We propose a definition of a non-invertible symmetry enriched topological order (NI-SETO), and we implement our definition for string net models. We do so in two ways, using full inclusions of unitary fusion categories (UFCs), as well as anyon condensation. In both cases, the NI-SETO is a relative center of UFCs. All NI-SETOs can be realized in either model, where we can use enriched UFCs to get chiral examples on the boundary of a 3D Walker-Wang model representing the anomaly. We describe several examples of NI-SETOs and compute the qualitative symmetry action on anyons and symmetry defects using tube algebra techniques.
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.
A bstract We develop a systematic framework for understanding symmetries in topological phases in 2 + 1 dimensions using the string-net model, encompassing both gauge invariances that preserve anyon types and global symmetries permuting anyon types, including both invertible symmetries describable by groups and noninvertible symmetries described by categories. As an archetypal example, we reveal the first noninvertible categorical gauge invariance of topological orders in 2 + 1 dimensions: the Fibonacci gauge invariance of the doubled Fibonacci topological order, described by the Fibonacci fusion 2-category. Our approach involves two steps: first, classifying and establishing dualities between different string-net models describing the same topological order; and second, constructing symmetry transformations within the same string-net model when the dual models have isomorphic input data, achieved by composing duality maps with isomorphisms of degrees of freedom between the dual models.
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 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 F and (ii) promotion via the Frobenius algebras of
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.
Although anyon condensation is a standard mechanism for relating topological orders, anyon condensation in symmetry-enriched topological (SET) phases is more intricate because the condensate must also be compatible with the global symmetry. We study symmetry-preserving anyon condensation in SET phases described by the enlarged Hu-Geer-Wu (HGW) string-net model with multifusion-category input data. We show that a
G-preserving condensation is characterized by a compatible grading of the input multifusion category, and that this grading constructs the multifusion-category input of the child SET phase. To make this construction concrete, we consider the case where the relevant data come from a finite group extension
E of
G by
N: an
E-graded fusion category induces a
G-SET input by passing to the quotient symmetry
G, and the resulting input naturally carries a compatible
N-grading that implements the further condensation inside the SET phase while preserving
G. We illustrate the construction using three quantum-double examples: the trivial extension
Z2×Z2, the non-Abelian semidirect product
S3, and the nontrivial central extension
Z4. The
Z4 example further shows that symmetry fractionalized anyons do not obstruct symmetry-preserving condensation once the condensate is treated as a physical coherent state.
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 A representing the Witt class of an anomaly, the article \cite{MR4640433} gave a commuting projector model associated to an A -enriched unitary fusion category X on a 2D boundary of the 3D Walker-Wang model associated to A . That article claimed that the boundary excitations were given by the enriched center/Müger centralizer ZA(X) of A in Z(X) .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 Z2(A) , and we construct bulk-to-boundary hopping operators Z2(A)→ZA(X) reflecting how the UMTC of boundary excitations ZA(X) is symmetric-braided enriched in Z2(A) .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.
Abstract Levin-Wen string-net models provide a construction of (2+1)D topologically ordered phases of matter with anyonic localized excitations described by the Drinfeld center of a unitary fusion category. Anyon condensation is a mechanism for phase transitions between (2+1)D topologically ordered phases. We construct an extension of Levin-Wen models in which tuning a parameter implements anyon condensation. We also describe the classification of anyons in Levin-Wen models via representation theory of the tube algebra, and use a variant of the tube algebra to classify low-energy localized excitations in the condensed phase.