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

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$(G^{\mathrm{\prime }},{\alpha }^{\mathrm{\prime }})$, 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 $D^{{\alpha }^{\mathrm{\prime }}}(G^{\mathrm{\prime }})$ and derive the map between the anyons of one model and those of the other.