Higher condensation theory

We develop a unified theory of defect condensations for topological orders in all dimensions based on higher categories, higher algebras and higher representations. We show that condensing a k-codimensional topological defect A in an n+1D (potentially anomalous) topological order Cⁿ⁺¹ amounts to a k-step process. In the first step, we condense A along one of the transversal directions, thus obtaining a $(k-1)$-codimensional defect Σ A, which can be further condensed as the second step, so on and so forth. In the k-th step, condensing Σᵏ⁻¹A along the only transversal direction defines a phase transition to a new phase Dⁿ⁺¹. Mathematically, a k-codimensional defect A is condensable if it is equipped with the structure of a condensable Eₖ-algebra. In this case, Σ A is naturally a condensable Eₖ₋₁-algebra, thus it can be further condensed. The condensed phase Dⁿ⁺¹ consists of all deconfined topological defects in Cⁿ⁺¹. A k-codimensional topological defect is deconfined if and only if it is equipped with a k-dimensional A-action, which defines an Eₖ-module over A. When Cⁿ⁺¹ is anomaly-free, the same condensation can be alternatively defined by replacing the last two steps by a single step of condensing the E₂-algebra Σᵏ⁻²A directly. The condensed phase Dⁿ⁺¹ is determined by the category of E₂-modules over Σᵏ⁻²A. When $n=2$, this modified last step is precisely a usual anyon condensation in a 2+1D topological order. The proofs of the most mathematical results will appear in a mathematical companion of this paper. We also briefly discuss some generalizations and applications that naturally arise from our condensation theory such as higher Morita theory, factorization homology and the condensation theory of non-topological defects.