Makeenko-Migdal equations for 2D Yang-Mills: from lattice to continuum

TL;DR

This paper proves that the discrete Makeenko-Migdal equations for 2D Yang-Mills theory on a lattice converge to the continuum equations, linking lattice models to the continuum limit through Wilson loop analysis.

Contribution

It establishes the rigorous convergence of lattice-based Makeenko-Migdal equations to their continuum form in 2D Yang-Mills theory.

Findings

Discrete equations converge to continuum equations

Identification of limits as area derivatives of Wilson loops

Provides a rigorous mathematical foundation for lattice-continuum transition

Abstract

In this paper, we prove the convergence of the discrete Makeenko-Migdal equations for the Yang-Mills model on $(\varepsilon \mathbf{Z})^{2}$ to their continuum counterparts on the plane, in an appropriate sense. The key step in the proof is identifying the limits of the contributions from deformations as the area derivatives of the Wilson loop expectations.