QCD Instantons and 2D Surfaces

TL;DR

This paper explores the connection between QCD instantons, topological vacua, and 2D surfaces, proposing an effective model involving WZW theory coupled to immersed Riemann surfaces in four-dimensional space.

Contribution

It elaborates on Atiyah's identification of instantons with holomorphic maps and suggests a new effective description of QCD using WZW models on immersed surfaces.

Findings

QCD vacua relate to self-intersecting Riemann surfaces in 4D

Effective QCD description may involve WZW models coupled to immersed surfaces

Connections between Kac-Moody coadjoint orbits and instantons with symmetry

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 $\Omega G$. Since then, Nair and Mazur, have associated the $\Theta $ 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 $\Theta$ 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.