Orbifolds and minimal modular extensions.(English)Zbl 07972366
Summary: Let \(V\) be a simple vertex operator algebra and \(G\) a finite automorphism group of \(V\) such that \(V^G\) is regular, and the conformal weight of any irreducible \(g\)-twisted \(V\)-module \(N\) for \(g \in G\) is nonnegative and is zero if and only if \(N = V\). It is established that if \(V\) is holomorphic, then the \(V^G\)-module category \(\mathcal{C}_{V^G}\) is a minimal modular extension of \(\mathcal{E} = \operatorname{Rep}(G)\), and is equivalent to the Drinfeld center \(\mathcal{Z}( \operatorname{Vec}_G^\alpha)\) as modular tensor categories for some \(\alpha \in H^3(G, S^1)\) with a canonical embedding of \(\mathcal{E} \). Moreover, the collection \(\mathcal{M}_v(\mathcal{E})\) of equivalence classes of the minimal modular extensions \(\mathcal{C}_{V^G}\) of \(\mathcal{E}\) for holomorphic vertex operator algebras \(V\) with a \(G\)-action forms a group, which is isomorphic to a subgroup of \(H^3(G, S^1)\). Furthermore, any pointed modular category \(\mathcal{Z}( \operatorname{Vec}_G^\alpha)\) is equivalent to \(\mathcal{C}_{V_L^G}\) for some positive definite even unimodular lattice \(L\). In general, for any rational vertex operator algebra \(U\) with a \(G\)-action, \( \mathcal{C}_{U^G}\) is a minimal modular extension of the braided fusion subcategory \(\mathcal{F}\) generated by the \(U^G\)-submodules of \(U\)-modules. Furthermore, the group \(\mathcal{M}_v(\mathcal{E})\) acts freely on the set of equivalence classes \(\mathcal{M}_v(\mathcal{F})\) of the minimal modular extensions \(\mathcal{C}_{W^G}\) of \(\mathcal{F}\) for any rational vertex operator algebra \(W\) with a \(G\)-action.
MSC:
| 17Bxx | Lie algebras and Lie superalgebras |
| 18Dxx | Categorical structures |
| 81Rxx | Groups and algebras in quantum theory |
Cite
References:
| [1] | Abe, T.; Buhl, G.; Dong, C., Rationality, regularity, and \(C_2\)-cofiniteness, Trans. Am. Math. Soc., 356, 3391-3402, 2004; Abe, T.; Buhl, G.; Dong, C., Rationality, regularity, and \(C_2\)-cofiniteness, Trans. Am. Math. Soc., 356, 3391-3402, 2004 ·Zbl 1070.17011 |
| [2] | Abe, T.; Dong, C.; Li, H., Fusion rules for the vertex operator algebra \(M(1)\) and \(V_L^+\), Commun. Math. Phys., 253, 171-219, 2005; Abe, T.; Dong, C.; Li, H., Fusion rules for the vertex operator algebra \(M(1)\) and \(V_L^+\), Commun. Math. Phys., 253, 171-219, 2005 ·Zbl 1207.17032 |
| [3] | Ai, C.; Dong, C.; Jiao, X.; Ren, L., The irreducible modules and fusion rules for the parafermion vertex operator algebras, Trans. Am. Math. Soc., 370, 5963-5981, 2018; Ai, C.; Dong, C.; Jiao, X.; Ren, L., The irreducible modules and fusion rules for the parafermion vertex operator algebras, Trans. Am. Math. Soc., 370, 5963-5981, 2018 ·Zbl 1433.17034 |
| [4] | Borcherds, R. E., Vertex algebras, Kac-Moody algebras, and the Monster, Proc. Natl. Acad. Sci. USA, 83, 3068-3071, 1986; Borcherds, R. E., Vertex algebras, Kac-Moody algebras, and the Monster, Proc. Natl. Acad. Sci. USA, 83, 3068-3071, 1986 ·Zbl 0613.17012 |
