3, 353-386 (1988; Zbl 0656.53078)].

46G12Measures and integration on abstract linear spaces
81S40Path integrals in quantum mechanics