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arXiv:2403.04446 · Journal of High Energy Physics

Weak Hopf symmetry and tube algebra of the generalized multifusion string-net model

Zhian Jia, Sheng Tan, Dagomir Kaszlikowski

Published Jul 23, 2024• 9 citations
arXivHTMLPDF

Abstract

A bstract We investigate the multifusion generalization of string-net ground states and lattice Hamiltonians, delving into their associated weak Hopf symmetries. For the multifusion string-net, the gauge symmetry manifests as a general weak Hopf algebra, leading to a reducible vacuum string label; the charge symmetry, serving as a quantum double of gauge symmetry, constitutes a connected weak Hopf algebra. This implies that the associated topological phase retains its characterization by a unitary modular tensor category (UMTC). The bulk charge symmetry can also be captured by a weak Hopf tube algebra. We offer an explicit construction of the weak Hopf tube algebra structure and thoroughly discuss its properties. The gapped boundary and domain wall models are extensively discussed, with these 1 d phases characterized by unitary multifusion categories (UMFCs). We delve into the gauge and charge symmetries of these 1 d phases, as well as the construction of the boundary and domain wall tube algebras. Additionally, we illustrate that the domain wall tube algebra can be regarded as a cross product of two boundary tube algebras. As an application of our model, we elucidate how to interpret the defective string-net as a restricted multifusion string-net.