| [5] | Bruillard, P.; Ng, S. H.; Rowell, E.; Wang, Z., Rank-finiteness for modular categories, J. Am. Math. Soc., 29, 857-881, 2016; Bruillard, P.; Ng, S. H.; Rowell, E.; Wang, Z., Rank-finiteness for modular categories, J. Am. Math. Soc., 29, 857-881, 2016 ·Zbl 1344.18008 |
| [6] | Carnahan, S.; Miyamoto, M., Regularity of fixed-point vertex operator subalgebras; Carnahan, S.; Miyamoto, M., Regularity of fixed-point vertex operator subalgebras |
| [7] | Creutzig, T.; Kanade, S.; McRae, R., Tensor categories for vertex operator superalgebra extensions; Creutzig, T.; Kanade, S.; McRae, R., Tensor categories for vertex operator superalgebra extensions ·Zbl 1540.17001 |
| [8] | Davydov, A.; Müger, M.; Nikshych, D.; Ostrik, V., The Witt group of nondegenerate braided fusion categories, J. Reine Angew. Math., 677, 135-177, 2013; Davydov, A.; Müger, M.; Nikshych, D.; Ostrik, V., The Witt group of nondegenerate braided fusion categories, J. Reine Angew. Math., 677, 135-177, 2013 ·Zbl 1271.18008 |
| [9] | Davydov, A.; Nikshych, D.; Ostrik, V., On the structure of the Witt group of braided fusion categories, Sel. Math. New Ser., 19, 2013, 237-269, 2013; Davydov, A.; Nikshych, D.; Ostrik, V., On the structure of the Witt group of braided fusion categories, Sel. Math. New Ser., 19, 2013, 237-269, 2013 ·Zbl 1345.18005 |
| [10] | Deligne, P., Catégories tannakiennes, (The Grothendieck Festschrift, Vol. II. The Grothendieck Festschrift, Vol. II, Progr. Math., vol. 87, 1990, Birkhäuser Boston: Birkhäuser Boston Boston, MA), 111-195; Deligne, P., Catégories tannakiennes, (The Grothendieck Festschrift, Vol. II. The Grothendieck Festschrift, Vol. II, Progr. Math., vol. 87, 1990, Birkhäuser Boston: Birkhäuser Boston Boston, MA), 111-195 ·Zbl 0727.14010 |
| [11] | Dijkgraaf, R.; Pasquier, V.; Roche, P., Quasi Hopf algebras, group cohomology and orbifold models, Nucl. Phys. B, Proc. Suppl., 18B, 60-72, 1990; Dijkgraaf, R.; Pasquier, V.; Roche, P., Quasi Hopf algebras, group cohomology and orbifold models, Nucl. Phys. B, Proc. Suppl., 18B, 60-72, 1990 ·Zbl 0957.81670 |
| [12] | Dong, C., Vertex algebras associated with even lattices, J. Algebra, 161, 245-265, 1993; Dong, C., Vertex algebras associated with even lattices, J. Algebra, 161, 245-265, 1993 ·Zbl 0807.17022 |
| [13] | Dong, C.; Jiao, X.; Xu, F., Quantum dimensions and quantum Galois theory, Trans. Am. Math. Soc., 365, 6441-6469, 2013; Dong, C.; Jiao, X.; Xu, F., Quantum dimensions and quantum Galois theory, Trans. Am. Math. Soc., 365, 6441-6469, 2013 ·Zbl 1337.17018 |
| [14] | Dong, C.; Lepowsky, J., Generalized Vertex Algebras and Relative Vertex Operators, Progress in Math, vol. 112, 1993, Birkhäuser: Birkhäuser Boston; Dong, C.; Lepowsky, J., Generalized Vertex Algebras and Relative Vertex Operators, Progress in Math, vol. 112, 1993, Birkhäuser: Birkhäuser Boston ·Zbl 0803.17009 |
| [15] | Dong, C.; Lepowsky, J., The algebraic structure of relative twisted vertex operators, J. Pure Appl. Algebra, 110, 259-295, 1996; Dong, C.; Lepowsky, J., The algebraic structure of relative twisted vertex operators, J. Pure Appl. Algebra, 110, 259-295, 1996 ·Zbl 0862.17021 |
