Dirac Strings and Monopoles in the Continuum Limit of SU(2) Lattice Gauge Theory

TL;DR

This paper explores the continuum limit of SU(2) lattice gauge theories, introducing singular gauge potentials to incorporate monopoles and Dirac strings, and constructs the 't Hooft loop operator to analyze monopole behavior.

Contribution

It presents a modified continuum formulation allowing monopoles with finite action by including singular gauge potentials, bridging lattice and continuum descriptions.

Findings

Monopoles with |Q_M|=1,2 can be represented in the continuum with singular potentials.

Zero-action monopole-antimonopole solutions exist with Dirac strings allowed.

The 't Hooft loop operator is constructed and its behavior analyzed at various distances and temperatures.

Abstract

Magnetic monopoles are known to emerge as leading non-perturbative fluctuations in the lattice version of non-Abelian gauge theories in some gauges. In terms of the Dirac quantization condition, these monopoles have magnetic charge |Q_M|=2. Also, magnetic monopoles with |Q_M|=1 can be introduced on the lattice via the 't Hooft loop operator. We consider the |Q_M|=1,2 monopoles in the continuum limit of the lattice gauge theories. To substitute for the Dirac strings which cost no action on the lattice, we allow for singular gauge potentials which are absent in the standard continuum version. Once the Dirac strings are allowed, it turns possible to find a solution with zero action for a monopole--antimonopole pair. This implies equivalence of the standard and modified continuum versions in perturbation theory. To imitate the nonperturbative vacuum, we introduce then a nonsingular background. The modified continuum version of the gluodynamics allows in this case for monopoles with finite non-vanishing action. Using similar techniques, we construct the 't Hooft loop operator in the continuum and predict its behavior at small and large distances both at zero and high temperatures.