Geometric Derivation of the Finite $N$ Master Loop Equation

TL;DR

This paper introduces a geometric method to derive the master loop equation for lattice Yang-Mills models with classical groups, simplifying the process by leveraging the intrinsic geometry of the structure group.

Contribution

It presents a novel geometric derivation of the master loop equation using integration by parts on the structure group, offering a simpler alternative to existing methods.

Findings

Derivation based on intrinsic geometry simplifies the proof.

Comparison shows equivalence with Schwinger-Dyson and stochastic approaches.

Applicable to SO(N), SU(N), and U(N) groups.

Abstract

In this paper we provide a geometric derivation of the master loop equation for the lattice Yang-Mills model with structure group $G \in \{SO(N),SU(N), U(N)\}$. This approach is based on integration by parts on $G$. In the appendix we compare our approach to that of \cite{Ch19a} and \cite{J16} based on Schwinger-Dyson equations, and \cite{SheSmZh22} based on stochastic analysis. In particular these approaches are all easily seen to be equivalent. The novelty in our approach is the use of intrinsic geometry of $G$ which we believe simplifies the derivation.