A state space for 3D Euclidean Yang-Mills theories

TL;DR

This paper proposes a framework to define Euclidean Yang-Mills theories on the 3D torus as random distributional gauge orbits, enabling the construction of regularized Wilson loop observables under the assumption that they behave like the Gaussian free field.

Contribution

It introduces a novel approach to define 3D Euclidean Yang-Mills theories as gauge orbits, leveraging the Yang-Mills heat flow and GFF-like behavior.

Findings

Existence of Yang-Mills heat flow from GFF-like initial data on 3D torus

Definition of regularized Wilson loop observables in 3D Yang-Mills

Framework for gauge theories as random distributional gauge orbits

Abstract

It is believed that Euclidean Yang-Mills theories behave like the massless Gaussian free field (GFF) at short distances. This makes it impossible to define the main observables for these theories - the Wilson loop observables - in dimensions greater than two, because line integrals of the GFF do not exist in such dimensions. Taking forward a proposal of Charalambous and Gross, this article shows that it is possible to define Euclidean Yang-Mills theories on the 3D unit torus as "random distributional gauge orbits", provided that they indeed behave like the GFF in a certain sense. One of the main technical tools is the existence of the Yang-Mills heat flow on the 3D torus starting from GFF-like initial data, which is established in a companion paper. A key consequence of this construction is that under the GFF assumption, one can define a notion of "regularized Wilson loop observables" for Euclidean Yang-Mills theories on the 3D unit torus.