Phases of theories with $\mathbb{Z}_N$ 1-form symmetry and the roles of center vortices and magnetic monopoles

TL;DR

This paper investigates the phases of theories with a microscopic rac{1}{N} 1-form symmetry, revealing the roles of monopoles and vortices in confinement across different dimensions, and constructing a lattice gauge theory with multiple photons.

Contribution

It provides a comprehensive analysis of the phases of rac{1}{N} 1-form symmetric theories, clarifying the conditions for confinement and constructing a novel lattice gauge theory with multiple photons.

Findings

In 2D, only the confined phase exists.

In 3D, both confined and topological BF phases occur.

In 4D, confined, topological BF, and photon phases are present.

Abstract

We analyze the phases of theories which only have a microscopic $\mathbb{Z}_N$ 1-form symmetry, starting with a topological BF theory and deforming it in accordance with microscopic symmetry. These theories have a well-defined notion of confinement. Prototypical examples are pure $SU(N)$ gauge theories and $\mathbb{Z}_N$ lattice gauge theories. Our analysis shows that the only generic phases are in $d=2$, only the confined phase; in $d=3$, both the confined phase and the topological BF phase; and in $d=4$, the confined phase, the topological BF phase, and a phase with a massless photon. We construct a $\mathbb{Z}_N$ lattice gauge theory with a deformation which, surprisingly, produces up to $(N-1)$ photons. We give an interpretation of these findings in terms of two competing pictures of confinement -- proliferation of monopoles and proliferation of center vortices -- and conclude that the proliferation of center vortices is a necessary but insufficient condition for confinement, while that of monopoles is both necessary and sufficient.