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Computing fusion rules for spherical G-extensions of fusion categories

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Abstract

A G-extension of a fusion category \({\mathcal {C}}\) yields a categorical action of G on the center \(Z({{\mathcal {C}}})\). If the extension admits a spherical structure, we provide a method for recovering its fusion rules in terms of the action. We then apply this to find closed formulas for the fusion rules of extensions of some group theoretical categories and of cyclic permutation crossed extensions of modular categories.

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References

  1. Bantay, P.: Permutation orbifolds. Nucl. Phys. B 633(3), 365–378 (2002)

    Article  MathSciNet  Google Scholar 

  2. Bartlett, B.: Fusion categories via string diagrams. Commun. Contemp. Math. 18(5), 1550080–39 (2016)

    Article  MathSciNet  Google Scholar 

  3. Barkeshli, M., Bonderson, P., Cheng, M., Wang, Z.: Symmetry fractionalization, defects, and gauging of topological phases (2014) arXiv:1410.4540

  4. Bischoff, M., Davydov, A.: Hopf algebra actions in tensor categories. Transform. Groups 26, 69–90 (2021)

  5. Bartlett, B., Douglas, C.L, Schommer-Pries, C.J., Vicary, J.: Modular categories as representations of the 3-dimensional bordism 2-category (2015). arXiv:1509.06811

  6. Barmeier, T., Fuchs, J., Runkel, I., Schweigert, C.: Module categories for permutation modular invariants. Int. Math. Res. Not. IMRN 16, 3067–3100 (2010)

    MathSciNet  MATH  Google Scholar 

  7. Borisov, L., Halpern, M.B., Schweigert, C.: Systematic approach to cyclic orbifolds. Int. J. Mod. Phys. A 13(1), 125–168 (1998)

    Article  MathSciNet  Google Scholar 

  8. Bischoff, M.: Generalized orbifold construction for conformal nets. Rev. Math. Phys. 29(1), 1750002–53 (2017)

    Article  MathSciNet  Google Scholar 

  9. Barkeshli, M., Jian, C.-M., Qi, X.-L.: Twist defects and projective non-abelian braiding statistics. Phys. Rev. B 87, 045130 (2013)

    Article  Google Scholar 

  10. Bakalov, B., Kirillov, A., Jr.: Lectures on Tensor Categories and Modular Functors. University Lecture Series, p. 21. American Mathematical Society, Providence (2001)

    MATH  Google Scholar 

  11. Bischoff, M., Kawahigashi, Y., Longo, R., Rehren, K.-H.: Tensor Categories and Endomorphisms of von Neumann Algebras—With Applications to Quantum Field Theory: Springer Briefs in Mathematical Physics, vol. 3. Springer, Berlin (2015)

    MATH  Google Scholar 

  12. Burciu, S., Natale, S.: Fusion rules of equivariantizations of fusion categories. J. Math. Phys. 54(1), 013511–21 (2013)

    Article  MathSciNet  Google Scholar 

  13. Barmeier, T., Schweigert, C.: A geometric construction for permutation equivariant categories from modular functors. Transform. Groups 16(2), 287–337 (2011)

    Article  MathSciNet  Google Scholar 

  14. Barrett, J.W., Westbury, B.W.: Spherical categories. Adv. Math. 143(2), 357–375 (1999)

    Article  MathSciNet  Google Scholar 

  15. Delaney, C.: Fusion rules for permutation extensions of modular tensor categories. Preprint (2019). arXiv:1909.03003

  16. Davydov, A., Nikshych, D.: The Picard crossed module of a braided tensor category. Algebra Number Theory 7(6), 1365–1403 (2013)

    Article  MathSciNet  Google Scholar 

  17. Day, B., Pastro, C.: Note on Frobenius monoidal functors. N. Y. J. Math. 14, 733–742 (2008)

    MathSciNet  MATH  Google Scholar 

  18. Davydov, A., Simmons, D.. A.: Third cohomology and fusion categories. Homol. Homotopy Appl. 20(1), 275–302 (2018)

    Article  MathSciNet  Google Scholar 

  19. Etingof, P., Galindo, C.: Reflection fusion categories. J. Algebra 516, 172–196 (2018)

    Article  MathSciNet  Google Scholar 

  20. Etingof, P., Gelaki, S., Nikshych, D., Ostrik, V.: Tensor Categories, Mathematical Surveys and Monographs, vol. 205. American Mathematical Society, Providence (2015)

