Topological field theories and symmetry protected topological phases with fusion category symmetries

Inamura, Kansei

doi:10.1007/jhep05(2021)204

Cited by 29 publications

(23 citation statements)

References 57 publications

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“…Let us show that the bulk Frobenius algebra H, namely the a = 1 case, is semisimple. This was already proven in [18], 29 but we repeat it here in order to generalize it to the a = 1 case. It is well-known that a finite-dimensional algebra is semisimple if and only if the following trace in the regular representation is non-degenerate:…”

Section: Jhep12(2021)028supporting

confidence: 62%

“…There is a vast literature [7][8][9][10][16][17][18][19][20][21][22][23][24][25][26][27][28][29] on two-dimensional topological field theories (TFTs) with varying amounts of structure. Like other types of TFTs, C-symmetric TFTs can be formulated axiomatically, either in the language of transition amplitudes associated to cobordisms [7], or in terms of the correlators of point-like operators [8].…”

Section: Jhep12(2021)028mentioning

confidence: 99%

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Construction of two-dimensional topological field theories with non-invertible symmetries

Huang

Lin

Seifnashri

2021

J. High Energ. Phys.

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We construct the defining data of two-dimensional topological field theories (TFTs) enriched by non-invertible symmetries/topological defect lines. Simple formulae for the three-point functions and the lasso two-point functions are derived, and crossing symmetry is proven. The key ingredients are open-to-closed maps and a boundary crossing relation, by which we show that a diagonal basis exists in the defect Hilbert spaces. We then introduce regular TFTs, provide their explicit constructions for the Fibonacci, Ising and Haagerup ℋ3 fusion categories, and match our formulae with previous bootstrap results. We end by explaining how non-regular TFTs are obtained from regular TFTs via generalized gauging.

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Section: Jhep12(2021)028supporting

confidence: 62%

Section: Jhep12(2021)028mentioning

confidence: 99%

Construction of two-dimensional topological field theories with non-invertible symmetries

Huang

Lin

Seifnashri

2021

J. High Energ. Phys.

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“…One of them is so-called "non-invertible symmetry." In 2 dimensions, such non-invertible symmetries are described by "fusion categories" and investigated actively in many papers including [2,3,4,5,6,7,8,9,10,11,12,13]. Non-invertible symmetries in higher dimensions are less understood than those in 2 dimensions.…”

Section: Introductionmentioning

confidence: 99%

Non-invertible topological defects in 4-dimensional $\mathbb{Z}_2$ pure lattice gauge theory

Koide

Nagoya

Yamaguchi

2021

Preprint

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We explore topological defects in the 4-dimensional pure Z 2 lattice gauge theory. This theory has 1-form Z 2 center symmetry as well as the Kramers-Wannier-Wegner (KWW) duality. We construct the KWW duality topological defects in the similar way to that constructed by Aasen, Mong, Fendley [1] for the 2-dimensional Ising model. These duality defects turn out to be non-invertible. We also construct the 1-form Z 2 symmetry defects as well as the junctions among KWW duality defects and 1-form Z 2 center symmetry defects. The crossing relations among these defects are derived. The expectation values of some configurations of these topological defects are calculated by using these crossing relations.

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“…In operator language, we demand that the allowed local operators commute with the operator appearing in eqn. (77). The algebra of the allowed local operators will give rise to emergent low energy symmetry.…”

Section: A 1d Bosonic Quantum System With a Z2 × Z2 Symmetry "Beyond ...mentioning

confidence: 99%

Algebra of local symmetric operators and braided fusion $n$-category -- symmetry is a shadow of topological order

Chatterjee¹,

Wen²

2022

Preprint

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Symmetry is usually defined via transformations described by a (higher) group. But a symmetry really corresponds to an algebra of local symmetric operators, which directly constrains the properties of the system. In particular, isomorphic operator algebras correspond to equivalent symmetries. In this paper, we pointed out that the algebra of local symmetry operators actually contains extended string-like, membrane-like, etc operators. The algebra of those extended operators in n-dimensional space gives rise to a non-degenerate braided fusion n-category, which happens to describe a topological order in one higher dimension. This allows us to show that the equivalent classes of finite symmetries actually correspond to topological orders in one higher dimension. Such a holographic theory not only describes (higher) symmetries, it also describes anomalous (higher) symmetries, non-invertible (higher) symmetries (also known as algebraic higher symmetries), and (invertible and non-invertible) gravitational anomalies, in a unified and systematic way. We demonstrate this unified holographic framework via some simple examples of (higher and/or anomalous) symmetries, as well as a symmetry that is neither anomalous nor anomaly-free. We also show the equivalence between Z2 × Z2 symmetry with mixed anomaly and Z4 symmetry, as well as between many other symmetries, in 1-dimensional space. CONTENTS(1) 2 1-symmetry in 2d space D. The equivalence between Z2 0-symmetry and Z(1) 21-symmetry in 2d space IX. 2d non-Abelian symmetry and its dual A. The G 0-symmetry in 2d space B. The Grep 1-symmetry in 2d space X. A review of holographic theory of (algebraic higher) symmetry A. Representation category B. A holographic point view of symmetry C. Transformation category -dual of the representation category D. A simple example 1. Holographic view of 2d Z2 0-symmetry 2. Holographic view of 2d Z2 1-symmetry 3. Symmetry R = 2Rep Z 2 and dual-symmetry R = 2VecZ 2 XI. Equivalent symmetries A. Some known examples B. Equivalence between anomalous and anomaly-free Zn and Zn 1 × Zn 2 symmetries in 1-dimensional space

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Topological field theories and symmetry protected topological phases with fusion category symmetries

Cited by 29 publications

References 57 publications

Construction of two-dimensional topological field theories with non-invertible symmetries

Construction of two-dimensional topological field theories with non-invertible symmetries

Non-invertible topological defects in 4-dimensional $\mathbb{Z}_2$ pure lattice gauge theory

Algebra of local symmetric operators and braided fusion $n$-category -- symmetry is a shadow of topological order

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