DOI
10.1007/jhep07(2024)207

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A bstract We investigate the multifusion generalization of string-net ground states and lattice Hamiltonians, delving into their associated weak Hopf symmetries. For the multifusion string-net, the gauge symmetry manifests as a general weak Hopf algebra, leading to a reducible vacuum string label; the charge symmetry, serving as a quantum double of gauge symmetry, constitutes a connected weak Hopf algebra. This implies that the associated topological phase retains its characterization by a unitary modular tensor category (UMTC). The bulk charge symmetry can also be captured by a weak Hopf tube algebra. We offer an explicit construction of the weak Hopf tube algebra structure and thoroughly discuss its properties. The gapped boundary and domain wall models are extensively discussed, with these 1 d phases characterized by unitary multifusion categories (UMFCs). We delve into the gauge and charge symmetries of these 1 d phases, as well as the construction of the boundary and domain wall tube algebras. Additionally, we illustrate that the domain wall tube algebra can be regarded as a cross product of two boundary tube algebras. As an application of our model, we elucidate how to interpret the defective string-net as a restricted multifusion string-net.
The tube algebra, which carries the structure of a C∗C^*C∗ weak Hopf algebra, is a fundamental tool for characterizing topological excitations in topological phases. In this work, we generalize the tube algebra framework to codimension-2 defects in (2+1)(2+1)(2+1)D gapped phases described by Turaev--Viro--Barrett--Westbury TQFTs, with particular emphasis on Levin--Wen string-net models. We show that such codimension-2 defects are described by a \emph{defect tube algebra}, which naturally carries the structure of a comodule algebra over the weak Hopf tube algebra associated with the topological excitations. Boundary and domain wall defects are then characterized by the representations of the corresponding boundary and domain wall defect tube algebras. In particular, a domain wall defect tube algebra can be regarded as a generalized Drinfeld double of the tube algebras associated with the two adjacent boundary defects. More generally, for a domain wall joining NNN bulk phases, the associated domain wall defect tube algebra can be viewed as an NNN-tuple algebra. This perspective extends naturally to general kkk-defects, namely codimension-2 defects joining kkk codimension-1 defects, for which the resulting generalized tube algebra carries the structure of a multicomodule algebra.
A bstract We investigate domain walls between 2 d gapped phases of Turaev-Viro type topological quantum field theories (TQFTs) by constructing domain wall tube algebras. We begin by analyzing the domain wall tube algebra associated with bimodule categories, and then extend the construction to multimodule categories over N base fusion categories. We prove that the resulting tube algebra is naturally equipped with a C ∗ weak Hopf algebra structure. We show that topological excitations localized on domain walls are classified by representations of the corresponding domain wall tube algebra, in the sense that the functor category of bimodules admits a fusion-preserving embedding into the representation category of the domain wall tube algebra. We further establish the folding trick and Morita theory in this context. Then, most crucially, we provide a rigorous construction of the Drinfeld quantum double from weak Hopf boundary tube algebras using a skew-pairing, and establish an isomorphism between domain wall tube algebra and Drinfeld quantum double of boundary tube algebras. Motivated by the correspondence between the domain wall tube algebra and the quantum double of the boundary tube algebras, we introduce the notion of an N -tuple algebra and demonstrate how it arises in the multimodule domain wall setting. Finally, we consider defects between two domain walls, showing that such defects can be characterized by representations of a domain wall defect tube algebra. We briefly outline how these defects can be systematically treated within this representation-theoretic framework.
A bstract We present a systematic framework for constructing exactly-solvable lattice models of symmetry-enriched topological (SET) phases based on an enlarged version of the string-net model. We also gauge the global symmetries of our SET models to obtain string-net models of pure topological phases. Without invoking externally imposed onsite symmetry actions, our approach promotes the string-net model of a pure topological order, specified by an input unitary fusion category F \mathcal{F} F F , to an SET model, specified by a multifusion category together with a set of isomorphisms. Two complementary construction strategies are developed in the main text: (i) promotion via outer automorphisms of F \mathcal{F} F F and (ii) promotion via the Frobenius algebras of F \mathcal{F} F F . The global symmetries derived via these two strategies are intrinsic to topological phases and are thus termed blood symmetries, as opposed to adopted symmetries, which can be arbitrarily imposed on topological phases. We propose the concept of symmetry-gauging family of topological phases, which are related by gauging their blood symmetries. With our approach, we construct the first explicit lattice realization of a nonabelian-symmetry-enriched topological phase — the S 3 symmetry-enriched ℤ 2 × ℤ 2 quantum-double phase. The approach further reveals the role of local excitations in SET phases and establishes their symmetry constraints.
A bstract We propose weak Hopf symmetry as a general framework to explore (1+1)D topological phases that exhibit non-invertible symmetries. Inspired by the Symmetry Topological Field Theory (SymTFT) description of quantum phases with non-invertible symmetry, we construct a lattice model by introducing two distinct topological boundary conditions for a weak Hopf lattice gauge theory. One boundary encodes the topological symmetry information, while the other incorporates the non-topological dynamics. The resulting model is termed the cluster ladder model. We demonstrate that the cluster state model is a special case of this broader class of lattice models exhibiting weak Hopf symmetry H × Ĥ , where H is a weak Hopf algebra and Ĥ is its dual weak Hopf algebra. On a closed manifold, the symmetry reduces to Cocom( H ) × Cocom( Ĥ ), corresponding to the cocommutative subalgebras of H × Ĥ . An essential weak Hopf sub-symmetry is Cocom( H ) × Rep( H ), which, in the finite group case, reduces to the familiar symmetry G × Rep( G ). To exactly solve the lattice model, we introduce a weak Hopf tensor network. Furthermore, we demonstrate how to construct the lattice realization of an arbitrary fusion category symmetry S \mathcal{S} S S via combining Tannaka-Krein reconstruction or weak Hopf tube algebra and the cluster ladder model.
Abstract The generalized quantum double lattice realization of 2d topological orders based on Hopf algebras is discussed in this work. Both left-module and right-module constructions are investigated. The ribbon operators and the classification of topological excitations based on the representations of the quantum double of Hopf algebras are discussed. To generalize the model to a 2d surface with boundaries and surface defects, we present a systematic construction of the boundary Hamiltonian and domain wall Hamiltonian. The algebraic data behind the gapped boundary and domain wall are comodule algebras and bicomodule algebras. The topological excitations in the boundary and domain wall are classified by bimodules over these algebras. The ribbon operator realization of boundary-bulk duality is also discussed. Finally, via the Hopf tensor network representation of the quantum many-body states, we solve the ground state of the model in the presence of the boundary and domain wall.
We use finite group topological lattice gauge theory, also known as the quantum double model, as a lens to explore a notion of topological order enriched by a non-invertible symmetry. For invertible symmetry enriched topological order, there is an established axiomatisation in terms of a G-crossed braided fusion category. We lay the foundations for a generalisation of this notion. By condensing an arbitrary algebra of charges in a quantum double model, we demonstrate that the category of localised excitations in the resulting theory forms a hypergroup-graded extension of the category of deconfined excitations. For every element in the hypergroup, the associated domain wall acts in a typically non-invertible way on these localised excitations. Both this action and the monoidal structure are compatible with the hypergroup grading. The actual categorical action is encoded in a Hopf monad on the category of localised excitations, and gauging the non-invertible symmetry amounts to computing the category of modules over this Hopf monad. Finally, we outline how this framework naturally extends to theories obtained by condensing algebras in a generic string-net model.