On generalized symmetries and structure of modular categories

Cui, Shawn X.; Zini, Modjtaba Shokrian; Wang, Zhenghan

doi:10.1007/s11425-018-9455-5

Cited by 9 publications

(9 citation statements)

References 30 publications

(76 reference statements)

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“…It is an important open question to find the inverse process to this more general condensation. The recent article [CZW18] provides an interesting step in this direction.…”

Section: (De)equivariantization Condensation and Gaugingmentioning

confidence: 99%

Spontaneous symmetry breaking from anyon condensation

Bischoff

Jones

Lü

et al. 2019

J. High Energ. Phys.

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In a physical system undergoing a continuous quantum phase transition, spontaneous symmetry breaking occurs when certain symmetries of the Hamiltonian fail to be preserved in the ground state. In the traditional Landau theory, a symmetry group can break down to any subgroup. However, this no longer holds across a continuous phase transition driven by anyon condensation in symmetry enriched topological orders (SETOs). For a SETO described by a Gcrossed braided extension C ⊆ C × G , we show that physical considerations require that a connected etale algebra A ∈ C admit a G-equivariant algebra structure for symmetry to be preserved under condensation of A. Given any categorical action G → Aut br ⊗ (C) such that g(A) ∼ = A for all g ∈ G, we show there is a short exact sequence whose splittings correspond to G-equivariant algebra structures. The non-splitting of this sequence forces spontaneous symmetry breaking under condensation of A. Furthermore, we show that if symmetry is preserved, there is a canonically associated SETO of C loc A , and gauging this symmetry commutes with anyon condensation.

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“…It is an important open question to find the inverse process to this more general condensation. The recent article [CZW18] provides an interesting step in this direction.…”

Section: (De)equivariantization Condensation and Gaugingmentioning

confidence: 99%

Spontaneous symmetry breaking from anyon condensation

Bischoff

Jones

Lü

et al. 2019

J. High Energ. Phys.

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“…The name "generalised orbifold" derives from the observation that example 3 is an actual orbifold by G, but that the same construction also covers examples 1 and 2. This is similar to the use of the term "generalised symmetries" in [6].…”

Section: Motivation From Three-dimensional Tqftmentioning

confidence: 71%

“…1. Examples 2. and 3. can also be obtained by a construction using Hopf monads developed in [6], but example 1 is in general not covered by that construction.…”

Section: Letmentioning

confidence: 99%

Constructing modular categories from orbifold data

Mulevicius

Runkel

2023

Quantum Topol.

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The notion of an orbifold datum A in a modular fusion category C was introduced as part of a generalised orbifold construction for Reshetikhin-Turaev TQFTs by Carqueville, Runkel, and Schaumann in 2018. In this paper, given a simple orbifold datum A in C , we introduce a ribbon category C A and show that it is again a modular fusion category. The definition of C A is motivated by properties of Wilson lines in the generalised orbifold. We analyse two examples in detail: (i) when A is given by a simple commutative -separable Frobenius algebra A in C; (ii) when A is an orbifold datum in C D Vect, built from a spherical fusion category Ã. We show that, in case (i), C A is ribbon-equivalent to the category of local modules of A, and, in case (ii), to the Drinfeld centre of Ã. The category C A thus unifies these two constructions into a single algebraic setting.

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“…While our main result shows that it is still possible to learn new things about MTCs through elementary considerations, already there has been work towards a more general extension theory of MTCs by mathematical objects with richer structure than groups, for example the Hopf monads of [8] or hypergroups of [3].The success of a classical approach here suggests it may too be possible to deduce fusion rules for more general symmetry-enriched categories using only the decategorified part of the symmetry.…”

Section: More General Symmetriesmentioning

confidence: 85%

Fusion rules for permutation extensions of modular tensor categories

Delaney

2019

Preprint

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We give a construction and algorithmic description of the fusion ring of permutation extensions of an arbitrary modular tensor category using a combinatorial approach inspired by the physics of anyons and symmetry defects in bosonic topological phases of matter. The definition is illustrated with examples, namely bilayer symmetry defects and S3-extensions of small modular tensor categories like the Ising and Fibonacci theories. An implementation of the fusion algorithm is provided in the form of a Mathematica package. We introduce the notions of confinement and deconfinement of anyons and defects, respectively, which develop the tools to generalize our approach to more general fusion rings of G-crossed extensions.

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On generalized symmetries and structure of modular categories

Cited by 9 publications

References 30 publications

Spontaneous symmetry breaking from anyon condensation

Spontaneous symmetry breaking from anyon condensation

Constructing modular categories from orbifold data

Fusion rules for permutation extensions of modular tensor categories

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