A theorem concerning twisted and untwisted partition functions in U(N) and SU(N) lattice gauge theories

TL;DR

This paper explores the relationship between vortex free energies in lattice gauge theories, deriving an equality that relates different lattice sizes and discussing implications for quark confinement and volume dependence.

Contribution

It introduces a new relation between vortex free energies on different lattice sizes and analyzes its implications for understanding confinement in gauge theories.

Findings

Derived an equality relating vortex free energies on different lattice sizes.

Showed the discrepancy in couplings vanishes in the thermodynamic limit.

Critically examined Tomboulis's proof of quark confinement.

Abstract

In order to get a clue to understanding the volume-dependence of vortex free energy (which is defined as the ratio of the twisted against the untwisted partition function), we investigate the relation between vortex free energies defined on lattices of different sizes. An equality is derived through a simple calculation which equates a general linear combination of vortex free energies defined on a lattice to that on a smaller lattice. The couplings in the denominator and in the numerator however shows a discrepancy, and we argue that it vanishes in the thermodynamic limit. Comparison between our result and the work of Tomboulis is also presented. In the appendix we carefully examine the proof of quark confinement by Tomboulis and summarize its loopholes.