Fusion categories of rank 2

Ostrik, Viktor

doi:10.4310/mrl.2003.v10.n2.a5

Cited by 77 publications

(81 citation statements)

References 13 publications

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“…Fusion categories containing only two simple objects were classified in [67]. In the TDL language, there is only one nontrivial TDL X in addition to the trivial line I, with the fusion relation X 2 = I + aX, where a = 0 or 1.…”

Section: On Fusion Categories Of Small Ranksmentioning

confidence: 99%

Topological defect lines and renormalization group flows in two dimensions

Chang

Lin

Shao

et al. 2019

J. High Energ. Phys.

284

425

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We consider topological defect lines (TDLs) in two-dimensional conformal field theories. Generalizing and encompassing both global symmetries and Verlinde lines, TDLs together with their attached defect operators provide models of fusion categories without braiding. We study the crossing relations of TDLs, discuss their relation to the 't Hooft anomaly, and use them to constrain renormalization group flows to either conformal critical points or topological quantum field theories (TQFTs). We show that if certain non-invertible TDLs are preserved along a RG flow, then the vacuum cannot be a non-degenerate gapped state. For various massive flows, we determine the infrared TQFTs completely from the consideration of TDLs together with modular invariance.

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Section: On Fusion Categories Of Small Ranksmentioning

confidence: 99%

Topological defect lines and renormalization group flows in two dimensions

Chang

Lin

Shao

et al. 2019

J. High Energ. Phys.

284

425

View full text Add to dashboard Cite

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“…We remark that the fusion rules (4) have categorifications (in terms of fusion categroies) only for n = 0, 1 [Ost03], while the Izumi-Haagerup categories are shown to exist for many n, including n = 9 [EG11]. It is important to remark that the "rules" (3,4) do not have a direct interpretation of fusion rules.…”

Section: Nilpotent Hypergroups and Intermediate Groups We Can Ask Ifmentioning

confidence: 99%

Generalized orbifold construction for conformal nets

Bischoff

2017

Rev. Math. Phys.

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Abstract. Let B be a conformal net. We give the notion of a proper action of a finite hypergroup acting by vacuum preserving unital completely positive (so-called stochastic) maps, which generalizes the proper actions of finite groups. Taking fixed points under such an action gives a finite index subnet B K of B, which generalizes the G-orbifold. Conversely, we show that if A ⊂ B is a finite inclusion of conformal nets, then A is a generalized orbifold A = B K of the conformal net B by a unique finite hypergroup K. There is a Galois correspondence between intermediate netsIn this case, the fixed point of B K ⊂ A is the generalized orbifold by the hypergroup of double cosets L\K/L.If A ⊂ B is an finite index inclusion of completely rational nets, we show that the inclusion A(I) ⊂ B(I) is conjugate to a Longo-Rehren inclusion. This implies that if B is a holomorphic net, and K acts properly on B, then there is a unitary fusion category F which is a categorification of K and Rep(B K

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“…For the sake of contradiction assume that C is a braided fusion category with Gr(C) = R n , n ≥ 2. Then the same argument as in proof of [O2,Corollary 2.2] shows that C is spherical. The lifting theory (see [EGNO,9.16]) shows that C must be degenerate, that is 1 + d 2 = 0 where d ∈ k is the dimension of the object X (as by the result of [O2], there is no categorifications of R n in characteristic zero).…”

Section: Symmetric and Braided Categorifications Of Fusion Rings Of Rmentioning

confidence: 65%

Computations in Symmetric Fusion Categories in Characteristicp

Etingof

Ostrik

Venkatesh

2016

Int Math Res Notices

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Abstract. We study properties of symmetric fusion categories in characteristic p. In particular, we introduce the notion of a super Frobenius-Perron dimension of an object X of such a category, and derive an explicit formula for the Verlinde fiber functor F (X) of X (defined by the second author) in terms of the usual and super Frobenius-Perron dimensions of X. We also compute the decomposition of symmetric powers of objects of the Verlinde category, generalizing a classical formula of Cayley and Sylvester for invariants of binary forms. Finally, we show that the Verlinde fiber functor is unique, and classify braided fusion categories of rank two and triangular semisimple Hopf algebras in any characteristic. IntroductionLet k be an algebraically closed field of characteristic p > 0. Let C be a symmetric fusion category over k. Then, according to the main result of [O], C admits a symmetric tensor functor F : C → Ver p into the Verlinde category Ver p (the quotient of Rep k (Z/pZ) by negligible morphisms); we prove that this functor is unique. We derive an explicit formula for the decomposition of F (X) into simple objects for each X ∈ C; it turns out that this decomposition is completely determined by just two parameters -the Frobenius-Perron dimension FPdim(X) and the super Frobenius-Perron dimension SFPdim(X) which we introduce in this paper. We use this formula to find the decomposition of symmetric powers of objects in Ver p , and in particular find the invariants in them -the "fusion" analog of the classical formula for polynomial invariants of binary forms. We also relate the super Frobenius-Perron dimension to the second Adams operation and to the p-adic dimension introduced in [EHO], and classify symmetric categorifications of fusion rings of rank 2. Finally, we classify triangular semisimple Hopf algebras in an arbitrary characteristic, generalizing the result of [EG] for characteristic zero, and classify braided fusion categories of rank two.The paper is organized as follows. Section 2 contains preliminaries. In Section 3 we prove the uniqueness of the Verlinde fiber functor. In Section 4, we define the super Frobenius-Perron dimension of an object 1 of a symmetric fusion category, and give a formula for this dimension in terms of the second Adams operation; here we also classify braided fusion categories of rank two. In Section 5, we prove a decomposition formula for the Verlinde fiber functor of an object X of a symmetric fusion category in terms of the ordinary and super Frobenius-Peron dimensions of X, and use this formula to compute the transcendence degrees of the symmetric and exterior algebra of X. In Section 6, we use the formula of Section 5 to find the decomposition of symmetric powers of objects in the Verlinde category, and in particular give a formula for their invariants. In Section 7, we compute the p-adic dimensions of an object in a symmetric fusion category. Finally, in Section 8, we classify semisimple triangular Hopf algebras in any characteristic.Acknowledgements. The work of P.E...

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Fusion categories of rank 2

Cited by 77 publications

References 13 publications

Topological defect lines and renormalization group flows in two dimensions

Topological defect lines and renormalization group flows in two dimensions

Generalized orbifold construction for conformal nets

Computations in Symmetric Fusion Categories in Characteristicp

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