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

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, 𝒞_A and 𝒞_B, with corresponding Lagrangian algebras A and B in a 2+1D topological order 𝒞, respectively, we propose that the critical point of the phase transition between these two phases, if it exists, corresponds to D = A ×_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.

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BibTeX RIS

BibTeX

@misc{lin2026fiber,
  title = {{Fiber product of condensable algebras and critical points of boundary phase transitions}},
  author = {Hanlin Lin and Hao Zheng},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.12079},
  eprint = {2610.12079},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.12079},
}

RIS

TY  - UNPB
TI  - Fiber product of condensable algebras and critical points of boundary phase transitions
AU  - Hanlin Lin
AU  - Hao Zheng
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2610.12079
UR  - https://arxiv.org/abs/2610.12079
AB  - 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, 𝒞_A and 𝒞_B, with corresponding Lagrangian algebras A and B in a 2+1D topological order 𝒞, respectively, we propose that the critical point of the phase transition between these two phases, if it exists, corresponds to D = A ×_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.
ER  -

Abstract

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