Connections and loops intertwinning
Yves Le Jan

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.
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Taxonomy
TopicsAdvanced Mathematical Theories and Applications · Advanced Differential Geometry Research · Mathematics and Applications
