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

TL;DR

This work establishes a rigorous bridge between lattice and continuum formulations of 2D Yang-Mills by proving that discrete Makeenko-Migdal (master loop) equations for lattice Wilson loops converge to their continuum counterparts on the plane under the scaling $\beta=\varepsilon^{-2}$. The authors integrate Driver's formula, Gaussian (Laplace) approximations, and Lie-group harmonic analysis to track the limiting behavior of deformation terms and show that appropriate linear combinations of lattice loop equations yield the continuum area-derivative structure, including explicit deformation integrals $I_m$. They first treat simple loops and then extend to general loops with crossings, and further extend the results to strings as well as to $SU(N)$ and $SO(N)$, incorporating twisting and expansions. The result provides a rigorous justification of the continuum Makeenko-Migdal equations from discrete lattice dynamics and offers a framework for analyzing area-law-type relations and large-$N$ limits in 2D Yang-Mills from lattice approximations. This advances the mathematical understanding of how lattice gauge theories converge to continuum field theories in two dimensions and supports broader investigations into surface sums and related probabilistic structures.

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.

Figures (6)

• Figure 1: An example of a loop and a neighborhood around a vertex $v$.
• Figure 2: The left picture illustrates part of a loop $l$ in a neighborhood of the crossing point $v$ as in Fig. \ref{['fig:DHKcase']}. The right picture is a lattice approximation of the loop $l$ with respect to $v$. The edge $e^{\varepsilon}$ can be viewed as the "lattice approximation" to $v$. The picture also shows the bonds $\boldsymbol{\epsilon}_1$, $\boldsymbol{\epsilon}_3$ in red in \ref{['e:triple']} (where $\boldsymbol{\epsilon}$ that is not drawn here is a bond in $e^\varepsilon$) which are used later in Lemmas \ref{['lem:ms1']}\ref{['lem:ms2']}.
• Figure 3: Loops obtained by deforming a bond $\boldsymbol{\epsilon}$ of $e^\varepsilon$. Here $|e^\varepsilon|=2\varepsilon$ and $|\boldsymbol{\epsilon}|=\varepsilon$ ($e^\varepsilon$ has two bonds where $\boldsymbol{\epsilon}$ is the upper one). The number of arrows on an edge or bond indicates how many times it shows up in the loop.
• Figure 4: Picture for $l_{F_4,-}^{\boldsymbol{\epsilon}_1}$. The green part is the tree used in the axial gauge fixing. The dashed line is where $\boldsymbol{\epsilon}_1$ was and it is replaced by $\boldsymbol{\epsilon}_1'$ (obviously a backtracking is formed but we just erase it as usual). The variable $a_1$ is the holonomy along the black part of $e_1^\varepsilon$. The pictures for $l_{F_4,+}^{\boldsymbol{\epsilon}_1}$, $l_{F_1,-}^{\boldsymbol{\epsilon}_1}$ and $l_{F_1,+}^{\boldsymbol{\epsilon}_1}$ are similar and we do not draw all of them here.
• Figure 5: Illustrations for $\{l_1,l_2\}$, their merger $l_{12}$ in Theorem \ref{['th:g2']}, and the lattice approximations. Here the dashed lines $A,B$ only mean generic sequences of edges and can allow arbitrary shapes.

Theorems & Definitions (40)

• Remark 1.1
• Remark 1.2
• Theorem 1.3
• Remark 1.4
• Remark 2.1
• Remark 2.2
• Remark 2.3
• Lemma 2.4
• proof
• Remark 2.5