Classification of Gapped Domain Walls in 2+1D Topological Orders through 2-Morita Equivalence

Abstract We classify condensable E₂-algebras in a modular tensor category $\mathcal{C}$ up to 2-Morita equivalence. Physically, this classification provides an explicit criterion to determine when distinct condensable E₂-algebras yield the same condensed topological phase under a two-dimensional anyon condensation process. The relations between different condensable algebras can be translated into their module categories, interpreted physically as gapped domain walls in topological orders. As concrete examples, we interpret the categories of quantum doubles of finite groups and examples beyond group symmetries. Our framework fully elucidates the interplay among condensable E₁-algebras in $\mathcal{C}$, condensable E₂-algebras in $\mathcal{C}$ up to 2-Morita equivalence, and Lagrangian algebras in $\mathcal{C}⊠\overline{\mathcal{C}}$.