Field Theories for Gauged Symmetry Protected Topological Phases: Abelian Gauge Theories with non-Abelian Quasiparticles

TL;DR

This work builds a continuum field-theoretic formulation of abelian Dijkgraaf-Witten theories in (2+1)D and focuses on type III twists, especially twisted $ ext{Z}_2^{ imes3}$ and its generalization to $ ext{Z}_N^{ imes3}$. By explicitly constructing line operators—Wilson lines $U_{n_1n_2n_3}$ and flux insertion operators $V_{n_1n_2n_3}$—and enforcing gauge invariance via auxiliary fields, the authors reveal that abelian DW models can host non-Abelian statistics and fusion structures. They identify 22 distinct line operators in the $ ext{Z}_2^{ imes3}$ case, compute their fusion and linking correlators, and show nontrivial topological spins, establishing a bridge between continuum DW field theories and quantum-double formalisms. The results extend to general $ ext{Z}_N^{ imes3}$ twists, enabling systematic analysis of non-Abelian topological order arising from abelian gauge data, with potential implications for higher dimensions and SPT gauging.

Abstract

Dijkgraaf-Witten (DW) theories are of recent interest to the condensed matter community, in part because they represent topological phases of matter, but also because they characterize the response theory of certain symmetry protected topological (SPT) phases. However, as yet there has not been a comprehensive treatment of the spectra of these models in the field theoretic setting -- the goal of this work is to fill the gap in the literature, at least for a selection of DW models with abelian gauge groups but non-abelian topological order. As applications, various correlation functions and fusion rules of line operators are calculated. We discuss for example the appearance of non-abelian statistics in DW theories with abelian gauge groups.

Figures (2)

• Figure 1: An illustration of a linking correlation of $V_{100}(\gamma_1)$ and $V_{010}(\gamma_2)$.
• Figure 2: An illustration of self-twist. The left panel is a ribbon without self-twists. The dotted line is the illustration of framing for the solid line. The right panel is a ribbon with one self-twist. The dotted line, the framing, winds around the solid line oncewitten1989. We can simply view the right panel as a link of the solid and dotted line.