Dynamics of Wilson Observables in Non-Commutative Gauge Theory

TL;DR

This paper derives equations governing Wilson loop averages in non-commutative Yang-Mills theory, revealing connections to ordinary Yang-Mills and introducing new gauge invariant observables involving open contours.

Contribution

It provides the first derivation of Wilson loop equations in non-commutative gauge theory, including finite N effects and correlators of open Wilson lines.

Findings

In the 't Hooft limit, equations reduce to ordinary Yang-Mills loop equations.

Finite N equations involve open contour observables.

Perturbative checks support the derived equations.

Abstract

An equation for the quantum average of the gauge invariant Wilson loop in non-commutative Yang-Mills theory with gauge group U(N) is obtained. In the 't Hooft limit, the equation reduces to the loop equation of ordinary Yang-Mills theory. At finite $N$, the equation involves the quantum averages of the additional gauge invariant observables of the non-commutative theory associated with open contours in space-time. We also derive equations for correlators of several gauge invariant (open or closed) Wilson lines. Finally, we discuss a perturbative check of our results.