Planar Markovian Holonomy Fields

TL;DR

This paper introduces planar Markovian holonomy fields, constructs Yang-Mills fields within this framework, and classifies their symmetries, providing a bridge between planar and non-planar theories with explicit loop law computations.

Contribution

It defines and constructs planar Markovian holonomy fields, proves their relation to Yang-Mills fields, and classifies these fields based on symmetry properties.

Findings

Construction of a family of planar Yang-Mills fields.

Any regular planar Markovian holonomy field is a planar Yang-Mills field.

Explicit computation of the law of contractible loops on surfaces.

Abstract

We study planar random holonomy fields which are processes indexed by paths on the plane which behave well under the concatenation and orientation-reversing operations on paths. We define the Planar Markovian Holonomy Fields as planar random holonomy fields which satisfy some independence and invariance by area-preserving homeomorphisms properties. We use the theory of braids in the framework of classical probabilities: for finite and infinite random sequences the notion of invariance by braids is defined and we prove a new version of the de-Finetti's Theorem. This allows us to construct a family of Planar Markovian Holonomy Fields, the Yang-Mills fields, and we prove that any regular Planar Markovian Holonomy Field is a planar Yang-Mills field. This family of planar Yang-Mills fields can be partitioned into three categories according to the degree of symmetry: we study some equivalent conditions in order to classify them. Finally, we recall the notion of Markovian Holonomy Fields and construct a bridge between the planar and non-planar theories. Using the results previously proved in the article, we compute, for any Markovian Holonomy Field, the "law" of any family of contractible loops drawn on a surface.