SU(N)-Gauge Theories in Polyakov Gauge on the Torus

TL;DR

This paper studies SU(N) gauge theories on a four-dimensional torus using Polyakov gauge, revealing how magnetic defects relate to instanton numbers and are situated on the Gribov horizon.

Contribution

It constructs fundamental domains for the gauge field $A_0$ and analyzes the singularities and magnetic defects in sectors with non-zero instanton number.

Findings

Magnetic defects are located on the Gribov horizon.

Singularities correspond to magnetic monopoles, strings, or walls.

Magnetic charges are quantized and related to the instanton number.

Abstract

We investigate the Abelian projection with respect to the Polyakov loop operator for SU(N) gauge theories on the four torus. The gauge fixed $A_0$ is time-independent and diagonal. We construct fundamental domains for $A_0$. In sectors with non-vanishing instanton number such gauge fixings are always singular. The singularities define the positions of magnetically charged monopoles, strings or walls. These magnetic defects sit on the Gribov horizon and have quantized magnetic charges. We relate their magnetic charges to the instanton number.