Abstract
Arbitrarily many pairwise inequivalent modular categories can share the same modular data. We exhibit a family of examples that are module categories over twisted Drinfeld doubles of finite groups, and thus in particular integral modular categories.
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Bakalov, B., Kirillov, Jr. A.: Lectures on tensor categories and modular functors, volume 21 of University Lecture Series. American Mathematical Society, Providence, RI (2001)
Bruillard, P., Ng, S.-H., Rowell, E.C., Wang, Z.: Rank-finiteness for modular categories. J. Am. Math. Soc. 29(3), 857–881 (2016)
Brown, K.S.: Cohomology of Groups. Graduate Texts in Mathematics, vol. 87. Springer, New York (1982)
Cartan, H., Eilenberg, S.: Homological Algebra. Princeton University Press, Princeton (1956)
Coste, A., Gannon, T.: Remarks on Galois symmetry in rational conformal field theories. Phys. Lett. B 323(3–4), 316–321 (1994)
Coste, A., Gannon, T., Ruelle, P.: Finite group modular data. Nuclear Phys. B 581(3), 679–717 (2000)
de Boer, J., Goeree, J.: Markov traces and \({\rm II}_1\) factors in conformal field theory. Commun. Math. Phys. 139(2), 267–304 (1991)
Davidovich, O., Hagge, T., Wang, Z.: On Arithmetic Modular Categories. ArXiv e-prints (May 2013)
Dong, C., Lin, X., Ng, S.-H.: Congruence property in conformal field theory. Algebra Number Theory 9(9), 2121–2166 (2015)
Dijkgraaf, R., Pasquier, V., Roche, P.: Quasi Hopf algebras, group cohomology and orbifold models. Nuclear Phys. B Proc. Suppl., 18B:60–72 (1991), 1990. Recent advances in field theory (Annecy-le-Vieux, 1990)
Etingof, P., Gelaki, S., Nikshych, D., Ostrik, V.: Tensor Categories. Mathematical Surveys and Monographs, vol. 205. American Mathematical Society, Providence, RI (2015)
Etingof, P., Nikshych, D., Ostrik, V.: On fusion categories. Ann. Math. (2) 162(2), 581–642 (2005)
Goff, C., Mason, G., Ng, S.-H.: On the gauge equivalence of twisted quantum doubles of elementary abelian and extra-special 2-groups. J. Algebra 312(2), 849–875 (2007)
Mac Lane, S.: Homology. Classics in Mathematics. Springer, Berlin (1995). (reprint of the 1975 edition)
Moore, G., Seiberg, N.: Classical and quantum conformal field theory. Commun. Math. Phys. 123(2), 177–254 (1989)
Müger, M.: On the structure of modular categories. Proc. Lond. Math. Soc. (3) 87(2), 291–308 (2003)
Naidu, D.: Categorical Morita equivalence for group-theoretical categories. Commun. Algebra 35(11), 3544–3565 (2007)
Nikshych, D.: Morita equivalence methods in classification of fusion categories. In: Hopf algebras and tensor categories, volume 585 of Contemp. Math., pp. 289–325. Amer. Math. Soc., Providence, RI (2013)
Ostrik, V.: Module categories, weak Hopf algebras and modular invariants. Transform. Groups 8(2), 177–206 (2003)
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Mignard, M., Schauenburg, P. Modular categories are not determined by their modular data. Lett Math Phys 111, 60 (2021). https://doi.org/10.1007/s11005-021-01395-0
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DOI: https://doi.org/10.1007/s11005-021-01395-0