scholarly journals ON EXOTIC MODULAR TENSOR CATEGORIES

2008 ◽  
Vol 10 (supp01) ◽  
pp. 1049-1074 ◽  
Author(s):  
SEUNG-MOON HONG ◽  
ERIC ROWELL ◽  
ZHENGHAN WANG
Keyword(s):  
Lie Group ◽  
Braid Groups ◽  
Fusion Category ◽  
Direct Products ◽  
Quantum Double ◽  
Tensor Categories ◽  
Topological Quantum ◽  
Modular Tensor Categories ◽  
A Finite Group ◽  
Chern Simons

It has been conjectured that every (2 + 1)-TQFT is a Chern-Simons-Witten (CSW) theory labeled by a pair (G, λ), where G is a compact Lie group, and λ ∈ H4(BG; ℤ) a cohomology class. We study two TQFTs constructed from Jones' subfactor theory which are believed to be counterexamples to this conjecture: one is the quantum double of the even sectors of the E6subfactor, and the other is the quantum double of the even sectors of the Haagerup subfactor. We cannot prove mathematically that the two TQFTs are indeed counterexamples because CSW TQFTs, while physically defined, are not yet mathematically constructed for every pair (G, λ). The cases that are constructed mathematically include: (1) G is a finite group — the Dijkgraaf-Witten TQFTs; (2) G is torus Tn; (3) G is a connected semi-simple Lie group — the Reshetikhin-Turaev TQFTs.We prove that the two TQFTs are not among those mathematically constructed TQFTs or their direct products. Both TQFTs are of the Turaev-Viro type: quantum doubles of spherical tensor categories. We further prove that neither TQFT is a quantum double of a braided fusion category, and give evidence that neither is an orbifold or coset of TQFTs above. Moreover, representation of the braid groups from the half E6TQFT can be used to build universal topological quantum computers, and the same is expected for the Haagerup case.

scholarly journals Witness algebra and anyon braiding

2020 ◽  
Vol 30 (3) ◽  
pp. 234-270
Author(s):  
Andreas Blass ◽  
Yuri Gurevich
Keyword(s):  
Quantum Computation ◽  
Category Theory ◽  
Two Dimensional ◽  
Mathematical Basis ◽  
Tensor Categories ◽  
Topological Quantum ◽  
Modular Tensor Categories ◽  
Topological Quantum Computation

AbstractTopological quantum computation employs two-dimensional quasiparticles called anyons. The generally accepted mathematical basis for the theory of anyons is the framework of modular tensor categories. That framework involves a substantial amount of category theory and is, as a result, considered rather difficult to understand. Is the complexity of the present framework necessary? The computations of associativity and braiding matrices can be based on a much simpler framework, which looks less like category theory and more like familiar algebra. We introduce that framework here.

scholarly journals CHERN–SIMONS APPROACH TO THREE-MANIFOLD INVARIANTS

1995 ◽  
Vol 10 (06) ◽  
pp. 487-493
Author(s):  
BOGUSŁAW BRODA
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Lie Group ◽  
Three Dimensional ◽  
Simons Theory ◽  
Quantum Field ◽  
Topological Quantum ◽  
Chern Simons ◽  
Direct Implementation ◽  
Chern Simons Theory

A new, formal, noncombinatorial approach to invariants of three-dimensional manifolds of Reshetikhin, Turaev and Witten in the framework of nonperturbative topological quantum Chern–Simons theory, corresponding to an arbitrary compact simple Lie group, is presented. A direct implementation of surgery instructions in the context of quantum field theory is proposed. An explicit form of the specialization of the invariant to the group SU(2) is shown.

scholarly journals Extensions of tensor categories by finite group fusion categories

2019 ◽  
pp. 1-29
Author(s):  
SONIA NATALE ◽  
Keyword(s):  
Finite Group ◽  
Exact Sequence ◽  
Fusion Category ◽  
Matched Pair ◽  
Tensor Categories ◽  
Semisimple Hopf Algebra ◽  
Fusion Categories ◽  
A Finite Group ◽  
Exact Sequences ◽  
Group Fusion

Abstract We study exact sequences of finite tensor categories of the form Rep G → 𝒞 → 𝒟, where G is a finite group. We show that, under suitable assumptions, there exists a group Γ and mutual actions by permutations ⊳ : Γ × G → G and ⊲ : Γ × G→ Γ that make (G, Γ) into matched pair of groups endowed with a natural crossed action on 𝒟 such that 𝒞 is equivalent to a certain associated crossed extension 𝒟(G,Γ) of 𝒟. Dually, we show that an exact sequence of finite tensor categories Vec G → 𝒞 → 𝒟 induces an Aut(G)-grading on 𝒞 whose neutral homogeneous component is a (Z(G), Γ)-crossed extension of a tensor subcategory of 𝒟. As an application we prove that such extensions 𝒞 of 𝒟 are weakly group-theoretical fusion categories if and only if 𝒟 is a weakly group-theoretical fusion category. In particular, we conclude that every semisolvable semisimple Hopf algebra is weakly group-theoretical.

scholarly journals Finiteness for mapping class group representations from twisted Dijkgraaf–Witten theory

2018 ◽  
Vol 27 (06) ◽  
pp. 1850043 ◽  
Author(s):  
Paul P. Gustafson
Keyword(s):  
Braid Group ◽  
Mapping Class Group ◽  
Class Group ◽  
Braid Groups ◽  
Fusion Category ◽  
Mapping Class ◽  
Group Representations ◽  
Colored Graphs ◽  
Finite Image ◽  
Spanning Set

We show that any twisted Dijkgraaf–Witten representation of a mapping class group of an orientable, compact surface with boundary has finite image. This generalizes work of Etingof et al. showing that the braid group images are finite [P. Etingof, E. C. Rowell and S. Witherspoon, Braid group representations from twisted quantum doubles of finite groups, Pacific J. Math. 234 (2008)(1) 33–42]. In particular, our result answers their question regarding finiteness of images of arbitrary mapping class group representations in the affirmative. Our approach is to translate the problem into manipulation of colored graphs embedded in the given surface. To do this translation, we use the fact that any twisted Dijkgraaf–Witten representation associated to a finite group [Formula: see text] and 3-cocycle [Formula: see text] is isomorphic to a Turaev–Viro–Barrett–Westbury (TVBW) representation associated to the spherical fusion category [Formula: see text] of twisted [Formula: see text]-graded vector spaces. The representation space for this TVBW representation is canonically isomorphic to a vector space of [Formula: see text]-colored graphs embedded in the surface [A. Kirillov, String-net model of Turaev-Viro invariants, Preprint (2011), arXiv:1106.6033 ]. By analyzing the action of the Birman generators [J. Birman, Mapping class groups and their relationship to braid groups, Comm. Pure Appl. Math. 22 (1969) 213–242] on a finite spanning set of colored graphs, we find that the mapping class group acts by permutations on a slightly larger finite spanning set. This implies that the representation has finite image.

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.

Sign in / Sign up

Export Citation Format

Share Document