A geometric approach to the Yang-Mills mass gap

Abstract

I provide a new idea based on geometric analysis to obtain a positive mass gap in pure non-abelian renormalizable Yang-Mills theory. The orbit space, that is the space of connections of Yang-Mills theory modulo gauge transformations, is equipped with a Riemannian metric that naturally arises from the kinetic part of reduced classical action and admits a positive definite sectional curvature. The corresponding regularized Bakry-Émery Ricci curvature (if positive) is shown to produce a mass gap for 2+1 and 3+1 dimensional Yang-Mills theory assuming the existence of a quantized Yang-Mills theory on (ℝ1+2, η) and (ℝ1+3, η), respectively. My result on the gap calculation, described at least as a heuristic one, applies to non-abelian Yang-Mills theory with any compact semi-simple Lie group in the aforementioned dimensions. In 2+1 dimensions, the square of the Yang-Mils coupling constant gYM2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {g}_{YM}^2 $$\\end{document} has the dimension of mass, and therefore the spectral gap of the Hamiltonian is essentially proportional to gYM2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {g}_{YM}^2 $$\\end{document} with proportionality constant being purely numerical as expected. Due to the dimensional restriction on 3+1 dimensional Yang-Mills theory, it seems one ought to introduce a length scale to obtain an energy scale. It turns out that a certain ‘trace’ operation on the infinite-dimensional geometry naturally introduces a length scale that has to be fixed by measuring the energy of the lowest glu-ball state. However, this remains to be understood in a rigorous way.