Covariant Symanzik identities

TL;DR

This paper extends classical isomorphism theorems to vector bundles over graphs with connections, linking Gaussian free fields and path holonomies, and provides new formulas for moments of complex random fields.

Contribution

It introduces distributional identities between Gaussian free vector fields and path holonomies on graphs with connections, generalizing existing isomorphism theorems.

Findings

Extended isomorphism identities to vector bundles of arbitrary rank.

Derived formulas for moments of non-Gaussian fields using path holonomies.

Connected classical results with gauge field concepts in a novel framework.

Abstract

Classical isomorphism theorems due to Dynkin, Eisenbaum, Le Jan, and Sznitman establish equalities between the correlation functions or distributions of occupation times of random paths or ensembles of paths and Markovian fields, such as the discrete Gaussian free field. We extend these results to the case of real, complex, or quaternionic vector bundles of arbitrary rank over graphs endowed with a connection, by providing distributional identities between functionals of the Gaussian free vector field and holonomies of random paths. As an application, we give a formula for computing moments of a large class of random, in general non-Gaussian, fields in terms of holonomies of random paths with respect to an annealed random gauge field, in the spirit of Symanzik's foundational work on the subject.