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Abstract We study a commutant-closed collection of von Neumann algebras acting on a common Hilbert space indexed by a poset with an order-reversing involution. We give simple geometric axioms for the poset which allow us to construct a braided tensor category of superselection sectors analogous to the construction of Gabbiani and Fröhlich for conformal nets. For cones in$$\mathbb {R}^2$$ $R^2$, we weaken our conditions to a bounded spread version of Haag duality and obtain similar results. We show that intertwined nets of algebras have isomorphic braided tensor categories of superselection sectors. Finally, we show that the categories constructed here are equivalent to those constructed by Naaijkens and Ogata for certain 2D quantum spin systems. 

Abstract We introduce a set of axioms for locally topologically ordered quantum spin systems in terms of nets of local ground state projections, and we show they are satisfied by Kitaev’s Toric Code and Levin-Wen type models. For a locally topologically ordered spin system on$$\mathbb {Z}^{k}$$, we define a local net of boundary algebras on$$\mathbb {Z}^{k-1}$$, which provides a mathematically precise algebraic description of the holographic dual of the bulk topological order. We construct a canonical quantum channel so that states on the boundary quasi-local algebra parameterize bulk-boundary states without reference to a boundary Hamiltonian. As a corollary, we obtain a new proof of a recent result of Ogata [Oga24] that the bulk cone von Neumann algebra in the Toric Code is of type$$\mathrm {II}$$, and we show that Levin-Wen models can have cone algebras of type$$\mathrm {III}$$. Finally, we argue that the braided tensor category of DHR bimodules for the net of boundary algebras characterizes the bulk topological order in (2+1)D, and can also be used to characterize the topological order of boundary states. 

Braided-enriched monoidal categories were introduced in the work of Morrison–Penneys, where they were characterized using braided central functors. The recent work of Kong–Yuan–Zhang–Zheng and Dell extended this characterization to an equivalence of 2-categories. Since their introduction, braided-enriched fusion categories have been used to describe certain phenomena in topologically ordered systems in theoretical condensed matter physics. While these systems are unitary, there was previously no general notion of unitarity for enriched categories in the literature. We supply the notion of unitarity for enriched categories and braided-enriched monoidal categories and extend the above 2-equivalence to the unitary setting. 

Abstract We show that the Levin-Wen model of a unitary fusion category$${\mathcal {C}}$$ $C$is a gauge theory with gauge symmetry given by the tube algebra$${\text {Tube}}({\mathcal {C}})$$ $\text{Tube}\left(C\right)$. In particular, we define a model corresponding to a$${\text {Tube}}({\mathcal {C}})$$ $\text{Tube}\left(C\right)$symmetry protected topological phase, and we provide a gauging procedure which results in the corresponding Levin-Wen model. In the case$${\mathcal {C}}=\textsf{Hilb}(G,\omega )$$ $C=\mathrm{Hilb}(G,\omega )$, we show how our procedure reduces to the twisted gauging of a trivalG-SPT to produce the Twisted Quantum Double. We further provide an example which is outside the bounds of the current literature, the trivial Fibbonacci SPT, whose gauge theory results in the doubled Fibonacci string-net. Our formalism has a natural topological interpretation with string diagrams living on a punctured sphere. We provide diagrams to supplement our mathematical proofs and to give the reader an intuitive understanding of the subject matter. 

The recent article by Jones et al. [arXiv:2307.12552 (2023)] gave local topological order (LTO) axioms for a quantum spin system, showed they held in Kitaev’s Toric Code and in Levin-Wen string net models, and gave a bulk boundary correspondence to describe bulk excitations in terms of the boundary net of algebras. In this article, we prove the LTO axioms for Kitaev’s Quantum Double model for a finite group G. We identify the boundary nets of algebras with fusion categorical nets associated to (Hilb(G),C[G]) or (Rep(G),CG) depending on whether the boundary cut is rough or smooth, respectively. This allows us to make connections to the work of Ogata [Ann. Henri Poincaré 25, 2353–2387 (2024)] on the type of the cone von Neumann algebras in the algebraic quantum field theory approach to topological superselection sectors. We show that the boundary algebras can also be calculated from a trivial G-symmetry protected topological phase (G-SPT), and that the gauging map preserves the boundary algebras. Finally, we compute the boundary algebras for the (3 + 1)D Quantum Double model associated to an Abelian group. 

Abstract In our previous article (http://arxiv.org/abs/1607.06041), we established an equivalence between pointed pivotal module tensor categories and anchored planar algebras. This article introduces the notion of unitarity for both module tensor categories and anchored planar algebras, and establishes the unitary analog of the above equivalence. Our constructions use Baez’s 2-Hilbert spaces (i.e., semisimple$$\textrm{C}^*$$ ${\text{C}}^{\ast }$-categories equipped with unitary traces), the unitary Yoneda embedding, and the notion of unitary adjunction for dagger functors between 2-Hilbert spaces. 

A unitary fusion category is called $$\mathbb{Z}/2\mathbb{Z}$$-quadratic if it has a $$\mathbb{Z}/2\mathbb{Z}$$ group of invertible objects and one other orbit of simple objects under the action of this group. We give a complete classification of $$\mathbb{Z}/2\mathbb{Z}$$-quadratic unitary fusion categories. The main tools for this classification are skein theory, a generalization of Ostrik's results on formal codegrees to analyze the induction of the group elements to the center, and a computation similar to Larson's rank-finiteness bound for $$\mathbb{Z}/3\mathbb{Z}$$-near group pseudounitary fusion categories. This last computation is contained in an appendix coauthored with attendees from the 2014 AMS MRC on Mathematics of Quantum Phases of Matter and Quantum Information. 

Boundaries of Walker-Wang models have been used to construct commuting projector models which realize chiral unitary modular tensor categories (UMTCs) as boundary excitations. Given a UMTC $\mathcal{A}$representing the Witt class of an anomaly, the article \cite{MR4640433} gave a commuting projector model associated to an $\mathcal{A}$-enriched unitary fusion category $\mathcal{X}$on a 2D boundary of the 3D Walker-Wang model associated to $\mathcal{A}$. That article claimed that the boundary excitations were given by the enriched center/Müger centralizer $Z^{\mathcal{A}}\left(\mathcal{X}\right)$of $\mathcal{A}$in $Z\left(\mathcal{X}\right)$.In this article, we give a rigorous treatment of this 2D boundary model, and we verify this assertion using topological quantum field theory (TQFT) techniques, including skein modules and a certain semisimple algebra whose representation category describes boundary excitations. We also use TQFT techniques to show the 3D bulk point excitations of the Walker-Wang bulk are given by the Müger center $Z_2\left(\mathcal{A}\right)$, and we construct bulk-to-boundary hopping operators $Z_2\left(\mathcal{A}\right)\to Z^{\mathcal{A}}\left(\mathcal{X}\right)$reflecting how the UMTC of boundary excitations $Z^{\mathcal{A}}\left(\mathcal{X}\right)$is symmetric-braided enriched in $Z_2\left(\mathcal{A}\right)$.This article also includes a self-contained comprehensive review of the Levin-Wen string net model from a unitary tensor category viewpoint, as opposed to the skeletal $6j$symbol viewpoint.