Electro-magnetic duality in 2D topological orders with gapped boundaries

We generalize the Electro-magnetic (EM) duality and the mapping to the Levin-Wen (LW) model of the quantum double (QD) model to the case of topological orders with gapped boundaries. To achieve our goal, we Fourier transform the extended QD model defined on the lattice with boundaries. The input data of the model is finite group $G$ and the boundary condition is characterized by subgroup $K \subseteq G$. After the Fourier transform, we find while the bulk degrees of freedom become $\rep_G$, the boundary condition is now characterized by Frobenius algebra $(\mathds{C}[G]/\mathds{C}[K])^*$, the quotient of the group algebra of $G$ over that of $K$. We also show that our Fourier-transformed extended QD model can be mapped to an extended LW model on the same lattice via enlarging the Hilbert space of the extended LW model. Moreover, our Fourier transform of the extended QD model provides a visualizable explanation of the phenomenon of splitting and partial condensation of anyons.