| [16] | Dong, C.; Li, H.; Mason, G., Compact automorphism groups of vertex operator algebras, Int. Math. Res. Not., 18, 913-921, 1996; Dong, C.; Li, H.; Mason, G., Compact automorphism groups of vertex operator algebras, Int. Math. Res. Not., 18, 913-921, 1996 ·Zbl 0873.17028 |
| [17] | Dong, C.; Li, H.; Mason, G., Regularity of rational vertex operator algebras, Adv. Math., 132, 148-166, 1997; Dong, C.; Li, H.; Mason, G., Regularity of rational vertex operator algebras, Adv. Math., 132, 148-166, 1997 ·Zbl 0902.17014 |
| [18] | Dong, C.; Li, H.; Mason, G., Twisted representations of vertex operator algebras, Math. Ann., 310, 571-600, 1998; Dong, C.; Li, H.; Mason, G., Twisted representations of vertex operator algebras, Math. Ann., 310, 571-600, 1998 ·Zbl 0890.17029 |
| [19] | Dong, C.; Li, H.; Mason, G., Modular-invariance of trace functions in orbifold theory and generalized moonshine, Commun. Math. Phys., 214, 1-56, 2000; Dong, C.; Li, H.; Mason, G., Modular-invariance of trace functions in orbifold theory and generalized moonshine, Commun. Math. Phys., 214, 1-56, 2000 ·Zbl 1061.17025 |
| [20] | Dong, C.; Li, H.; Xu, F.; Yu, N., Fusion products of twisted modules in permutation orbifolds, Trans. Am. Math. Soc., 377, 1717-1760, 2024; Dong, C.; Li, H.; Xu, F.; Yu, N., Fusion products of twisted modules in permutation orbifolds, Trans. Am. Math. Soc., 377, 1717-1760, 2024 ·Zbl 07876003 |
| [21] | Dong, C.; Lin, X.; Ng, S.-H., Congruence property in conformal field theory, Algebra Number Theory, 9, 2121-2166, 2015; Dong, C.; Lin, X.; Ng, S.-H., Congruence property in conformal field theory, Algebra Number Theory, 9, 2121-2166, 2015 ·Zbl 1377.17025 |
| [22] | Dong, C.; Mason, G., Nonabelian orbifolds and the boson-fermion correspondence, Commun. Math. Phys., 163, 523-559, 1994; Dong, C.; Mason, G., Nonabelian orbifolds and the boson-fermion correspondence, Commun. Math. Phys., 163, 523-559, 1994 ·Zbl 0808.17019 |
| [23] | Dong, C.; Mason, G., On quantum Galois theory, Duke Math. J., 86, 305-321, 1997; Dong, C.; Mason, G., On quantum Galois theory, Duke Math. J., 86, 305-321, 1997 ·Zbl 0890.17031 |
| [24] | Dong, C.; Ng, S.-H.; Ren, L., Vertex operator superalgebras and 16-fold way; Dong, C.; Ng, S.-H.; Ren, L., Vertex operator superalgebras and 16-fold way ·Zbl 1481.17042 |
| [25] | Dong, C.; Ren, L., Congruence property in orbifold theory, Proc. Am. Math. Soc., 146, 497-506, 2018; Dong, C.; Ren, L., Congruence property in orbifold theory, Proc. Am. Math. Soc., 146, 497-506, 2018 ·Zbl 1432.17030 |
| [26] | Dong, C.; Ren, L.; Xu, F., On orbifold theory, Adv. Math., 32, 1-30, 2017; Dong, C.; Ren, L.; Xu, F., On orbifold theory, Adv. Math., 32, 1-30, 2017 ·Zbl 1419.17031 |
| [27] | Dong, C.; Wang, H., Hopf actions on vertex operator algebras, J. Algebra, 514, 310-329, 2018; Dong, C.; Wang, H., Hopf actions on vertex operator algebras, J. Algebra, 514, 310-329, 2018 ·Zbl 1446.17037 |
| [28] | Dong, C.; Yu, N., Z-graded weak modules and regularity, Commun. Math. Phys., 316, 269-277, 2012; Dong, C.; Yu, N., Z-graded weak modules and regularity, Commun. Math. Phys., 316, 269-277, 2012 ·Zbl 1315.17021 |
