2-Morita Equivalent Condensable Algebras and Domain Walls in 2+1D Topological Orders
Rongge Xu, Holiverse Yang
TL;DR
This work develops a complete framework for classifying 2‑Morita equivalent $E_2$ condensable algebras in a modular tensor category $\mathcal{C}$ by connecting three equivalent viewpoints: (i) left/right/full centers of condensable algebras and their induced $E_2$-modules, (ii) lagrangian algebras in the Drinfeld center $\mathfrak{Z}(\mathcal{C})$, and (iii) $1$-Morita classes of 1d domain walls in $\mathcal{C}$. The authors prove that two condensable algebras $A_1,A_2$ are 2‑Morita equivalent iff there exists a 1d algebra $B$ with $Z_l(B)\simeq A_1$ and $Z_r(B)\simeq A_2$, equivalently that $\mathcal{C}_{A_1}^{loc}\simeq\mathcal{C}_{A_2}^{loc}$ and $\mathfrak{Z}(\mathcal{C}_{A_1})\simeq\mathfrak{Z}(\mathcal{C}_{A_2})$. The framework uses a two-step condensation and folding to map domain-wall data to boundaries of $\mathfrak{Z}(\mathcal{C})$, establishing a universal Trinity of reversible arrows among condensable algebras, walls, and lagrangians. Concrete examples with group doubles (e.g., $\mathfrak{Z}(\mathrm{Vec}_{G})$ for $G=\mathbb{Z}_2,\mathbb{Z}_4,S_3$) demonstrate the method, including lattice realizations of 1d condensables, left/right centers, and lagrangians. The results extend to Witt-equivalent MTCs and open avenues for 0d defects, generalized Morita notions, and broader topological-operator formalisms, offering a robust, model-independent route to understand domain walls and phase transitions in 2+1D topological orders.
Abstract
We classify $E_2$ condensable algebras in a modular tensor category $\mathcal{C}$ up to 2-Morita equivalence. From a physical perspective, this is equivalent to providing a criterion for when different $E_2$ condensable algebras result in the same condensed topological phase in a 2d anyon condensation process. By considering the left and right centers of $E_1$ condensable algebras in $\mathcal{C}$, we exhaust all 2-Morita equivalent $E_2$ condensable algebras in $\mathcal{C}$ and provide a method to recover $E_1$ condensable algebras from 2-Morita equivalent $E_2$ condensable algebras. We also prove that intersecting Lagrangian algebras in $\mathcal{C} \boxtimes \overline{\mathcal{C}}$ with its left and right components generates all 2-Morita equivalent $E_2$ condensable algebras in $\mathcal{C}$. This paper establishes a complete interplay between $E_1$ condensable algebras in $\mathcal{C}$, 2-Morita equivalent $E_2$ condensable algebras in $\mathcal{C}$, and Lagrangian algebras in $\mathcal{C} \boxtimes \overline{\mathcal{C}}$. The relations between different condensable algebras can be translated into their module categories, which correspond to domain walls in topological orders. We introduce a two-step condensation process and study the fusion of domain walls. We also show that an automorphism of an $E_2$ condensable algebra may lead to a nontrivial braided autoequivalence in the condensed phase. As concrete examples, we interpret the categories of quantum doubles of finite groups. We also discuss examples beyond group symmetries. Moreover, our results can be generalized to Witt-equivalent modular tensor categories.