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A new derivation of the finite $N$ master loop equation for lattice Yang-Mills

Hao Shen, Scott A. Smith, Rongchan Zhu

Abstract

We give a new derivation of the finite $N$ master loop equation for lattice Yang-Mills theory with structure group $SO(N)$, $U(N)$ or $SU(N)$. The $SO(N)$ case was initially proved by Chatterjee in \cite{Cha}, and $SU(N)$ was analyzed in a follow-up work by Jafarov \cite{Jafar}. Our approach is based on the Langevin dynamic, an SDE on the manifold of configurations, and yields a simple proof via Itô's formula.

A new derivation of the finite $N$ master loop equation for lattice Yang-Mills

Abstract

We give a new derivation of the finite master loop equation for lattice Yang-Mills theory with structure group , or . The case was initially proved by Chatterjee in \cite{Cha}, and was analyzed in a follow-up work by Jafarov \cite{Jafar}. Our approach is based on the Langevin dynamic, an SDE on the manifold of configurations, and yields a simple proof via Itô's formula.
Paper Structure (4 sections, 4 theorems, 35 equations)

This paper contains 4 sections, 4 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Yang Mills SDE
  4. Proof of the main theorem

Key Result

Theorem 1.1

Let $s$ be as above, and suppose that all vertices that are at distance $\leqslant 1$ from any $l_i$ belong to $\Lambda$. Then for $G=SO(N)$ (Cha) For $G=SU(N)$ (Jafar) Compared to Jafar we use a different notation ${\mathbb M}_{U}^{\pm}$ for the sets of merger terms to distinguish it from $SO(N)$ case. For $G=U(N)$,

Theorems & Definitions (10)

  • Theorem 1.1
  • Remark 2.1
  • Remark 2.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • proof : Proof of Theorem \ref{['theo:main']}