Categories of quantum liquids II
Liang Kong, Hao Zheng
TL;DR
This work constructs a unified categorical framework for gapped quantum liquids by extending separable higher categories with center functors, centralizers, and modular extensions, and by introducing orthogonal higher categories to accommodate anti-unitary symmetries. It provides a systematic classification of gapped quantum liquids with finite and spacetime symmetries, predicting new time-reversal SPT orders in spacetime dimension $\ge 3$ and proving a crystalline equivalence principle that equates crystalline and onsite-symmetry classifications. The methodology leverages condensation completion and a cascade of center-related reductions (from $E_0$ to $E_1$ and beyond) to translate high-dimensional problems into tractable algebraic data such as center equivalences and perfect pairings. The results illuminate the structure of group-theoretical higher categories, representations of finite $n$-groups, and pointed fusion $n$-categories, connecting them directly to SPT/SET orders, symmetry actions, and crystalline phases with rigorous categorical language. Overall, the paper provides a rigorous mathematical scaffold for understanding and classifying a wide spectrum of gapped phases in higher dimensions, including time-reversal and spatial symmetries, with potential implications for topological quantum matter and beyond-group-cohomology phenomena.
Abstract
We continue to develop the theory of separable higher categories, including center functors, higher centralizers, modular extensions and group theoretical higher fusion categories. Moreover, we outline a theory of orthogonal higher categories to treat anti-unitary symmetries. Using these results we derive a systematic classification of gapped quantum liquids and predict many new SPT orders in spacetime dimension $\ge3$.