Monopoles, Polyakov-Loops and Gauge Fixing on the Torus

TL;DR

This paper explores gauge fixing in pure Yang Mills theory on a four-dimensional torus, introducing transition functions for instanton sectors and an Abelian projection related to Polyakov loops, highlighting non-perturbative singularities linked to monopoles.

Contribution

It presents a comprehensive set of transition functions for all instanton sectors and develops an extended Abelian projection based on Polyakov loops for gauge fixing.

Findings

Transition functions encompass all instanton sectors.

Gauge fixing in non-perturbative sectors involves singularities.

Singularities are confined to Dirac string-like defects.

Abstract

We consider pure Yang Mills theory on the four torus. A set of non-Abelian transition functions is presented which encompass all instanton sectors. It is argued that these transition functions are a convenient starting point for gauge fixing. In particular, we give an extended Abelian projection with respect to the Polyakov loop, where $A_0$ is independent of time and in the Cartan subalgebra. In the non-perturbative sectors such gauge fixings are necessarily singular. These singularities can be restricted to Dirac strings joining monopole and anti-monopole like ``defects''.