A Geometric Approach to the Yang-Mills Mass Gap

TL;DR

This paper proposes a geometric analysis approach to establish a positive mass gap in pure non-abelian Yang-Mills theory by examining the orbit space with a Riemannian metric and Ricci curvature, assuming the existence of a quantum theory.

Contribution

It introduces a novel geometric framework using the orbit space and Ricci curvature to heuristically derive a mass gap in Yang-Mills theory across 2+1 and 3+1 dimensions.

Findings

Mass gap is proportional to the square of the Yang-Mills coupling in 2+1 dimensions.

The geometric approach suggests a natural length scale in 3+1 dimensions.

The method applies to any compact semi-simple Lie group.

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 \textit{Bakry-\'Emery} 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 $(\mathbb{R}^{1+2},\eta)$ and $(\mathbb{R}^{1+3},\eta)$, 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 $g^{2}_{YM}$ has the dimension of mass, and therefore the spectral gap of the Hamiltonian is essentially proportional to $g^{2}_{YM}$ 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.