Higher condensation theory

Abstract: We develop a unified mathematical theory of defect condensations for topological orders in all dimensions based on higher categories, higher algebras and higher representations. A k-codimensional topological defect $A$ in an n+1D (potentially anomalous) topological order $C^{n+1}$ is condensable if it is equipped with the structure of a condensable $E_k$-algebra. Condensing such a defect $A$ amounts to a k-step process. In the first step, we condense the defect $A$ along one of its transversal directions, thus obtaining a (k-1)-codimensional defect $\Sigma A$, which is naturally equipped with the structure of a condensable $E_{k-1}$-algebra. In the second step, we condense the defect $\Sigma A$ in one of the remaining transversal directions, thus obtaining a (k-2)-codimensional defect $\Sigma^2 A$, so on and so forth. In the k-th step, we condense the 1-codimensional defect $\Sigma^{k-1}A$ along the only transversal direction, thus defining a phase transition from $C^{n+1}$ to a new n+1D topological order $D^{n+1}$. We give precise mathematical descriptions of each step in above process, including the precise mathematical characterization of the condensed phase $D^{n+1}$. When $C^{n+1}$ is anomaly-free, the same phase transition can be alternatively defined by replacing the last two steps by a single step of condensing the $E_2$-algebra $\Sigma^{k-2}A$ directly along the remaining two transversal directions. When n=2, this modified last step is precisely a usual anyon condensation in a 2+1D topological order. We derive many new mathematical results physically along the way. Some of them will be proved in a mathematical companion of this paper. We also briefly discuss questions, generalizations and applications that naturally arise from our theory, including higher Morita theory, a theory of integrals and the condensations of liquid-like gapless defects in topological orders.