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arXiv:2610.12079

Fiber product of condensable algebras and critical points of boundary phase transitions

Hanlin Lin, Hao Zheng

Published Oct 8, 2026• 0 citations
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Abstract

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 $G$.

DOI
10.48550/arxiv.2610.12079

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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, CA\mathcal{C}_ACA​ and CB\mathcal{C}_BCB​, with corresponding Lagrangian algebras AAA and BBB in a 2+1D topological order C\mathcal{C}C, respectively, we propose that the critical point of the phase transition between these two phases, if it exists, corresponds to D=A×MBD = A \times_{M} BD=A×M​B, which is the fiber product of AAA and BBB over AAA-BBB-algebra MMM. We prove that DDD is also a condensable algebra. Additionally, we present an alternative classification of condensable algebras in Z(VecGω)Z(Vec_G^ω)Z(VecGω​) 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(VecGω)Z(Vec_G^ω)Z(VecGω​), covering both Abelian and non-Abelian groups GGG.
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