    Book  Google Scholar 

  21. Edie-Michell, C., Jones, C., Plavnik, J., Fusion rules for \({\mathbb{Z}}/ 2{\mathbb{Z}}\) permutation gauging (2018). arXiv:1804.01657

  22. Etingof, P., Nikshych, D., Ostrik, V.: On fusion categories. Ann. Math. (2) 162(2), 581–642 (2005)

    Article  MathSciNet  Google Scholar 

  23. Etingof, P., Nikshych, D., Ostrik, V.: Fusion categories and homotopy theory. Quantum Topol. 1(3), 209–273 (2010). (With an appendix by Ehud Meir)

    Article  MathSciNet  Google Scholar 

  24. Fuchs, J., Runkel, I., Schweigert, C.: TFT construction of RCFT correlators I. Partition functions. Nucl. Phys. B 646(3), 353–497 (2002)

    Article  MathSciNet  Google Scholar 

  25. Gannon, T., Jones, C.: Vanishing of categorical obstructions for permutation orbifolds. Commun. Math. Phys. 369(1), 245–259 (2019)

    Article  MathSciNet  Google Scholar 

  26. Gelaki, S., Naidu, D., Nikshych, D.: Centers of graded fusion categories. Algebra Number Theory 3(8), 959–990 (2009)

    Article  MathSciNet  Google Scholar 

  27. Huang, Y..-Z., Lepowsky, J.: Tensor categories and the mathematics of rational and logarithmic conformal field theory. J. Phys. A 46(49), 494009–21 (2013)

    Article  MathSciNet  Google Scholar 

  28. Jones, C., Morrison, S., Penneys, D., Plavnik, J.: Extension Theory for Braided-Enriched Fusion Categories, 202107, International Mathematics Research Notices. https://academic.oup.com/imrn/advance-article-pdf/doi/10.1093/imrn/rnab133/38851094/rnab133.pdf. https://doi.org/10.1093/imrn/rnab133,

  29. Kelly, G.M.: Doctrinal Adjunction, Category Seminar (Proceedings of Seminar in Sydney, 1972/1973). Lecture Notes in Mathematics, vol. 420, pp. 257– 280 (1974)

  30. Kirillov, A., Jr.: Modular categories and orbifold models. Commun. Math. Phys. 229(2), 309–335 (2002)

    Article  MathSciNet  Google Scholar 

  31. Kitaev, A.: Anyons in an exactly solved model and beyond. Ann. Phys. 321(1), 2–111 (2006)

    Article  MathSciNet  Google Scholar 

  32. Kirillov Jr, A., Balsam, B., Turaev–Viro invariants as an extended tqft (2010). arXiv:1004.1533

  33. Kawahigashi, Y., Longo, R.: Subfactors and modularity of representations in conformal field theory. Commun. Math. Phys. 219, 631–669 (2001)

    Article  MathSciNet  Google Scholar 

  34. Kac, V.G., Longo, R., Xu, F.: Solitons in affine and permutation orbifolds. Commun. Math. Phys. 253(3), 723–764 (2005)

    Article  MathSciNet  Google Scholar 

  35. Kong, L., Runkel, I.: Morita classes of algebras in modular tensor categories. Adv. Math. 219(5), 1548–1576 (2008)

    Article  MathSciNet  Google Scholar 

  36. Liu, G., Ng, S.-H.: On Total Frobenius–Schur Indicators. Recent Advances in Representation Theory, Quantum Groups, Algebraic Geometry, and Related Topics, vol. 623, pp. 193–213 (2014)

  37. Longo, R., Xu, F.: Topological sectors and a dichotomy in conformal field theory. Commun. Math. Phys. 251(2), 321–364 (2004)

    Article  MathSciNet  Google Scholar 

  38. Müger, M.: On the structure of modular categories, 2002-01. Mathematics e-prints. arXiv:math/0201017

  39. Müger, M.: From subfactors to categories and topology. I. Frobenius algebras in and Morita equivalence of tensor categories. J. Pure Appl. Algebra 180(1–2), 81–157 (2003)

    Article  MathSciNet  Google Scholar 

  40. Müger, M.: From subfactors to categories and topology. II. The quantum double of tensor categories and subfactors,. J. Pure Appl. Algebra 180(1–2), 159–219 (2003)

