量子液体的范畴 I
Categories of quantum liquids I
AI总结：
本文基于凝聚完备化与拓扑 Wick 旋转建立可分离及酉高阶范畴理论，并统一刻画包含拓扑序、SPT/SET 序等的量子液体。
摘要：
我们在 Gaiotto 和 Johnson-Freyd 关于凝聚完备化工作的基础上，发展了一套可分离高阶范畴的数学理论。基于该理论，我们证明了关于 $E_m$-多融合高阶范畴及其高阶中心的若干基本结果。我们还基于凝聚完备化的 $*$-版本概述了一套酉高阶范畴理论。在完成这些数学准备后，我们依据拓扑 Wick 旋转的思想，发展了一套涵盖所有量子液体的统一数学理论，其中包括拓扑序、SPT/SET 序、对称性破缺序以及类 CFT 无隙相。我们解释了量子液体由拓扑骨架和局部量子对称性两部分组成，并证明所有 $n$D 量子液体构成一个 $*$-凝聚完备高阶范畴，其等价类型可由一个简单的余切片 1-范畴显式计算。
英文摘要：
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 $*$-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 $*$-condensation complete higher category whose equivalence type can be computed explicitly from a simple coslice 1-category.
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