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In the context of rough path theory (RPT), the theories of Hairer (2014) andGubinelli--Imkeller--Perkowski (2015) (GIP theory) gave new methods forconstruction of $\Phi_{3}^{4}$ model. Roughly, their results state that aquantum field in a $\Phi_{3}^{4}$ model can be smoothly approximated. Considerthe following question: Can RPT be applied to quantum Yang--Mills (YM) gaugefield theories to show that any YM theory can be smoothly approximated? In thispaper 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. Weprove that a (quantum) $SU(n)$ YM theory on $\mathbb{R}^{2}$ in axial gauge canbe smoothly approximated for some class of Wilson loops.  While our study is inspired by the theories of Hairer and GIP, instead we usethe RPT framework of Friz--Victoir (2010) in proving the theorem.

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