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. We also establish the connections among various notions of "gauging" symmetries. 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.