Loop Equation and Wilson line Correlators in Non-commutative Gauge Theories

TL;DR

This paper derives new loop equations for Wilson line correlators in non-commutative gauge theories, revealing how they relate to closed Wilson loops and highlighting non-planar effects that do not vanish as non-commutativity goes to zero.

Contribution

It introduces a novel loop equation for Wilson line correlators in non-commutative gauge theories, connecting them to closed Wilson loops and analyzing non-planar contributions.

Findings

Joining Wilson lines relates their correlators to closed Wilson loops.

Non-planar contributions do not smoothly vanish as non-commutativity approaches zero.

Perturbative checks support the derived loop equations.

Abstract

We investigate Schwinger-Dyson equations for correlators of Wilson line operators in non-commutative gauge theories. We point out that, unlike what happens for closed Wilson loops, the joining term survives in the planar equations. This fact may be used to relate the correlator of an arbitrary number of Wilson lines eventually to a set of {\it closed} Wilson loops, obtained by joining the individual Wilson lines together by a series of well-defined cutting and joining manipulations. For closed loops, we find that the non-planar contributions do not have a smooth limit in the limit of vanishing non-commutativity and hence the equations do not reduce to their commutative counterparts. We use the Schwinger-Dyson equations to derive loop equations for the correlators of Wilson observables. In the planar limit, this gives us a {\it new} loop equation which relates the correlators of Wilson lines to the expectation values of closed Wilson loops. We discuss perturbative verification of the loop equation for the 2-point function in some detail. We also suggest a possible connection between Wilson line based on an arbitrary contour and the string field of closed string.