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A FINITENESS PROPERTY FOR BRAIDED FUSION CATEGORIES
"... We introduce a finiteness property for braided fusion categories, describe a conjecture that would characterize categories possessing this, and verify the conjecture in a number of important cases. In particular we say a category has property F if the associated braid group representations factor o ..."
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We introduce a finiteness property for braided fusion categories, describe a conjecture that would characterize categories possessing this, and verify the conjecture in a number of important cases. In particular we say a category has property F if the associated braid group representations factor over a finite group, and suggest that categories of integral FrobeniusPerron dimension are precisely those with property F.
Congruence subgroups and generalized FrobeniusSchur indicators
"... Abstract. We define generalized FrobeniusSchur indicators for objects in a linear pivotal category C. An equivariant indicator of an object is defined as a functional on the Grothendieck algebra of the quantum double Z(C) of C using the values of the generalized FrobeniusSchur indicators. In a sph ..."
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Abstract. We define generalized FrobeniusSchur indicators for objects in a linear pivotal category C. An equivariant indicator of an object is defined as a functional on the Grothendieck algebra of the quantum double Z(C) of C using the values of the generalized FrobeniusSchur indicators. In a spherical fusion category C with FrobeniusSchur exponent N, we prove that the set of all equivariant indicators admits a natural action of the modular group, and the kernel of the canonical modular representation of Z(C) is a congruence subgroup of level N. Moreover, if C is modular, then the kernel of the projective modular representation of C is also a congruence subgroup of level N, and every modular representation of C has a finite image.
The classification of subfactors of index at most 5
"... Abstract. A subfactor is an inclusion N ⊂ M of von Neumann algebras with trivial centers. The simplest example comes from the fixed points of a group action MG ⊂M, and subfactors can be thought of as fixed points of more general grouplike algebraic structures. These algebraic structures are closely ..."
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Abstract. A subfactor is an inclusion N ⊂ M of von Neumann algebras with trivial centers. The simplest example comes from the fixed points of a group action MG ⊂M, and subfactors can be thought of as fixed points of more general grouplike algebraic structures. These algebraic structures are closely related to tensor categories and have played important roles in knot theory, quantum groups, statistical mechanics, and topological quantum field theory. There’s a measure of size of a subfactor, called the index. Remarkably the values of the index below 4 are quantized, which suggests that it may be possible to classify subfactors of small index. Subfactors of index at most 4 were classified in the ’80s and early ’90s. The possible index values above 4 are not quantized, but once you exclude a certain family it turns out that again the possibilities are quantized. Recently the classification of subfactors has been extended up to index 5, and (outside of the infinite families) there are only 10 subfactors of index between 4 and 5. We give a summary of the key ideas in this classification and discuss what is known about these special small subfactors. Contents
Noncyclotomic fusion categories
 Trans. Amer. Math. Soc
"... can be completely defined over a cyclotomic field. We show that this is not the case: in particular one of the fusion categories coming from the Haagerup subfactor [1] and one coming from the newly constructed extended Haagerup subfactor [2] can not be completely defined over a cyclotomic field. On ..."
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can be completely defined over a cyclotomic field. We show that this is not the case: in particular one of the fusion categories coming from the Haagerup subfactor [1] and one coming from the newly constructed extended Haagerup subfactor [2] can not be completely defined over a cyclotomic field. On the other hand, we show that the Drinfel’d center of the even part of the Haagerup subfactor is completely defined over a cyclotomic field. We identify the minimal field of definition for each of these fusion categories, compute the Galois groups, and identify their Galois conjugates. AMS Classification 18D10; 46L37
GENERALIZED AND QUASILOCALIZATIONS OF BRAID GROUP REPRESENTATIONS
"... We develop a theory of localization for braid group representations associated with objects in braided fusion categories and, more generally, to YangBaxter operators in monoidal categories. The essential problem is to determine when a family of braid representations can be uniformly modelled upon a ..."
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We develop a theory of localization for braid group representations associated with objects in braided fusion categories and, more generally, to YangBaxter operators in monoidal categories. The essential problem is to determine when a family of braid representations can be uniformly modelled upon a tensor power of a fixed vector space in such a way that the braid group generators act “locally”. Although related to the notion of (quasi)fiber functors for fusion categories, remarkably, such localizations can exist for representations associated with objects of nonintegral dimension. We conjecture that such localizations exist precisely when the object in question has dimension the squareroot of an integer and prove several key special cases of the conjecture.
Tensor categories: A selective guided tour
, 2008
"... These are the – only lightly edited – lecture notes for a short course on tensor categories. The coverage in these notes is relatively nontechnical, focussing on the essential ideas. They are meant to be accessible for beginners, but it is hoped that also some of the experts will find something int ..."
