Elsevier

Physics Letters B

Volume 297, Issues 1–2, 24 December 1992, Pages 82-88
Physics Letters B

Topological “observables” in semiclassical field theories

https://doi.org/10.1016/0370-2693(92)91073-IGet rights and content

Abstract

We give a geometrical set-up for the semiclassical approximation to euclidean field theories having families of minima (instantons) parametrized by suitable moduli spaces M. The standard examples are of course Yang-Mills theory and non-linear σ-models. The relevant space here is a family of measure spaces Ñ→M, with standard fibre a distribution space, given by a suitable extension of the normal bundle to M in the space of the smooth fields. Over Ñ there is a probability measure dμ given by the twisted product of the (normalized) volume element on M and the family of gaussian measures with covariance given by the tree propagator Cφ in the background of an instanton φϵM. The space of “observables”, i.e., measurable functions on (Ñ, dμ), is studied and it is shown to contain a topological sector, corresponding to the intersection theory on M. The expectation value of these topological “observables” does not depend on the covariance; it is therefore exact at all orders in perturbation theory and can moreover be computed in the topological regime by setting the covariance to zero.

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