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We generalize the Electric-magnetic (EM) duality in the quantum double (QD)models to the case of topological orders with gapped boundaries. We also mapthe QD models with boundaries to the Levin-Wen (LW) models with boundaries. Tothis end, we Fourier transform and rewrite the extended QD model with a finitegauge group $G$ on a trivalent lattice with a boundary. Gapped boundaryconditions of the model before the transformation are known to be characterizedby the subgroups $K \subseteq G$. We find that after the transformation, theboundary conditions are then characterized by the Frobenius algebras $A_{G,K}$in $\mathrm{Rep}_G$. An $A_{G,K}$ is the dual space of the quotient of thegroup algebra of $G$ over that of $K$, and $\mathrm{Rep}_G$ is the category ofthe representations of $G$. The EM duality on the boundary is revealed bymapping the $K$'s to $A_{G,K}$'s. We also show that our transformed extended QDmodel can be mapped to an extended LW model on the same lattice via enlargingthe Hilbert space of the extended LW model. Moreover, our transformed extendedQD model elucidates the phenomenon of anyon splitting in anyon condensation.

Electric-magnetic duality in the quantum double models of topological orders with gapped boundaries