Poisson Algebra of Wilson Loops and Derivations of Free Algebras

TL;DR

This paper introduces a finite analogue of the Poisson algebra of Wilson loops in Yang-Mills theory, revealing its connection to non-commutative geometry and constructing its quantized deformation in the large N_c limit.

Contribution

It establishes a novel finite Poisson algebra for Wilson loops and links it to non-commutative geometry, also deriving its quantized deformation at large N_c.

Findings

Finite analogue of Wilson loop Poisson algebra identified

Connection to Lie algebra of vector fields on non-commutative space shown

Deformation of the loop algebra constructed via quantization in large N_c limit

Abstract

We describe a finite analogue of the Poisson algebra of Wilson loops in Yang-Mills theory. It is shown that this algebra arises in an apparently completely different context; as a Lie algebra of vector fields on a non-commutative space. This suggests that non-commutative geometry plays a fundamental role in the manifestly gauge invariant formulation of Yang-Mills theory. We also construct the deformation of the loop algebra induced by quantization, in the large N_c limit.