| [29] | Drinfeld, V.; Gelaki, S.; Nikshych, D.; Ostrik, V., On braided fusion categories. I, Sel. Math., 16, 2010, 1-119, 2010; Drinfeld, V.; Gelaki, S.; Nikshych, D.; Ostrik, V., On braided fusion categories. I, Sel. Math., 16, 2010, 1-119, 2010 ·Zbl 1201.18005 |
| [30] | Eilenberg, S.; MacLane, S., Cohomology theory of Abelian groups and homotopy theory. I, Proc. Natl. Acad. Sci. USA, 36, 443-447, 1950; Eilenberg, S.; MacLane, S., Cohomology theory of Abelian groups and homotopy theory. I, Proc. Natl. Acad. Sci. USA, 36, 443-447, 1950 ·Zbl 0037.39504 |
| [31] | Eilenberg, S.; MacLane, S., Cohomology theory of Abelian groups and homotopy theory. II, Proc. Natl. Acad. Sci. USA, 36, 657-663, 1950; Eilenberg, S.; MacLane, S., Cohomology theory of Abelian groups and homotopy theory. II, Proc. Natl. Acad. Sci. USA, 36, 657-663, 1950 ·Zbl 0039.19002 |
| [32] | van Ekeren Jethro, J.; Möller, S.; Scheithauer, N., Construction and classification of holomorphic vertex operator algebras, J. Reine Angew. Math., 759, 61-99, 2020; van Ekeren Jethro, J.; Möller, S.; Scheithauer, N., Construction and classification of holomorphic vertex operator algebras, J. Reine Angew. Math., 759, 61-99, 2020 ·Zbl 1447.81181 |
| [33] | Etingof, P.; Gelaki, S.; Nikshych, D.; Ostrik, V., Tensor Categories, Mathematical Surveys and Monographs, vol. 205, 2015, American Mathematical Society: American Mathematical Society Providence, RI; Etingof, P.; Gelaki, S.; Nikshych, D.; Ostrik, V., Tensor Categories, Mathematical Surveys and Monographs, vol. 205, 2015, American Mathematical Society: American Mathematical Society Providence, RI ·Zbl 1365.18001 |
| [34] | Etingof, P.; Nikshych, D.; Ostrik, V., On fusion categories, Ann. Math. (2), 162, 581-642, 2005; Etingof, P.; Nikshych, D.; Ostrik, V., On fusion categories, Ann. Math. (2), 162, 581-642, 2005 ·Zbl 1125.16025 |
| [35] | Evans, D.; Gannon, T., Reconstruction and local extensions for twisted group doubles, and permutation orbifolds, Trans. Am. Math. Soc., 375, 2789-2826, 2022; Evans, D.; Gannon, T., Reconstruction and local extensions for twisted group doubles, and permutation orbifolds, Trans. Am. Math. Soc., 375, 2789-2826, 2022 ·Zbl 1486.81155 |
| [36] | Frenkel, I.; Huang, Y.; Lepowsky, J., On axiomatic approaches to vertex operator algebras and modules, Mem. Am. Math. Soc., 104, 1993; Frenkel, I.; Huang, Y.; Lepowsky, J., On axiomatic approaches to vertex operator algebras and modules, Mem. Am. Math. Soc., 104, 1993 ·Zbl 0789.17022 |
| [37] | Frenkel, I.; Lepowsky, J.; Meurman, A., Vertex Operator Algebras and the Monster, Pure and Applied Math., vol. 134, 1988, Academic Press: Academic Press Boston; Frenkel, I.; Lepowsky, J.; Meurman, A., Vertex Operator Algebras and the Monster, Pure and Applied Math., vol. 134, 1988, Academic Press: Academic Press Boston ·Zbl 0674.17001 |
| [38] | Frenkel, I.; Zhu, Y., Vertex operator algebras associated to representations of affine and Virasoro algebras, Duke Math. J., 66, 123-168, 1992; Frenkel, I.; Zhu, Y., Vertex operator algebras associated to representations of affine and Virasoro algebras, Duke Math. J., 66, 123-168, 1992 ·Zbl 0848.17032 |
