QCD Instantons and 2D Surfaces

Abstract

Some time ago, Atiyah showed that there exists a natural identification between the k-instantons of a Yang-Mills theory with gauge group $G$ and the holomorphic maps from $CP_1$ to $ΩG$. Since then, Nair and Mazur, have associated the $Θ$ vacua structure in QCD with self-intersecting Riemann surfaces immersed in four dimensions. From here they concluded that these 2D surfaces correspond to the non-perturbative phase of QCD and carry the topological information of the $Θ$ vacua. In this paper we would like to elaborate on this point by making use of Atiyah's identification. We will argue that an effective description of QCD may be more like a $WZW$ model coupled to the induced metric of an immersion of a 2-D Riemann surface in $R^4$. We make some further comments on the relationship between the coadjoint orbits of the Kac-Moody group on $G$ and instantons with axial symmetry and monopole charge.