Gauge Fixing and Extended Abelian Monopoles in SU(2) Gauge Theory in 2+1 Dimensions

TL;DR

This paper investigates extended Abelian monopoles in SU(2) lattice gauge theory in 2+1 dimensions, analyzing their distribution, scaling behavior, and potential relevance to continuum physics like confinement.

Contribution

It provides a detailed analysis of monopole distributions in various gauges, clarifies the reasons for non-scaling in some gauges, and explores the scale dependence of monopole properties.

Findings

Monopole distribution in the maximal Abelian gauge appears random.

Non-scaling in local gauges is mainly due to short-distance correlations.

Increasing monopole size reduces scale violation, suggesting relevance to continuum physics.

Abstract

Extended Abelian monopoles are investigated in SU(2) lattice gauge theory in three dimensions. Monopoles are computed by Abelian projection in several gauges, including the maximal Abelian gauge. The number $N_m$ of extended monopoles in a cube of size $m^3$ (in lattice units) is defined as the number of elementary ($1^3$) monopoles minus antimonopoles in the cube ($m=1,2,\ldots$). The distribution of $1^3$ monopoles in the nonlocal maximal Abelian gauge is shown to be essentially random, while nonscaling of the density of $1^3$ monopoles in some local gauges, which has been previously observed, is shown to be mainly due to strong short-distance correlations. The density of extended monopoles in local gauges is studied as a function of $\beta$ for monopoles of fixed physical ``size'' ($m / \beta = {\rm fixed}$); the degree of scale violation is found to decrease substantially as the monopole size is increased. The possibility therefore remains that long distance properties of monopoles in local gauges may be relevant to continuum physics, such as confinement.