Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Hopf symmetry breaking and confinement in (2+1)-dimensional gauge theory

F. A. Bais, B. J. Schroers, J. K. Slingerland

Abstract

Gauge theories in 2+1 dimensions whose gauge symmetry is spontaneously broken to a finite group enjoy a quantum group symmetry which includes the residual gauge symmetry. This symmetry provides a framework in which fundamental excitations (electric charges) and topological excitations (magnetic fluxes) can be treated on equal footing. In order to study symmetry breaking by both electric and magnetic condensates we develop a theory of symmetry breaking which is applicable to models whose symmetry is described by a quantum group (quasitriangular Hopf algebra). Using this general framework we investigate the symmetry breaking and confinement phenomena which occur in (2+1)-dimensional gauge theories. Confinement of particles is linked to the formation of string-like defects. Symmetry breaking by an electric condensate leads to magnetic confinement and vice-versa. We illustrate the general formalism with examples where the symmetry is broken by electric, magnetic and dyonic condensates.

Hopf symmetry breaking and confinement in (2+1)-dimensional gauge theory

Abstract

Gauge theories in 2+1 dimensions whose gauge symmetry is spontaneously broken to a finite group enjoy a quantum group symmetry which includes the residual gauge symmetry. This symmetry provides a framework in which fundamental excitations (electric charges) and topological excitations (magnetic fluxes) can be treated on equal footing. In order to study symmetry breaking by both electric and magnetic condensates we develop a theory of symmetry breaking which is applicable to models whose symmetry is described by a quantum group (quasitriangular Hopf algebra). Using this general framework we investigate the symmetry breaking and confinement phenomena which occur in (2+1)-dimensional gauge theories. Confinement of particles is linked to the formation of string-like defects. Symmetry breaking by an electric condensate leads to magnetic confinement and vice-versa. We illustrate the general formalism with examples where the symmetry is broken by electric, magnetic and dyonic condensates.

Paper Structure

This paper contains 40 sections, 16 theorems, 181 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Discrete gauge theories: physical setting
  3. Hopf symmetry in (2+1) dimensions
  4. The quantum double of a finite group
  5. The double and its dual
  6. Irreducible representations
  7. Matrix elements and Characters
  8. Examples of quantum doubles
  9. $D(H)$ for Abelian $H$
  10. $D(D_{2m+1})$
  11. Symmetry Breaking
  12. Hopf symmetry breaking
  13. Hopf subalgebras and Hopf quotients
  14. Hopf subalgebras of quantum doubles
  15. Confinement
  16. ...and 25 more sections

Key Result

Theorem 1

Let $F(X\times H)$ be a transformation group algebra and let $\{\mathcal{O}_{A}\}$ be the collection of $H$-orbits in $X$ ($A$ takes values in some index set). For each $A$, choose some $\xi_A \in \mathcal{O}_{A}$ and let $N_A$ be the stabilizer of $\xi_A$ in $H$. Then, for each pair $(\mathcal{O}_{ Moreover, all unitary irreducible representations of $F(X\times H)$ are of this form and irreps $\t

Figures (2)

  • Figure 1: Flux-charge lattice for a $D(\mathbb{Z}_9)$ theory. We assume that a condensate of particles with flux $r^3 \equiv 3$ and charge $\alpha_{-3}\equiv-3\equiv 6$ forms. The condensed irrep is indicated as a square dot. The residual symmetry algebra $\mathcal{T}^{3}_{-3}$ is a group algebra $\mathbb{C}(\mathbb{Z}_9\times\mathbb{Z}_3)$. Two $D(H)$-irreps in the lattice are equivalent as $\mathcal{T}^{3}_{-3}$-irreps if one can be reached from the other through translations by the "condensate vector" $(-3,3)$. This way, $D(H)$-irreps are identified in trios. The trio in the picture corresponds to the $\mathcal{T}$-irrep $\chi_{5,1}$. The shaded region contains one representative from each trio and is thus a diagram of all $\mathcal{T}$-irreps. The small white circles indicate the three unconfined irreps of $\mathcal{T}$, which correspond to the irreps of $\mathcal{U}\cong\mathbb{C}\mathbb{Z}_3$.
  • Figure 2: Schematic picture of the structures that play a role in this paper

Theorems & Definitions (25)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1
  • Definition 4
  • Definition 5
  • Definition 6
  • Proposition 1
  • Corollary 1
  • Corollary 2
  • ...and 15 more