Electric-Magnetic duality in twisted quantum double model of topological orders

TL;DR

This paper develops a partial electric-magnetic duality transformation for twisted quantum double models, enabling explicit isomorphisms between different topological order models and their anyon structures.

Contribution

It introduces a PEM duality transformation for TQD models with Abelian normal subgroups, providing a method to relate different models and their anyon content.

Findings

Constructed explicit isomorphisms between TQD algebras.

Derived the map between anyons of dual models.

Showed equivalence of TQD models via PEM duality.

Abstract

We derive a partial electric-magnetic (PEM) duality transformation of the twisted quantum double (TQD) model TQD$(G,\alpha)$---discrete Dijkgraaf-Witten model---with a finite gauge group $G$, which has an Abelian normal subgroup $N$, and a three-cocycle $\alpha \in H^3(G,U(1))$. Any equivalence between two TQD models, say, TQD$(G,\alpha)$ and TQD$(G',\alpha')$, can be realized as a PEM duality transformation, which exchanges the $N$-charges and $N$-fluxes only. Via the PEM duality, we construct an explicit isomorphism between the corresponding TQD algebras $D^\alpha(G)$ and $D^{\alpha'}(G')$ and derive the map between the anyons of one model and those of the other.