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These are the – only lightly edited – lecture notes for a short course on tensor categories. The coverage in these notes is relatively nontechnical, focussing on the essential ideas. They are meant to be accessible for beginners, but it is hoped that also some of the experts will find something interesting in them. Once the basic definitions are given, the focus is mainly on klinear categories with finite dimensional homspaces. Connections with quantum groups and low dimensional topology are pointed out, but these notes have no pretension to cover the latter subjects at any depth. Essentially, these notes should be considered as annotations to the extensive bibliography. We also recommend the recent review [33], which covers less ground in a deeper way.
LOCALIZATION OF UNITARY BRAID GROUP REPRESENTATIONS
"... Abstract. Governed by locality, we explore a connection between unitary braid group representations associated to a unitary Rmatrix and to a simple object in a unitary braided fusion category. Unitary Rmatrices, namely unitary solutions to the YangBaxter equation, afford explicitly local unitary ..."
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Abstract. Governed by locality, we explore a connection between unitary braid group representations associated to a unitary Rmatrix and to a simple object in a unitary braided fusion category. Unitary Rmatrices, namely unitary solutions to the YangBaxter equation, afford explicitly local unitary representations of braid groups. Inspired by topological quantum computation, we study whether or not it is possible to reassemble the irreducible summands appearing in the unitary braid group representations from a unitary braided fusion category with possibly different positive multiplicities to get representations that are uniformly equivalent to the ones from a unitary Rmatrix. Such an equivalence will be called a localization of the unitary braid group representations. We show that the q = e πi/6 specialization of the unitary Jones representation of the braid groups can be localized by a unitary 9 × 9 Rmatrix. Actually this Jones representation is the first one in a family of theories (SO(N), 2) for an odd prime N> 1, which are conjectured to be localizable. We formulate several general conjectures and discuss possible connections to physics and computer science. 1.
Identifying Topological Order by Entanglement Entropy,” arXiv:1205.4289
"... Topological phases are unique states of matter incorporating longrange quantum entanglement, hosting exotic excitations with fractional quantum statistics. We report a practical method to identify topological phases in arbitrary realistic models by accurately calculating the Topological Entanglem ..."
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Topological phases are unique states of matter incorporating longrange quantum entanglement, hosting exotic excitations with fractional quantum statistics. We report a practical method to identify topological phases in arbitrary realistic models by accurately calculating the Topological Entanglement Entropy (TEE) using the Density Matrix Renormalization Group (DMRG). We argue that the DMRG algorithm naturally produces a minimally entangled state, from amongst the quasidegenerate ground states in a topological phase. This proposal both explains the success of this method, and the absence of ground state degeneracy found in prior DMRG sightings of topological phases. We demonstrate the effectiveness of the calculational procedure by obtaining the TEE for several microscopic models, with an accuracy of order 10−3 when the circumference of the cylinder is around ten times the correlation length. As an example, we definitively show the ground state of the quantum S = 1/2 antiferromagnet on the kagome ́ lattice is a topological spin liquid, and strongly constrain the full identification of this phase of matter. 1 ar
ON THE CLASSIFICATION OF THE GROTHENDIECK RINGS OF NONSELFDUAL MODULAR CATEGORIES
"... Abstract. We develop a symbolic computational approach to classifying lowrank modular fusion categories, up to finite ambiguity. By a generalized form of Ocneanu rigidity due to Etingof, Ostrik and Nikshych, it is enough to classify modular fusion algebras of a given rank–that is, to determine the p ..."
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Abstract. We develop a symbolic computational approach to classifying lowrank modular fusion categories, up to finite ambiguity. By a generalized form of Ocneanu rigidity due to Etingof, Ostrik and Nikshych, it is enough to classify modular fusion algebras of a given rank–that is, to determine the possible Grothendieck rings with modular realizations. We use this technique to classify modular categories of rank at most 5 that are nonselfdual, i.e. those for which some object is not isomorphic to its dual object. 1.
BRAID REPRESENTATIONS FROM QUANTUM GROUPS OF EXCEPTIONAL LIE TYPE
"... Abstract. We study the problem of determining if the braid group representations obtained from quantum groups of types E, F and G at roots of unity have infinite image or not. In particular we show that when the fusion categories associated with these quantum groups are not weakly integral, the brai ..."
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Abstract. We study the problem of determining if the braid group representations obtained from quantum groups of types E, F and G at roots of unity have infinite image or not. In particular we show that when the fusion categories associated with these quantum groups are not weakly integral, the braid group images are infinite. This provides further evidence for a recent conjecture that weak integrality is necessary and sufficient for the braid group representations associated with any braided fusion category to have finite image. 1.