Condensation Completion and Defects in 2+1D Topological Orders

Abstract

We review the condensation completion of a modular tensor category $\mathcal{C}$, which yields a fusion 2-category $\hat{I}\pounds \mathcal{C}$ of separable algebras, bimodules over algebras and bimodule maps in $\mathcal{C}$. Physically, $\hat{I}\pounds \mathcal{C}$ is the fusion 2-category of codimension-1 defects, codimension-2 defects and instantons in the $2+1$D topological order $\mathcal{C}$. We realize the rough-rough wall and $e$-$m$ exchange wall in Toric Code model on the lattice by deforming the Hamiltonian based on the corresponding algebraic data. We apply condensation completion to Toric Code, $3\mathbf{F}$, two-laryer semion and ${\mathbb{Z}}_4$ topological orders, and explicitly enumerate their $1$d and $0$d defects along with fusion rules. We also mention other applications of condensation completion: alternative interpretations of condensation completion of a braided fusion category; condensation completion of the category of symmetry charges and its correspondence to gapped phases with symmetry; for a topological order $\mathcal{C}$, one can find all gapped boundaries of the stacking of $\mathcal{C}$ with its time-reversal conjugate through computing the condensation completion of $\mathcal{C}$.

We have added more examples and remarks and revised the conventions to enhance readability. We also added a section about constructing 1d gapped defects in toric code model on the lattice based on the data of the corresponding separable algebras