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Journal articles on the topic 'Modular tensor categories'

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1

HONG, SEUNG-MOON, ERIC ROWELL, and ZHENGHAN WANG. "ON EXOTIC MODULAR TENSOR CATEGORIES." Communications in Contemporary Mathematics 10, supp01 (2008): 1049–74. http://dx.doi.org/10.1142/s0219199708003162.

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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
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2

Lyubashenko, V. "Modular transformations for tensor categories." Journal of Pure and Applied Algebra 98, no. 3 (1995): 279–327. http://dx.doi.org/10.1016/0022-4049(94)00045-k.

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3

Rowell, Eric, Richard Stong, and Zhenghan Wang. "On Classification of Modular Tensor Categories." Communications in Mathematical Physics 292, no. 2 (2009): 343–89. http://dx.doi.org/10.1007/s00220-009-0908-z.

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4

Liu, Zhengwei, and Feng Xu. "Jones-Wassermann subfactors for modular tensor categories." Advances in Mathematics 355 (October 2019): 106775. http://dx.doi.org/10.1016/j.aim.2019.106775.

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5

Giorgetti, Luca, and Karl-Henning Rehren. "Bantay's trace in unitary modular tensor categories." Advances in Mathematics 319 (October 2017): 211–23. http://dx.doi.org/10.1016/j.aim.2017.08.018.

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6

Creutzig, Thomas, and Terry Gannon. "Logarithmic conformal field theory, log-modular tensor categories and modular forms." Journal of Physics A: Mathematical and Theoretical 50, no. 40 (2017): 404004. http://dx.doi.org/10.1088/1751-8121/aa8538.

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7

Kirillov, Alexander A. "On an inner product in modular tensor categories." Journal of the American Mathematical Society 9, no. 4 (1996): 1135–69. http://dx.doi.org/10.1090/s0894-0347-96-00210-x.

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8

Edie-Michell, Cain. "Simple current auto-equivalences of modular tensor categories." Proceedings of the American Mathematical Society 148, no. 4 (2019): 1415–28. http://dx.doi.org/10.1090/proc/14795.

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9

Kong, Liang, and Ingo Runkel. "Morita classes of algebras in modular tensor categories." Advances in Mathematics 219, no. 5 (2008): 1548–76. http://dx.doi.org/10.1016/j.aim.2008.07.004.

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10

Al-Shomrani, M. M., and E. J. Beggs. "Making nontrivially associated modular categories from finite groups." International Journal of Mathematics and Mathematical Sciences 2004, no. 42 (2004): 2231–64. http://dx.doi.org/10.1155/s0161171204308203.

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We show that the double𝒟of the nontrivially associated tensor category constructed from left coset representatives of a subgroup of a finite groupXis a modular category. Also we give a definition of the character of an object in this category as an element of a braided Hopf algebra in the category. This definition is shown to be adjoint invariant and multiplicative on tensor products. A detailed example is given. Finally, we show an equivalence of categories between the nontrivially associated double𝒟and the trivially associated category of representations of the Drinfeld double of the groupD(
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11

TURAEV, VLADIMIR G. "MODULAR CATEGORIES AND 3-MANIFOLD INVARIANTS." International Journal of Modern Physics B 06, no. 11n12 (1992): 1807–24. http://dx.doi.org/10.1142/s0217979292000876.

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The aim of this paper is to give a concise introduction to the theory of knot invariants and 3-manifold invariants which generalize the Jones polynomial and which may be considered as a mathematical version of the Witten invariants. Such a theory was introduced by N. Reshetikhin and the author on the ground of the theory of quantum groups. Here we use more general algebraic objects, specifically, ribbon and modular categories. Such categories in particular arise as the categories of representations of quantum groups. The notion of modular category, interesting in itself, is closely related to
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12

Huang, Y. Z. "Vertex operator algebras, the Verlinde conjecture, and modular tensor categories." Proceedings of the National Academy of Sciences 102, no. 15 (2005): 5352–56. http://dx.doi.org/10.1073/pnas.0409901102.

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13

Müger, Michael. "Galois Theory for Braided Tensor Categories and the Modular Closure." Advances in Mathematics 150, no. 2 (2000): 151–201. http://dx.doi.org/10.1006/aima.1999.1860.

