Yang-Mills theory as a deformation of topological field theory, dimensional reduction, and quark confinement

Abstract

We propose a reformulation of Yang-Mills theory as a perturbative deformation of a novel topological (quantum) field theory. We prove that this reformulation of four-dimensional QCD leads to quark confinement in the sense of an area law of the Wilson loop. First, Yang-Mills theory with a non-Abelian gauge group G is reformulated as a deformation of a novel topological field theory. Next, a special class of topological field theories is defined by both Becchi-Rouet-Stora-Tyupin (BRST) and anti-BRST exact actions corresponding to the maximal Abelian gauge leaving the maximal torus group Hof G invariant. Then we find topological field theory $(D>2)$ has a hidden supersymmetry for a choice of maximal Abelian gauge. As a result, the D-dimensional topological field theory is equivalent to the $(D\ensuremath{-}2)$-dimensional coset $G/H$ nonlinear sigma model in the sense of the Parisi-Sourlas dimensional reduction. After maximal Abelian gauge fixing, the topological property of the magnetic monopole and antimonopole of four-dimensional Yang-Mills theory is translated into that of an instanton and anti-instanton in a two-dimensional equivalent model. It is shown that the linear static potential in four dimensions follows from the instanton--anti-instanton gas in the equivalent two-dimensional nonlinear sigma model obtained from the four-dimensional topological field theory by dimensional reduction, while the remaining Coulomb potential comes from the perturbative part in four-dimensional Yang-Mills theory. The dimensional reduction opens a path for applying various exact methods developed in two-dimensional quantum field theory to study the nonperturbative problem in low-energy physics of four-dimensional quantum field theories.