Yang-Mills Theory as a Deformation of Topological Field Theory, Dimensional Reduction and Quark Confinement

We prove quark confinement in four-dimensional QCD based on the criterion of area law of 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 BRST and anti-BRST exact action corresponding to the maximal abelian gauge leaving the maximal torus group H of G invariant. Then we find the 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-2)-dimensional coset G/H non-linear sigma model in the sense of Parisi and Sourlas dimensional reduction. After maximal abelian gauge fixing, the topological property of magnetic monopole and antimonopole of four-dimensional Yang-Mills theory is translated into that of instanton and anti-instanton in two-dimensional equivalent model. It is shown that the linear static potential in four-dimensions follows from dilute instanton–anti-instanton gas in the equivalent two-dimensional non-linear sigma model obtained from the fourdimensional topological field theory by dimensional reduction, while the remaining Coulomb potential comes from the perturbative part in four-dimensions. The dimensional reduction opens a path for applying various exact methods developed in two-dimensional quantum field theory to study the non-perturbative problem in fourdimensional quantum field theories.