QCD Instantons and 2d Surfaces

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 CP1 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 2D Riemann surface in R4. 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.