Discrete gauge theories

TL;DR

The notes develop a comprehensive framework for planar gauge theories where a continuous group G is Higgs-broken to a finite subgroup H, yielding a nontrivial long-distance discrete H gauge theory. The approach centers on the quantum double D(H), which unifies the spectrum of charges, magnetic vortices, and dyons with their spin, braid, and fusion properties, including abelian and nonabelian anyon-like statistics. Flux metamorphosis, Alice fluxes, and Cheshire charges illustrate rich nonperturbative topological effects, while truncated braid groups and Verlinde data provide a concrete computational toolkit. The explicit rac{D}{ar{D}_2} example demonstrates the emergence of nonabelian braid statistics and topological scattering, highlighting potential connections to Chern-Simons deformations and broader 2+1D topological phases with finite gauge groups. Overall, the work emphasizes a Hopf-algebraic view of symmetry breaking where long-distance physics is governed by D(H) and its representations.

Abstract

In these lecture notes, we present a self-contained discussion of planar gauge theories broken down to some finite residual gauge group H via the Higgs mechanism. The main focus is on the discrete H gauge theory describing the long distance physics of such a model. The spectrum features global H charges, magnetic vortices and dyonic combinations. Due to the Aharonov-Bohm effect, these particles exhibit topological interactions. Among other things, we review the Hopf algebra related to this discrete H gauge theory, which provides an unified description of the spin, braid and fusion properties of the particles in this model. Exotic phenomena such as flux metamorphosis, Alice fluxes, Cheshire charge, (non)abelian braid statistics, the generalized spin-statistics connection and nonabelian Aharonov-Bohm scattering are explained and illustrated by representative examples. Preface: Broken symmetry revisited, 1 Basics: 1.1 Introduction, 1.2 Braid groups, 1.3 Z_N gauge theory, 1.3.1 Coulomb screening, 1.3.2 Survival of the Aharonov-Bohm effect, 1.3.3 Braid and fusion properties of the spectrum, 1.4 Nonabelian discrete gauge theories, 1.4.1 Classification of stable magnetic vortices, 1.4.2 Flux metamorphosis, 1.4.3 Including matter, 2 Algebraic structure: 2.1 Quantum double, 2.2 Truncated braid groups, 2.3 Fusion, spin, braid statistics and all that..., 3 \bar{D}_2 gauge theory: 3.1 Alice in physics, 3.2 Scattering doublet charges off Alice fluxes, 3.3 Nonabelian braid statistics, 3.A Aharonov-Bohm scattering, 3.B B(3,4) and P(3,4), Concluding remarks and outlook

Figures (13)

• Figure 1: The braid operator $\tau_i$ establishes a counterclockwise interchange of the particles $i$ and $i+1$ in a set of $n$ numbered indistinguishable particles in the plane.
• Figure 2: Pictorial presentation of the braid relation $\tau_1 \tau_2 \tau_1= \tau_2 \tau_1 \tau_2$. The particle trajectories corresponding to the composition of exchanges $\tau_1 \tau_2 \tau_1$ (diagram at the l.h.s.) can be continuously deformed into the trajectories associated with the composition of exchanges $\tau_2 \tau_1 \tau_2$ (r.h.s. diagram).
• Figure 3: The braid relation $\tau_1 \tau_3 = \tau_3 \tau_1$ expresses the fact that the particle trajectories displayed in the l.h.s. diagram can be continuously deformed into the trajectories in the r.h.s. diagram.
• Figure 4: The monodromy operator $\gamma_{ij}$ takes particle $i$ counterclockwise around particle $j$.
• Figure 5: Taking a screened external charge $q$ around a magnetic vortex $\phi$ in the Higgs medium generates the Aharonov-Bohm phase $\exp (\imath q \phi)$. We have emphasized the extended structure of these sources, although this structure will not be probed in the low energy regime to which we confine ourselves here. The shaded region around the external point charge $q$ represents the cloud of screening charge of characteristic size $1/M_A$. The flux of the vortex is confined to the shaded circle bounded by the core at the distance $1/M_H$ from its centre. The string attached to the core represents the Dirac string of the flux, i.e. the strip in which the nontrivial parallel transport in the gauge fields takes place.