# Smooth approximation of Yang--Mills theory on $\mathbb{R}^{2}$: a rough path approach

## TL;DR

This paper demonstrates that quantum Yang--Mills theory on two-dimensional Euclidean space can be smoothly approximated using rough path theory, extending methods from quantum field theory to gauge fields.

## Contribution

It applies rough path theory to Euclidean Yang--Mills on , showing smooth approximation for Wilson loops, a novel approach in gauge field quantum theories.

## Key findings

- Yang--Mills on  can be approximated smoothly
- RPT framework effectively applied to gauge fields
- Wilson loops are well-approximated in this setting

## Abstract

In the context of rough path theory (RPT), the theories of Hairer (2014) and Gubinelli--Imkeller--Perkowski (2015) (GIP theory) gave new methods for construction of $\Phi_{3}^{4}$ model. Roughly, their results state that a quantum field in a $\Phi_{3}^{4}$ model can be smoothly approximated. Consider the following question: Can RPT be applied to quantum Yang--Mills (YM) gauge field theories to show that any YM theory can be smoothly approximated? In this paper we consider this problem in the simplest case of Euclidean YM theory, i.e. YM on $\mathbb{R}^{2}$ with the usual Euclidean metric, as a test case. We prove that a (quantum) $SU(n)$ YM theory on $\mathbb{R}^{2}$ in axial gauge can be smoothly approximated for some class of Wilson loops. While our study is inspired by the theories of Hairer and GIP, instead we use the RPT framework of Friz--Victoir (2010) in proving the theorem.