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

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 $Z(Ve{c}_G^{\omega })$ 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 $Z(Ve{c}_G^{\omega })$, covering both Abelian and non-Abelian groups $G$.