    Article  MathSciNet  Google Scholar 

  41. Müger, M.: Theories, conformal orbifold, crossed, braided, G-categories. Commun. Math. Phys. 260, 727–762 (2005)

    Article  Google Scholar 

  42. Mason, G., Ng, S.-H.: Group cohomology and gauge equivalence of some twisted quantum doubles. Trans. Am. Math. Soc. 353(9), 3465–3509 (2001). ((electronic))

    Article  MathSciNet  Google Scholar 

  43. Moore, G., Seiberg, N.: Classical and quantum conformal field theory. Commun. Math. Phys. 123(2), 177–254 (1989)

    Article  MathSciNet  Google Scholar 

  44. Moore, G., Seiberg, N., Lectures on RCFT, Superstrings ’89 (Trieste, 1989), pp. 1–129. World Sci. Publ, River Edge (1990)

  45. Nayak, C., Simon, S.H., Stern, A., Freedman, M., Das Sarma, S.: Non-abelian anyons and topological quantum computation. Rev. Mod. Phys. 80(3), 1083–1159 (2008)

    Article  MathSciNet  Google Scholar 

  46. Passegger, A.G.: Permutation actions on modular tensor categories of topological multilayer phases (2018). arXiv:1808.06574

  47. Selinger, P.: A survey of graphical languages for monoidal categories. In: Coecke, B. (ed.) New Structures for Physics, vol. 813, pp. 289–355. Springer, Berlin, Heidelberg (2011)

  48. Turaev, V.: Homotopy quantum field theory, EMS Tracts in Mathematics, European Mathematical Society (EMS), Zürich, Appendix 5 by Michael Müger and Appendices 6 and 7 by Alexis Virelizier, vol. 10 (2010)

  49. Turaev, V.G.: Quantum Invariants of Knots and 3-Manifolds. Walter de Gruyter, Berlin (1994)

    Book  Google Scholar 

  50. Wang, Z.: Topological quantum computation, CBMS Regional Conference Series in Mathematics, Published for the Conference Board of the Mathematical Sciences, vol. 112. Washington, DC; by the American Mathematical Society, Providence, RI (2010)

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Acknowledgements

The authors would like to thank Colleen Delaney, Cain Edie-Michell, Dave Penneys and Julia Plavnik for very useful discussions and comments on an early draft. We also thank Colleen Delaney for sharing an early draft of [15] with us and for coordinating arXiv submissions. Marcel Bischoff was supported by NSF grant DMS-1700192/1821162. Corey Jones was supported by NSF Grant DMS-1901082.

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  1. Department of Mathematics, Ohio University, Athens, OH, 45701, USA

    Marcel Bischoff

  2. Department of Mathematics, North Carolina State University, Raleigh, NC, 27695, USA

    Corey Jones

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Appendix A: Fusion rules for \(\mathbb {Z}/4\mathbb {Z}\) permutation extensions of Fib

Appendix A: Fusion rules for \(\mathbb {Z}/4\mathbb {Z}\) permutation extensions of Fib

Let \({\mathcal {C}}\) be the Fibonacci category with \({{\,\mathrm{Irr}\,}}({\mathcal {C}})=\{{\mathbb {1}},\tau \}\) with \(\tau \otimes \tau \cong {\mathbb {1}}\oplus \tau \) and \({\mathcal {D}}={\mathcal {C}}\wr {\mathbb {Z}}/4{\mathbb {Z}}\). Then \({{\,\mathrm{Irr}\,}}({\mathcal {D}}_i)=\{(i,{\mathbb {1}}),(i,\tau )\}\) for \(i=1,3\) and \({{\,\mathrm{Irr}\,}}({\mathcal {D}}_2)=\{(2,{\mathbb {11}}),(2,{\mathbb {1}}\tau ),(2,\tau {\mathbb {1}}),(2,\tau \tau )\}\) and

where we write “\(\cdots \)” for obvious permutation of objects (Table 1).

Table 1 The fusion coefficients \({}^gN_{\tau ^n}\) are given by the Fibonacci and Lucas numbers for \(g=0,1\), respectively
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Bischoff, M., Jones, C. Computing fusion rules for spherical G-extensions of fusion categories. Sel. Math. New Ser. 28, 26 (2022). https://doi.org/10.1007/s00029-021-00725-3

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