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14

Pfeiffer, Hendryk. "Tannaka–Kreıˇn reconstruction and a characterization of modular tensor categories". Journal of Algebra 321, № 12 (2009): 3714–63. http://dx.doi.org/10.1016/j.jalgebra.2009.02.026.

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15

HUANG, YI-ZHI. "RIGIDITY AND MODULARITY OF VERTEX TENSOR CATEGORIES." Communications in Contemporary Mathematics 10, supp01 (2008): 871–911. http://dx.doi.org/10.1142/s0219199708003083.

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Let V be a simple vertex operator algebra satisfying the following conditions: (i) V(n)= 0 for n < 0, V(0)= ℂ1 and V′ is isomorphic to V as a V-module. (ii) Every ℕ-gradable weak V-module is completely reducible. (iii) V is C2-cofinite. (In the presence of Condition (i), Conditions (ii) and (iii) are equivalent to a single condition, namely, that every weak V-module is completely reducible.) Using the results obtained by the author in the formulation and proof of the general version of the Verlinde conjecture and in the proof of the Verlinde formula, we prove that the braided tensor categor
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16

Schopieray, Andrew. "Prime decomposition of modular tensor categories of local modules of type D." Quantum Topology 11, no. 3 (2020): 489–524. http://dx.doi.org/10.4171/qt/140.

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17

Masdeu, Marc, and Marco Adamo Seveso. "Dirac operators in tensor categories and the motive of quaternionic modular forms." Advances in Mathematics 313 (June 2017): 628–88. http://dx.doi.org/10.1016/j.aim.2017.03.034.

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18

Gui, Bin. "Unitarity of the Modular Tensor Categories Associated to Unitary Vertex Operator Algebras, II." Communications in Mathematical Physics 372, no. 3 (2019): 893–950. http://dx.doi.org/10.1007/s00220-019-03534-0.

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19

Gui, Bin. "Unitarity of the Modular Tensor Categories Associated to Unitary Vertex Operator Algebras, I." Communications in Mathematical Physics 366, no. 1 (2019): 333–96. http://dx.doi.org/10.1007/s00220-019-03326-6.

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20

Gainutdinov, Azat M., and Ingo Runkel. "Projective objects and the modified trace in factorisable finite tensor categories." Compositio Mathematica 156, no. 4 (2020): 770–821. http://dx.doi.org/10.1112/s0010437x20007034.

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For ${\mathcal{C}}$ a factorisable and pivotal finite tensor category over an algebraically closed field of characteristic zero we show:(1)${\mathcal{C}}$ always contains a simple projective object;(2)if ${\mathcal{C}}$ is in addition ribbon, the internal characters of projective modules span a submodule for the projective $\text{SL}(2,\mathbb{Z})$-action;(3)the action of the Grothendieck ring of ${\mathcal{C}}$ on the span of internal characters of projective objects can be diagonalised;(4)the linearised Grothendieck ring of ${\mathcal{C}}$ is semisimple if and only if ${\mathcal{C}}$ is semi
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21

Schopieray, Andrew. "Level bounds for exceptional quantum subgroups in rank two." International Journal of Mathematics 29, no. 05 (2018): 1850034. http://dx.doi.org/10.1142/s0129167x18500349.

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There is a long-standing belief that the modular tensor categories [Formula: see text], for [Formula: see text] and finite-dimensional simple complex Lie algebras [Formula: see text], contain exceptional connected étale algebras (sometimes called quantum subgroups) at only finitely many levels [Formula: see text]. This premise has known implications for the study of relations in the Witt group of nondegenerate braided fusion categories, modular invariants of conformal field theories, and the classification of subfactors in the theory of von Neumann algebras. Here, we confirm this conjecture wh
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22

Harvey, Jeffrey A., Yichen Hu, and Yuxiao Wu. "Galois symmetry induced by Hecke relations in rational conformal field theory and associated modular tensor categories." Journal of Physics A: Mathematical and Theoretical 53, no. 33 (2020): 334003. http://dx.doi.org/10.1088/1751-8121/ab8e03.

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23

Lepowsky, J. "From the representation theory of vertex operator algebras to modular tensor categories in conformal field theory." Proceedings of the National Academy of Sciences 102, no. 15 (2005): 5304–5. http://dx.doi.org/10.1073/pnas.0501135102.

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24

Blass, Andreas, and Yuri Gurevich. "Witness algebra and anyon braiding." Mathematical Structures in Computer Science 30, no. 3 (2020): 234–70. http://dx.doi.org/10.1017/s0960129520000055.

