Stochastic Feynman Rules for Yang-Mills Theory on the Plane

TL;DR

This paper introduces a novel discretization approach for 2D Yang-Mills theory using an algebraic stochastic calculus analogue, revealing gauge-invariance restoration and Lie-theoretic identities.

Contribution

It develops a new algebraic stochastic calculus framework for Yang-Mills on the plane, connecting different gauge formulations and deriving Lie-theoretic identities.

Findings

Equivalence of Wilson loop expectations across gauge choices

Restoration of gauge invariance in the continuum limit

Derivation of Lie-theoretic identities involving heat kernels

Abstract

We analyze quantum Yang-Mills theory on $\mathbb{R}^2$ using a novel discretization method based on an algebraic analogue of stochastic calculus. Such an analogue involves working with "Gaussian" free fields whose covariance matrix is indefinite rather than positive definite. Specifically, we work with Lie-algebra valued fields on a lattice and exploit an approximate gauge-invariance that is restored when taking the continuum limit. This analysis is applied to show the equivalence between Wilson loop expectations computed using partial axial-gauge, complete axial-gauge, and the Migdal-Witten lattice formulation. As a consequence, we obtain intriguing Lie-theoretic identities involving heat kernels and iterated integrals.