| [39] | Goddard, P.; Kent, A.; Olive, D., Unitary representations of the Virasoro algebra and super-Virasoro algebras, Commun. Math. Phys., 103, 105-119, 1986; Goddard, P.; Kent, A.; Olive, D., Unitary representations of the Virasoro algebra and super-Virasoro algebras, Commun. Math. Phys., 103, 105-119, 1986 ·Zbl 0588.17014 |
| [40] | 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; 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 ·Zbl 1171.16021 |
| [41] | Griess, R. L.; Höhn, G., Virasoro frames and their stabilizers for the \(E_8\) lattice type vertex operator algebra, J. Reine Angew. Math., 561, 1-37, 2003; Griess, R. L.; Höhn, G., Virasoro frames and their stabilizers for the \(E_8\) lattice type vertex operator algebra, J. Reine Angew. Math., 561, 1-37, 2003 ·Zbl 1074.17006 |
| [42] | Hanaki, A.; Miyamoto, M.; Tambara, D., Quantum Galois theory for finite groups, Duke Math. J., 97, 541-544, 1999; Hanaki, A.; Miyamoto, M.; Tambara, D., Quantum Galois theory for finite groups, Duke Math. J., 97, 541-544, 1999 ·Zbl 0977.17029 |
| [43] | Huang, Y.-Z., Rigidity and modularity of vertex tensor categories, Commun. Contemp. Math., 10, suppl. 1, 871-911, 2008; Huang, Y.-Z., Rigidity and modularity of vertex tensor categories, Commun. Contemp. Math., 10, suppl. 1, 871-911, 2008 ·Zbl 1169.17019 |
| [44] | Huang, Y.; Kirillov, A.; Lepowsky, J., Braided tensor categories and extensions of vertex operator algebras, Commun. Math. Phys., 337, 1143-1159, 2015; Huang, Y.; Kirillov, A.; Lepowsky, J., Braided tensor categories and extensions of vertex operator algebras, Commun. Math. Phys., 337, 1143-1159, 2015 ·Zbl 1388.17014 |
| [45] | Huang, Y.; Lepowsky, J., A theory of tensor products for module categories for a vertex operator algebra, I, Sel. Math. New Ser., 1, 699-756, 1995; Huang, Y.; Lepowsky, J., A theory of tensor products for module categories for a vertex operator algebra, I, Sel. Math. New Ser., 1, 699-756, 1995 ·Zbl 0854.17032 |
| [46] | Huang, Y.; Lepowsky, J., A theory of tensor products for module categories for a vertex operator algebra, II, Sel. Math. New Ser., 1, 756-786, 1995; Huang, Y.; Lepowsky, J., A theory of tensor products for module categories for a vertex operator algebra, II, Sel. Math. New Ser., 1, 756-786, 1995 ·Zbl 0854.17033 |
| [47] | Huang, Y.; Lepowsky, J., A theory of tensor products for module categories for a vertex operator algebra, III, J. Pure Appl. Algebra, 100, 141-171, 1995; Huang, Y.; Lepowsky, J., A theory of tensor products for module categories for a vertex operator algebra, III, J. Pure Appl. Algebra, 100, 141-171, 1995 ·Zbl 0841.17014 |
| [48] | Humphreys, J. E., Introduction to Lie Algebras and Representation Theory, Graduate Texts in Mathematics, vol. 9, 1978, Springer-Verlag: Springer-Verlag New York-Berlin; Humphreys, J. E., Introduction to Lie Algebras and Representation Theory, Graduate Texts in Mathematics, vol. 9, 1978, Springer-Verlag: Springer-Verlag New York-Berlin ·Zbl 0447.17001 |
| [49] | Joyal, A.; Street, R., Braided tensor categories, Adv. Math., 102, 20-78, 1993; Joyal, A.; Street, R., Braided tensor categories, Adv. Math., 102, 20-78, 1993 ·Zbl 0817.18007 |
