Poisson brackets of Wilson loops and derivations of free algebras

Abstract

We describe a finite analog 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 noncommutative space. This suggests that noncommutative geometry plays a fundamental role in the manifestly gauge invariant formulation of Yang–Mills theory. We also construct the deformation of the algebra of loops induced by quantization, in the large-Nc limit.