Extremal Curves in 2+1-Dimensional Yang-Mills Theory

TL;DR

This paper investigates the structure of extremal energy curves in 2+1D Yang-Mills theory on a torus, providing insights into the mass gap and the spectrum's finiteness.

Contribution

It introduces a method to identify extremal potential energy curves in orbit space and analyzes their properties and implications for the mass gap in Yang-Mills theory.

Findings

Extremal curves have finite length, implying a gapped spectrum.

Explicit construction of extremal curves and their intersection with the Gribov horizon.

Dependence of extremal curve length on torus dimensions.

Abstract

We examine the structure of the potential energy of 2+1-dimensional Yang-Mills theory on a torus with gauge group SU(2). We use a standard definition of distance on the space of gauge orbits. A curve of extremal potential energy in orbit space defines connections satisfying a certain partial differential equation. We argue that the energy spectrum is gapped because the extremal curves are of finite length. Though classical gluon waves satisfy our differential equation, they are not extremal curves. We construct examples of extremal curves and find how the length of these curves depends on the dimensions of the torus. The intersections with the Gribov horizon are determined explicitly. The results are discussed in the context of Feynman's ideas about the origin of the mass gap.