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

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.