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On generalized symmetries and structure of modular categories

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Abstract

Pursuing a generalization of group symmetries of modular categories to category symmetries in topological phases of matter, we study linear Hopf monads. The main goal is a generalization of extension and gauging group symmetries to category symmetries of modular categories, which include also categorical Hopf algebras as special cases. As an application, we propose an analogue of the classification of finite simple groups to modular categories, where we define simple modular categories as the prime ones without any nontrivial normal algebras.

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References

  1. Aissaoui S, Makhlouf A. On classification of finite-dimensional superbialgebras and Hopf superalgebras. ArXiv:1301. 0838, 2013

    Google Scholar 

  2. Barkeshli M, Bonderson P, Cheng M, et al. Symmetry, defects, and gauging of topological phases. ArXiv:1410.4540, 2014

    Google Scholar 

  3. Barter D, Bridgeman J C, Jones C. Domain walls in topological phases and the Brauer-Picard ring for Vec(Z/pZ). ArXiv:1806.01279, 2018

    Google Scholar 

  4. Bischoff M, Kawahigashi Y, Longo R, et al. Tensor Categories and Endomorphisms of Von Neumann Algebras: With Applications to Quantum Field Theory, Volume 3. New York: Springer, 2015

    Book  MATH  Google Scholar 

  5. Böhm G, Lack S, Street R. Weak bimonads and weak Hopf monads. J Algebra, 2011, 328: 1–30

    Article  MathSciNet  MATH  Google Scholar 

  6. Bonderson P, Delaney C, Galindo C, et al. On invariants of modular categories beyond modular data. J Pure Appl Algebra, 2018, in press

    Google Scholar 

  7. Bruguieres A, Lack S, Virelizier A. Hopf monads on monoidal categories. Adv Math, 2011, 227: 745–800

    Article  MathSciNet  MATH  Google Scholar 

  8. Bruguieres A, Natale S. Exact sequences of tensor categories. Int Math Res Not IMRN, 2011, 24: 5644–5705

    MathSciNet  MATH  Google Scholar 

  9. Bruguieres A, Virelizier A. Hopf monads. Adv Math, 2007, 215: 679–733

    Article  MathSciNet  MATH  Google Scholar 

  10. Bruillard P, Ng S-H, Rowell E, et al. Rank-finiteness for modular categories. J Amer Math Soc, 2016, 29: 857–881

    Article  MathSciNet  MATH  Google Scholar 

  11. Cong I, Cheng M, Wang Z. Topological quantum computation with gapped boundaries. ArXiv:1609.02037, 2016

    Google Scholar 

  12. Cui S X, Galindo C, Plavnik J Y, et al. On gauging symmetry of modular categories. Comm Math Phys, 2016, 348: 1043–1064

    Article  MathSciNet  MATH  Google Scholar 

  13. Cui S X, Hong S-M, Wang Z. Universal quantum computation with weakly integral anyons. Quantum Inf Process, 2015, 14: 2687–2727

    Article  MathSciNet  MATH  Google Scholar 

  14. Delaney C, Tran A. A systematic search of knot and link invariants beyond modular data. ArXiv:1806.02843, 2018

    Google Scholar 

  15. Etingof P, Gelaki S, Nikshych D, et al. Tensor Categories, Volume 205. Providence: Amer Math Soc, 2016

    Google Scholar 

  16. Etingof P, Nikshych D, Ostrik V, et al. Fusion categories and homotopy theory. Quantum Topol, 2010, 1: 209–273

    Article  MathSciNet  MATH  Google Scholar 

  17. Evans D E, Gannon T. The exoticness and realisability of twisted Haagerup-Izumi modular data. Comm Math Phys, 2011, 307: 463

    Article  MathSciNet  MATH  Google Scholar 

  18. Evans D E, Gannon T. Near-group fusion categories and their doubles. Adv Math, 2014, 255: 586–640

    Article  MathSciNet  MATH  Google Scholar 

  19. Galindo C, Jaramillo N. Solutions of the hexagon equation for Abelian anyons. Rev Colombiana Mat, 2016, 50: 277–298

    Article  MathSciNet  MATH  Google Scholar 

  20. Grossman P, Snyder N. Quantum subgroups of the Haagerup fusion categories. Comm Math Phys, 2012, 311: 617–643

    Article  MathSciNet  MATH  Google Scholar 

  21. Izumi M. A cuntz algebra approach to the classification of near-group categories. In: Proceedings of the 2014 Maui and 2015 Qinhuangdao Conferences in Honour of Vaughan FR Jones’ 60th Birthday. Centre for Mathematics and Its Applications, Mathematical Sciences Institute. Canberra: The Australian National University, 2017, 222–343

    MATH  Google Scholar 

  22. Kirillov Jr A, Ostrik V. On a q-analogue of the McKay correspondence and the ade classification of sl2 conformal field theories. Adv Math, 2002, 171: 183–227

    Article  MathSciNet  MATH  Google Scholar 

  23. Larson H K. Pseudo-unitary non-self-dual fusion categories of rank 4. J Algebra, 2014, 415: 184–213

    Article  MathSciNet  MATH  Google Scholar 

  24. Mignard M, Schauenburg P. Modular categories are not determined by their modular data. ArXiv:1708.02796, 2017

    Google Scholar 

  25. Müger M. From subfactors to categories and topology II: The quantum double of tensor categories and subfactors. J Pure Appl Algebra, 2003, 180: 159–219

    Article  MathSciNet  MATH  Google Scholar 

  26. Müger M. On the structure of modular categories. Proc Lond Math Soc (3), 2003, 87: 291–308

    Article  MathSciNet  MATH  Google Scholar 

  27. Ostrik V. Module categories over the Drinfeld double of a finite group. Int Math Res Not IMRN, 2003, 2003: 1507–1520

    Article  MathSciNet  MATH  Google Scholar 

  28. Ostrik V. Module categories, weak Hopf algebras and modular invariants. Transform Groups, 2003, 8: 177–206

    Article  MathSciNet  MATH  Google Scholar 

  29. Siehler J. Near-group categories. Algebr Geom Topol, 2003, 3: 719–775

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The first author was supported by the Simons Foundation. The third author was sup- ported by National Science Foundation of USA (Grants Nos. DMS-1411212 and FRG-1664351).

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Authors and Affiliations

  1. Stanford Institute for Theoretical Physics, Stanford University, Stanford, CA, 94305, USA

    Shawn Xingshan Cui

  2. Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA, 24061, USA

    Shawn Xingshan Cui

  3. Department of Mathematics, University of California at Santa Barbara, Santa Barbara, CA, 93106, USA

    Modjtaba Shokrian Zini & Zhenghan Wang

  4. Microsoft Station Q, Santa Barbara, CA, 93106, USA

    Zhenghan Wang

Authors
  1. Shawn Xingshan Cui
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  2. Modjtaba Shokrian Zini
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  3. Zhenghan Wang
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Correspondence to Zhenghan Wang.

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Cui, S.X., Zini, M.S. & Wang, Z. On generalized symmetries and structure of modular categories. Sci. China Math. 62, 417–446 (2019). https://doi.org/10.1007/s11425-018-9455-5

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  • DOI: https://doi.org/10.1007/s11425-018-9455-5

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