Topological Observables in Semiclassical Field Theories

TL;DR

This paper develops a geometric framework for semiclassical Euclidean field theories with instantons, revealing a topological sector in observables linked to intersection theory, which remains exact across perturbation orders.

Contribution

It introduces a geometric setup for semiclassical approximations in field theories with instantons, highlighting a topological sector in observables related to intersection theory.

Findings

Topological observables are independent of covariance.

The topological sector is exactly computable at all perturbation orders.

Expectation values in the topological sector are computable by setting covariance to zero.

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 ${\cal M}$. The standard examples are of course Yang-Mills theory and non-linear $\sigma$-models. The relevant space here is a family of measure spaces $\tilde {\cal N} \ra {\cal M}$, with standard fibre a distribution space, given by a suitable extension of the normal bundle to ${\cal M}$ in the space of smooth fields. Over $\tilde {\cal N}$ there is a probability measure $d\mu$ given by the twisted product of the (normalized) volume element on ${\cal M}$ and the family of gaussian measures with covariance given by the tree propagator $C_\phi$ in the background of an instanton $\phi \in {\cal M}$. The space of ``observables", i.e. measurable functions on ($\tilde {\cal N}, \, d\mu$), is studied and it is shown to contain a topological sector, corresponding to the intersection theory on ${\cal M}$. 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.