Categories of quantum liquids III
Liang Kong, Hao Zheng
TL;DR
Kong and Zheng develop a unified higher-categorical framework for quantum liquids by introducing topological nets and defect nets, encapsulated in the symmetric monoidal $*$-$(n+1)$-category $ ext{Net}^n$ and its subcategory $ ext{LQS}^n$, from which the category of $n$D quantum liquids $ ext{QL}^n$ is extracted via a (co)slice construction. They provide concrete realizations from onsite finite-group symmetries and Levin–Wen models, and establish a condensation theory that proves $ ext{LQS}^n$ is $*$-condensation-complete, yielding equivalences $ ext{LQS}^n \ucong (n+1) ext{Hilb}$ and $ ext{QL}^n u ext{QL}_{ ext{sk}}^n$. The paper also develops nets on manifolds, sewing operations, and defect fusion, enabling explicit construction and composition of higher morphisms, as well as a mechanism to detect and encode local quantum symmetries through transparent walls. Overall, the work generalizes conformal nets to higher dimensions, providing a robust mathematical platform to describe gapped and gapless quantum phases and their domain walls across all spacetime dimensions.
Abstract
We continue our study of the categories of quantum liquids started in a previous work. We combine local quantum symmetries with topological skeletons into a single mathematical theory of topological nets and defect nets. In particular, we introduce the notion of a topological net, which is motivated from and generalizes that of a conformal net, and the notion of a defect net which generalizes that of a defect between conformal nets. We give explicit examples of them. Moreover, we construct the category of topological $n$-nets with $k$-morphisms defined by defect $n$-nets of codimension $k$, and show that the category of $n$D quantum liquids can be extracted from it and computed explicitly via the condensation theory of topological nets.