A Geometric and Metric-Measure Framework for the Yang–Mills Mass Gap

Abstract

This paper proves no mass gap. Its main result is a conditional theorem, and the existence of a mass gap in four-dimensional Yang–Mills theory remains open. The conditional theorem assumes two things: 1. the ground-state measures of regularized Yang–Mills Hamiltonians on the Gribov region satisfy a uniform Bakry–Émery curvature-dimension bound CD(K,∞); 2. the Hamiltonians converge in the strong resolvent sense. Under these hypotheses, the continuum Hamiltonian has a spectral gap of at least g²K/2 above the vacuum. Whether the Gribov–Zwanziger construction supplies such a bound is open. The paper recalls the Gribov and refined Gribov–Zwanziger frameworks with their sources, and it states the area law as a conjecture.