Topological interactions in broken gauge theories

TL;DR

The work develops a comprehensive, mathematically grounded framework for topological interactions in 2+1D broken gauge theories, where a continuous group G is spontaneously broken to a finite subgroup H. It shows that the long-distance physics is captured by discrete gauge theories described by quantum doubles D(H), which deform to D^ω(H) when a Chern-Simons term is present; the CS data are encoded in H^3(H,U(1)) via ω, and the spectrum comprises vortices, charges, and dyons with well-defined fusion, spin, and braid properties. The thesis provides explicit constructions and examples for abelian and nonabelian H (notably Z_N, Z_N×Z_N, Z_2^3, D_N, and bar{D}_2), including fusion rules, modular data, Cheshire charges, flux metamorphosis, truncated braid groups, and dualities (e.g., between Z_2^3 CS theories and D_4-type theories). It further connects these topological structures to Dijkgraaf-Witten invariants and discusses the obstructions for obtaining certain cocycles via symmetry breaking from U(1)^k theories, highlighting the rich landscape of abelian and nonabelian discrete Chern-Simons theories. Overall, the work deepens the understanding of topological order, anyonic statistics, and dualities in gauge theories with broken continuous symmetries. Picked results have potential implications for condensed matter realizations of nonabelian anyons and for the broader interplay between topology, quantum groups, and gauge theory.

Abstract

This thesis deals with planar gauge theories in which some gauge group G is spontaneously broken to a finite subgroup H. The spectrum consists of magnetic vortices, global H charges and dyonic combinations exhibiting topological Aharonov-Bohm interactions. Among other things, we review the Hopf algebra D(H) related to this residual discrete H gauge theory, which provides an unified description of the spin, braid and fusion properties of the aforementioned particles. The implications of adding a Chern-Simons (CS) term to these models are also addressed. We recall that the CS actions for a compact gauge group G are classified by the cohomology group H^4(BG,Z). For finite groups H this classification boils down to the cohomology group H^3(H,U(1)). Thus the different CS actions for a finite group H are given by the inequivalent 3-cocycles of H. It is argued that adding a CS action for the broken gauge group G leads to additional topological interactions for the vortices governed by a 3-cocycle for the residual finite gauge group H determined by a natural homomorphism from H^4(BG,Z) to H^3(H,U(1)). Accordingly, the related Hopf algebra D(H) is deformed into a quasi-Hopf algebra. These general considerations are illustrated by CS theories in which the direct product of some U(1) gauge groups is broken to a finite subgroup H. It turns out that not all conceivable 3-cocycles for finite abelian gauge groups H can be obtained in this way. Those that are not reached are the most interesting. A Z_2 x Z_2 x Z_2 CS theory given by such a 3-cocycle, for instance, is dual to an ordinary gauge theory with nonabelian gauge group the dihedral group of order eight. Finally, the CS theories with nonabelian finite gauge group a dihedral or double dihedral group are also discussed in full detail.

Figures (19)

• 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.