Magnetic Z(N) symmetry in 2+1 dimensions

TL;DR

This review explores how magnetic symmetry influences confinement in 2+1 dimensional gauge theories, emphasizing spontaneous symmetry breaking and the role of magnetic vortices in low-energy dynamics.

Contribution

It provides an explicit derivation of the effective theory for magnetic vortices and discusses the impact of dynamical matter and Chern-Simons terms on magnetic symmetry.

Findings

Magnetic symmetry is spontaneously broken in confining theories without matter.

The effective low-energy dynamics are governed by magnetic vortices.

Confinement arises from long-range interactions between topological solitons.

Abstract

This review describes the role of magnetic symmetry in 2+1 dimensional gauge theories. In confining theories without matter fields in fundamental representation the magnetic symmetry is spontaneously broken. Under some mild assumptions, the low-energy dynamics is determined universally by this spontaneous breaking phenomenon. The degrees of freedom in the effective theory are magnetic vortices. Their role in confining dynamics is similar to that played by pions and sigma in the chiral symmetry breaking dynamics. I give an explicit derivation of the effective theory in (2+1)-dimensional weakly coupled confining models and argue that it remains qualitatively the same in strongly coupled (2+1)-dimensional gluodynamics. Confinement in this effective theory is a very simple classical statement about the long range interaction between topological solitons, which follows (as a result of a simple direct classical calculation) from the structure of the effective Lagrangian. I show that if fundamentally charged dynamical fields are present the magnetic symmetry becomes local rather than global. The modifications to the effective low energy description in the case of heavy dynamical fundamental matter are discussed. This effective lagrangian naturally yields a bag like description of baryonic excitations. I also discuss the fate of the magnetic symmetry in gauge theories with the Chern-Simons term.