Higher condensation theory
arxiv(2024)
摘要
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 𝖢^n+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 Σ^k-1A along the only
transversal direction defines a phase transition to a new phase 𝖣^n+1. Mathematically, a k-codimensional defect A is condensable if it
is equipped with the structure of a condensable E_k-algebra. In this case,
Σ A is naturally a condensable E_k-1-algebra, thus it can be further
condensed. The condensed phase 𝖣^n+1 consists of all deconfined
topological defects in 𝖢^n+1. A k-codimensional topological
defect is deconfined if and only if it is equipped with a k-dimensional
A-action, which defines an E_k-module over A. When 𝖢^n+1 is
anomaly-free, the same condensation can be alternatively defined by replacing
the last two steps by a single step of condensing the E_2-algebra
Σ^k-2A directly. The condensed phase 𝖣^n+1 is determined
by the category of E_2-modules over Σ^k-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.
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