发表机构
Qiuzhen College, Tsinghua University; Institute for Applied Mathematics, Tsinghua University; Beijing Institute of Mathematical Sciences and Applications（清华大学邱成栋数学科学中心; 清华大学应用数学研究所; 北京国际数学研究中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究针对2+1维拓扑序边界相变临界点，提出以可凝聚代数的纤维积描述，证明其为可凝聚代数，还对$Z(Vec_G^\boldsymbol{\textit{\text{ω}}})$的可凝聚代数分类并计算相关例子。
AI 中文摘要
在本研究中，我们对2+1维拓扑序中边界相变的临界点提出了数学描述。给定2+1维拓扑序$\boldsymbol{\textit{C}}$中两个边界相$\boldsymbol{\textit{C}}_A$和$\boldsymbol{\textit{C}}_B$，它们分别对应拉格朗日代数$A$和$B$，我们提出这两个相之间的相变临界点（若存在）对应于$\boldsymbol{\textit{D}} = A \times_{M} B$，即$A$与$B$在$A$-$B$-代数$M$上的纤维积。我们证明$\boldsymbol{\textit{D}}$也是一个可凝聚代数。此外，我们对$Z(Vec_G^\boldsymbol{\textit{\text{ω}}})$中的可凝聚代数提出了另一种分类，并揭示了某些代数凝聚过程中出现的分裂现象。我们还对$Z(Vec_G^\boldsymbol{\textit{\text{ω}}})$的多个例子中的可凝聚代数进行了显式计算，涵盖了阿贝尔群和非阿贝尔群$\boldsymbol{\textit{G}}$。
英文摘要
In this work, we propose a mathematical description of the critical point of boundary phase transitions in 2+1D topological orders. Given two boundary phases, $\mathcal{C}_A$ and $\mathcal{C}_B$, with corresponding Lagrangian algebras $A$ and $B$ in a 2+1D topological order $\mathcal{C}$, respectively, we propose that the critical point of the phase transition between these two phases, if it exists, corresponds to $D = A \times_{M} B$, which is the fiber product of $A$ and $B$ over $A$-$B$-algebra $M$. We prove that $D$ is also a condensable algebra. Additionally, we present an alternative classification of condensable algebras in $Z(Vec_G^ω)$ and reveal splitting phenomenon that arises during the condensation of certain algebras. We also perform explicit computations for the condensable algebras within several examples of $Z(Vec_G^ω)$, covering both Abelian and non-Abelian groups $G$.
