Connections and loops intertwinning

TL;DR

This paper establishes a mathematical relationship between holonomies, loop ensembles, and Casimir operators on finite graphs, extending to a self-adjoint form under Yang-Mills measure.

Contribution

It introduces a novel intertwining relation linking holonomies and Casimir operators, including a deformation that ensures self-adjointness in the Yang-Mills setting.

Findings

Trace of holonomies determines intertwining relations.

Extension to self-adjoint Casimir operators with deformation.

Applicable to finite graph structures in gauge theories.

Abstract

On a finite graph, we prove that trace of holonomies determine an intertwining relation between merge-and-split generators on collections of geodesic loops ensembles and Casimir operators on unitary connections. By adding a deformation part to the generator on loops, this result is extended to the Casimir operator modified in order to be self adjoint with respect to Yang- Mills measure.