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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.
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25

Kirillov, Alexander A. "On inner product in modular tensor categories II: Inner product on conformal blocks and affine inner product identities." Advances in Theoretical and Mathematical Physics 2, no. 1 (1998): 155–80. http://dx.doi.org/10.4310/atmp.1998.v2.n1.a6.

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26

Wang, Zhenghan. "Beyond anyons." Modern Physics Letters A 33, no. 28 (2018): 1830011. http://dx.doi.org/10.1142/s0217732318300112.

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The theory of anyon systems, as modular functors topologically and unitary modular tensor categories algebraically, is mature. To go beyond anyons, our first step is the interplay of anyons with conventional group symmetry due to the paramount importance of group symmetry in physics. This led to the theory of symmetry-enriched topological order. Another direction is the boundary physics of topological phases, both gapless as in the fractional quantum Hall physics and gapped as in the toric code. A more speculative and interesting direction is the study of Banados–Teitelboim–Zanelli (BTZ) black
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27

Lochak, Pierre. "LECTURES ON TENSOR CATEGORIES AND MODULAR FUNCTORS (University Lecture Series 21) By BOJKO BAKALOV and ALEXANDER KIRILLOV, JR: 221 pp., US$29.00, ISBN 0-8218-2686-7 (American Mathematical Society, Providence, RI, 2001)." Bulletin of the London Mathematical Society 34, no. 3 (2002): 374–84. http://dx.doi.org/10.1112/s0024609302211145.

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28

Dong, Chongying, Siu-Hung Ng, and Li Ren. "Vertex operator superalgebras and the 16-fold way." Transactions of the American Mathematical Society, July 19, 2021. http://dx.doi.org/10.1090/tran/8454.

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Let V V be a vertex operator superalgebra with the natural order 2 automorphism σ \sigma . Under suitable conditions on V V , the σ \sigma -fixed subspace V 0 ¯ V_{\bar 0} is a vertex operator algebra and the V 0 ¯ V_{\bar 0} -module category C V 0 ¯ \mathcal {C}_{V_{\bar 0}} is a modular tensor category. In this paper, we prove that C V 0 ¯ \mathcal {C}_{V_{\bar 0}} is a fermionic modular tensor category and the Müger centralizer C V 0 ¯ 0 \mathcal {C}_{V_{\bar 0}}^0 of the fermion in C V 0 ¯ \mathcal {C}_{V_{\bar 0}} is generated by the irreducible V 0 ¯ V_{\bar 0} -submodules of the V V -mo
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29

Nandakumar, Vinoth, and Gufang Zhao. "Categorification via blocks of modular representations for." Canadian Journal of Mathematics, May 19, 2020, 1–29. http://dx.doi.org/10.4153/s0008414x20000346.

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Abstract Bernstein, Frenkel, and Khovanov have constructed a categorification of tensor products of the standard representation of $\mathfrak {sl}_2$ , where they use singular blocks of category $\mathcal {O}$ for $\mathfrak {sl}_n$ and translation functors. Here we construct a positive characteristic analogue using blocks of representations of $\mathfrak {s}\mathfrak {l}_n$ over a field $\mathbf {k}$ of characteristic p with zero Frobenius character, and singular Harish-Chandra character. We show that the aforementioned categorification admits a Koszul graded lift, which is equivalent to a ge
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30

Etingof, Pavel, and Victor Ostrik. "On the Frobenius functor for symmetric tensor categories in positive characteristic." Journal für die reine und angewandte Mathematik (Crelles Journal), October 8, 2020. http://dx.doi.org/10.1515/crelle-2020-0033.

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AbstractWe develop a theory of Frobenius functors for symmetric tensor categories (STC) {\mathcal{C}} over a field {\boldsymbol{k}} of characteristic p, and give its applications to classification of such categories. Namely, we define a twisted-linear symmetric monoidal functor {F:\mathcal{C}\to\mathcal{C}\boxtimes{\rm Ver}_{p}}, where {{\rm Ver}_{p}} is the Verlinde category (the semisimplification of {\mathop{\mathrm{Rep}}\nolimits_{\mathbf{k}}(\mathbb{Z}/p)}); a similar construction of the underlying additive functor appeared independently in [K. Coulembier, Tannakian categories in positive
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