scholarly journals The center of the extended Haagerup subfactor has 22 simple objects

2017 ◽  
Vol 28 (01) ◽  
pp. 1750009 ◽  
Author(s):  
Scott Morrison ◽  
Kevin Walker
Keyword(s):  
Fusion Category ◽  
Dimension Function ◽  
Fusion Ring ◽  
Fusion Categories ◽  
Modular Data ◽  
Grothendieck Rings ◽  
Combinatorial Data

We explain a technique for discovering the number of simple objects in [Formula: see text], the center of a fusion category [Formula: see text], as well as the combinatorial data of the induction and restriction functors at the level of Grothendieck rings. The only input is the fusion ring [Formula: see text] and the dimension function [Formula: see text]. In particular, we apply this to deduce that the center of the extended Haagerup subfactor has 22 simple objects, along with their decompositions as objects in either of the fusion categories associated to the subfactor. This information has been used subsequently in [T. Gannon and S. Morrison, Modular data for the extended Haagerup subfactor (2016), arXiv:1606.07165 .] to compute the full modular data. This is the published version of arXiv:1404.3955 .

scholarly journals String-net models for nonspherical pivotal fusion categories

2020 ◽  
Vol 29 (06) ◽  
pp. 2050035
Author(s):  
Ingo Runkel
Keyword(s):  
Field Theory ◽  
Marked Point ◽  
Simple Object ◽  
Fusion Category ◽  
Mapping Class ◽  
State Spaces ◽  
Spin Structures ◽  
Conformal Field ◽  
Fusion Categories ◽  
Topological Quantum

A string-net model associates a vector space to a surface in terms of graphs decorated by objects and morphisms of a pivotal fusion category modulo local relations. String-net models are usually considered for spherical fusion categories, and in this case, the vector spaces agree with the state spaces of the corresponding Turaev–Viro topological quantum field theory. In the present work, some effects of dropping the sphericality condition are investigated. In one example of nonspherical pivotal fusion categories, the string-net space counts the number of [Formula: see text]-spin structures on a surface and carries an isomorphic representation of the mapping class group. Another example concerns the string-net space of a sphere with one marked point labeled by a simple object [Formula: see text] of the Drinfeld center. This space is found to be nonzero iff [Formula: see text] is isomorphic to a nonunit simple object determined by the nonspherical pivotal structure. The last example mirrors the effect of deforming the stress tensor of a two-dimensional conformal field theory, such as in the topological twist of a supersymmetric theory.

scholarly journals Frobenius property for fusion categories of small integral dimension

2014 ◽  
Vol 14 (02) ◽  
pp. 1550011 ◽  
Author(s):  
Jingcheng Dong ◽  
Sonia Natale ◽  
Leandro Vendramin
Keyword(s):  
Characteristic Zero ◽  
Fusion Category ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Fusion Categories

Let k be an algebraically closed field of characteristic zero. In this paper, we prove that fusion categories of Frobenius–Perron dimensions 84 and 90 are of Frobenius type. Combining this with previous results in the literature, we obtain that every weakly integral fusion category of Frobenius–Perron dimension less than 120 is of Frobenius type.

scholarly journals On fusion rules and solvability of a fusion category

2017 ◽  
Vol 20 (1) ◽  
Author(s):  
Melisa Escañuela González ◽  
Sonia Natale
Keyword(s):  
Hopf Algebras ◽  
Fusion Category ◽  
Alternating Groups ◽  
Fusion Rules ◽  
Fusion Categories

AbstractWe address the question whether or not the condition on a fusion category being solvable is determined by its fusion rules. We prove that the answer is affirmative for some families of non-solvable examples arising from representations of semisimple Hopf algebras associated to exact factorizations of the symmetric and alternating groups. In the context of spherical fusion categories, we also consider the invariant provided by the

scholarly journals Generalized orbifold construction for conformal nets

2017 ◽  
Vol 29 (01) ◽  
pp. 1750002 ◽  
Author(s):  
Marcel Bischoff
Keyword(s):  
Fixed Point ◽  
Finite Index ◽  
Fusion Category ◽  
Proper Action ◽  
Completely Positive ◽  
Double Cosets ◽  
Fusion Ring ◽  
Galois Correspondence ◽  
A Finite Group ◽  
Orbifold Construction

Let [Formula: see text] be a conformal net. We give the notion of a proper action of a finite hypergroup [Formula: see text] acting by vacuum preserving unital completely positive (so-called stochastic) maps on [Formula: see text] which generalizes the proper action of a finite group [Formula: see text]. Taking the fixed point under such an action gives a finite index subnet [Formula: see text] of [Formula: see text], which generalizes the [Formula: see text]-orbifold net. Conversely, we show that if [Formula: see text] is a finite inclusion of conformal nets, then [Formula: see text] is a generalized orbifold [Formula: see text] of the conformal net [Formula: see text] by a unique finite hypergroup [Formula: see text]. There is a Galois correspondence between intermediate nets [Formula: see text] and subhypergroups [Formula: see text] given by [Formula: see text]. In this case, the fixed point of [Formula: see text] is the generalized orbifold by the hypergroup of double cosets [Formula: see text]. If [Formula: see text] is a finite index inclusion of completely rational nets, we show that the inclusion [Formula: see text] is conjugate to an intermediate subfactor of a Longo–Rehren inclusion. This implies that if [Formula: see text] is a holomorphic net, and [Formula: see text] acts properly on [Formula: see text], then there is a unitary fusion category [Formula: see text] which is a categorification of [Formula: see text] and [Formula: see text] is braided equivalent to the Drinfel’d center [Formula: see text]. More generally, if [Formula: see text] is a completely rational conformal net and [Formula: see text] acts properly on [Formula: see text], then there is a unitary fusion category [Formula: see text] extending [Formula: see text], such that [Formula: see text] is given by the double cosets of the fusion ring of [Formula: see text] by the Verlinde fusion ring of [Formula: see text] and [Formula: see text] is braided equivalent to the Müger centralizer of [Formula: see text] in the Drinfel’d center [Formula: see text].

