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

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. Work supported by PSC-CUNY Research Award Program grants nos. 668460, 61508-00 and 69466-00 and CUNY Collaborative Incentive Grant 91915-00-06. Work supported by NSERC of Canada and the Niels Bohr Fund of Denmark.