Categories of quantum liquids I

TL;DR

This work develops a comprehensive higher-categorical framework for quantum liquids by introducing separable and unitary $n$-categories built from condensation completion, and then applying them to unify gapped/gapless quantum phases, SPT/SET orders, and 2D rational CFTs. Central constructions include $E_m$-multi-fusion $n$-categories, their centers, and a $*$-condensation theory that yields unitary higher categories, enabling a macroscopic, category-theoretic description of quantum liquids. The authors define two complementary data types for a liquid, the local quantum symmetry and the topological skeleton, and use topological Wick rotation to connect these data to gapped/gapless boundaries and bulk phases, deriving a unified classification framework and explicit skeleta in low dimensions. They propose that the liquid category $ ext{QL}^n$ is equivalent to its skeleton $ ext{QL}_{ ext{sk}}^n$ and develop a program to detect local quantum symmetries via enriched higher categories, laying groundwork for a broad, mathematically rigorous theory of quantum phases across dimensions.

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 $*$-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.

Figures (6)

• Figure 1: macroscopic observables on the worldsheet of a 1+1D CFT
• Figure 2: the idea of topological Wick rotation
• Figure 3: The idea of topological Wick rotation (TWR): Before TWR, slightly abusing notations, $\mathcal{P}$ also denotes the category of all topological defects on the gapped wall and $\mathcal{B}$ also denotes that of topological defects of codimension 2 and higher in the $n+$1D topological order $\mathcal{B}$. After TWR, they become the category of topological sectors of states and that of operators, respectively (see KWZ21 for more details). The enriched higher category ${}^{\overline{\mathcal{B}}}\mathcal{P}$ summarizes all topological observables (or the topological skeleton) in the $n$D spacetime of the quantum liquid.
• Figure 4: the topological skeleton of the boundary of a quantum liquid
• Figure 5: Condensation completion of $\mathcal{QL}^n$

Theorems & Definitions (142)

• Remark 1.1
• Remark 2.1
• Remark 2.2
• Example 2.3
• Remark 2.4
• Remark 2.5
• Remark 2.6
• Remark 2.7
• Remark 2.8
• Lemma 2.9