scholarly journals State sum invariants of three manifolds from spherical multi-fusion categories

2017 ◽  
Vol 26 (14) ◽  
pp. 1750104 ◽  
Author(s):  
Shawn X. Cui ◽  
Zhenghan Wang
Keyword(s):  
Field Theories ◽  
Fusion Category ◽  
Usual Sense ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Quantum Invariants ◽  
Fusion Categories ◽  
Topological Quantum ◽  
Weakened Version ◽  
Theory Formula

We define a family of quantum invariants of closed oriented [Formula: see text]-manifolds using spherical multi-fusion categories (SMFCs). The state sum nature of this invariant leads directly to [Formula: see text]-dimensional topological quantum field theories ([Formula: see text]s), which generalize the Turaev–Viro–Barrett–Westbury ([Formula: see text]) [Formula: see text]s from spherical fusion categories. The invariant is given as a state sum over labeled triangulations, which is mostly parallel to, but richer than the [Formula: see text] approach in that here the labels live not only on [Formula: see text]-simplices but also on [Formula: see text]-simplices. It is shown that a multi-fusion category in general cannot be a spherical fusion category in the usual sense. Thus, we introduce the concept of a SMFC by imposing a weakened version of sphericity. Besides containing the [Formula: see text] theory, our construction also includes the recent higher gauge theory [Formula: see text]-[Formula: see text]s given by Kapustin and Thorngren, which was not known to have a categorical origin before.

THE CASIMIR NUMBER AND THE DETERMINANT OF A FUSION CATEGORY

2020 ◽  
pp. 1-13
Author(s):  
ZHIHUA WANG ◽  
GONGXIANG LIU ◽  
LIBIN LI
Keyword(s):  
Fusion Category ◽  
Complex Numbers ◽  
Closed Field ◽  
Arbitrary Characteristic ◽  
Fusion Categories ◽  
Cauchy Theorem ◽  
Prime Factors ◽  
Numerical Invariants ◽  
Characteristic Two ◽  
Positive Integers

Abstract Let $\mathcal{C}$ be a fusion category over an algebraically closed field $\mathbb{k}$ of arbitrary characteristic. Two numerical invariants of $\mathcal{C}$ , that is, the Casimir number and the determinant of $\mathcal{C}$ are considered in this paper. These two numbers are both positive integers and admit the property that the Grothendieck algebra $(\mathcal{C})\otimes_{\mathbb{Z}}K$ over any field K is semisimple if and only if any of these numbers is not zero in K. This shows that these two numbers have the same prime factors. If moreover $\mathcal{C}$ is pivotal, it gives a numerical criterion that $\mathcal{C}$ is nondegenerate if and only if any of these numbers is not zero in $\mathbb{k}$ . For the case that $\mathcal{C}$ is a spherical fusion category over the field $\mathbb{C}$ of complex numbers, these two numbers and the Frobenius–Schur exponent of $\mathcal{C}$ share the same prime factors. This may be thought of as another version of the Cauchy theorem for spherical fusion categories.

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

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Kansei Inamura
Keyword(s):  
Topological Field Theories ◽  
Time Reversal ◽  
Equivalence Classes ◽  
Field Theories ◽  
Fusion Category ◽  
Topological Phases ◽  
Fusion Categories ◽  
Self Duality ◽  
Topological Field ◽  
Symmetry Protected

Abstract Fusion category symmetries are finite symmetries in 1+1 dimensions described by unitary fusion categories. We classify 1+1d time-reversal invariant bosonic symmetry protected topological (SPT) phases with fusion category symmetry by using topological field theories. We first formulate two-dimensional unoriented topological field theories whose symmetry splits into time-reversal symmetry and fusion category symmetry. We then solve them to show that SPT phases are classified by equivalence classes of quintuples (Z, M, i, s, ϕ) where (Z, M, i) is a fiber functor, s is a sign, and ϕ is the action of orientation- reversing symmetry that is compatible with the fiber functor (Z, M, i). We apply this classification to SPT phases with Kramers-Wannier-like self-duality.

scholarly journals On Braided and Ribbon Unitary Fusion Categories

2014 ◽  
Vol 57 (3) ◽  
pp. 506-510 ◽  
Author(s):  
César Galindo
Keyword(s):  
Fusion Category ◽  
Fusion Categories ◽  
Ribbon Structure ◽  
Braided Fusion Category

AbstractWe prove that every braiding over a unitary fusion category is unitary and every unitary braided fusion category admits a unique unitary ribbon structure.

scholarly journals Slightly Degenerate Categories and $\mathbb{Z}$-Modular Data

2019 ◽  
Author(s):  
Abel Lacabanne
Keyword(s):  
Reflection Group ◽  
Fusion Category ◽  
Complex Reflection ◽  
Complex Reflection Group ◽  
Modular Data

Abstract Given a slightly degenerate braided pivotal fusion category $\mathscr{C}$, we explain how it naturally gives rise to a $\mathbb{Z}$-modular data. We do not restrict to spherical categories and work with pivotal categories. Finally, we give an interpretation in this framework of the Bonnafé–Rouquier categorification of the $\mathbb{Z}$-modular datum associated to nontrivial family of the cyclic complex reflection group.

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