Loop Equation in Two-dimensional Noncommutative Yang-Mills Theory

TL;DR

This paper extends the classical Makeenko-Migdal loop equation analysis to two-dimensional noncommutative Yang-Mills theory, deriving differential equations similar to the commutative case but with notable differences in solution methods.

Contribution

It generalizes the loop equation framework to noncommutative geometry, addressing subtleties and showing the equations' form remains consistent with the commutative case at large N.

Findings

Loop equations reduce to usual differential equations in noncommutative case.

Open Wilson line contributions cancel out in the differential equations.

Factorization properties used in commutative case are not applicable here.

Abstract

The classical analysis of Kazakov and Kostov of the Makeenko-Migdal loop equation in two-dimensional gauge theory leads to usual partial differential equations with respect to the areas of windows formed by the loop. We extend this treatment to the case of U(N) Yang-Mills defined on the noncommutative plane. We deal with all the subtleties which arise in their two-dimensional geometric procedure, using where needed results from the perturbative computations of the noncommutative Wilson loop available in the literature. The open Wilson line contribution present in the non-commutative version of the loop equation drops out in the resulting usual differential equations. These equations for all N have the same form as in the commutative case for N to infinity. However, the additional supplementary input from factorization properties allowing to solve the equations in the commutative case is no longer valid.