Categories of quantum liquids I

Abstract

We develop a mathematical theory of separable higher categories based on Gaiotto and Johnson-Freyd's work on condensation completion. Based on this theory, we prove some fundamental results on $E_m$-multi-fusion higher categories and their higher centers. We also outline a theory of unitary higher categories based on a $\ast$-version of condensation completion. After these mathematical preparations, based on the idea of topological Wick rotation, we develop a unified mathematical theory of all quantum liquids, which include topological orders, SPT/SET orders, symmetry-breaking orders and CFT-like gapless phases. We explain that a quantum liquid consists of two parts, the topological skeleton and the local quantum symmetry, and show that all $n$D quantum liquids form a $\ast$-condensation complete higher category whose equivalence type can be computed explicitly from a simple coslice 1-category.

arXiv comment36 pages, a minor revision, including a correction of an typo in Theorem 5.5, an illustrative picture in Hypothesis 5.3 and refinement of a few sentences

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