| [50] | Kirillov, A., Modular categories and orbifold models, Commun. Math. Phys., 229, 309-335, 2002; Kirillov, A., Modular categories and orbifold models, Commun. Math. Phys., 229, 309-335, 2002 ·Zbl 1073.17011 |
| [51] | Kirillov, A.; Ostrik, V., On a q-analogue of the McKay correspondence and the ADE classification of \(s l_2\) conformal field theories, Adv. Math., 171, 183-227, 2002; Kirillov, A.; Ostrik, V., On a q-analogue of the McKay correspondence and the ADE classification of \(s l_2\) conformal field theories, Adv. Math., 171, 183-227, 2002 ·Zbl 1024.17013 |
| [52] | Krauel, M.; Miyamoto, M., A modular invariance property of multivariable trace functions for regular vertex operator algebras, J. Algebra, 444, 124-142, 2015; Krauel, M.; Miyamoto, M., A modular invariance property of multivariable trace functions for regular vertex operator algebras, J. Algebra, 444, 124-142, 2015 ·Zbl 1358.17032 |
| [53] | Lan, T.; Kong, L.; Wen, X.-G., Modular extensions of unitary braided fusion categories and \(2 + 1 \operatorname{D}\) topological/SPT orders with symmetries, Commun. Math. Phys., 351, 709-739, 2017; Lan, T.; Kong, L.; Wen, X.-G., Modular extensions of unitary braided fusion categories and \(2 + 1 \operatorname{D}\) topological/SPT orders with symmetries, Commun. Math. Phys., 351, 709-739, 2017 ·Zbl 1361.81184 |
| [54] | Lan, T.; Long, L.; Wen, X.-G., Classification of (2+1)-dimensional topological order and symmetry-protected topological order for bosonic and fermionic systems with on-site symmetries, Phys. Rev. B, 95, Article 235140 pp., 2017; Lan, T.; Long, L.; Wen, X.-G., Classification of (2+1)-dimensional topological order and symmetry-protected topological order for bosonic and fermionic systems with on-site symmetries, Phys. Rev. B, 95, Article 235140 pp., 2017 |
| [55] | Lepowsky, J., Calculus of twisted vertex operators, Proc. Natl. Acad. Sci. USA, 82, 8295-8299, 1985; Lepowsky, J., Calculus of twisted vertex operators, Proc. Natl. Acad. Sci. USA, 82, 8295-8299, 1985 ·Zbl 0579.17010 |
| [56] | Lin, X., Mirror extensions of rational vertex operator algebras, Trans. Am. Math. Soc., 369, 3821-3840, 2017; Lin, X., Mirror extensions of rational vertex operator algebras, Trans. Am. Math. Soc., 369, 3821-3840, 2017 ·Zbl 1433.17036 |
| [57] | Mason, G.; Ng, S.-H., Group cohomology and gauge equivalence of some twisted quantum doubles, Trans. Am. Math. Soc., 353, 9, 3465-3509, 2001; Mason, G.; Ng, S.-H., Group cohomology and gauge equivalence of some twisted quantum doubles, Trans. Am. Math. Soc., 353, 9, 3465-3509, 2001 ·Zbl 0968.57030 |
| [58] | Mason, G.; Ng, S.-H., Cleft extensions and quotients of twisted quantum doubles, (Developments and Retrospectives in Lie Theory. Developments and Retrospectives in Lie Theory, Dev. Math., vol. 38, 2014, Springer: Springer Cham), 229-246; Mason, G.; Ng, S.-H., Cleft extensions and quotients of twisted quantum doubles, (Developments and Retrospectives in Lie Theory. Developments and Retrospectives in Lie Theory, Dev. Math., vol. 38, 2014, Springer: Springer Cham), 229-246 ·Zbl 1320.16017 |
| [59] | McRae, R., Twisted modules and G-equivariantization in logarithmic conformal field theory; McRae, R., Twisted modules and G-equivariantization in logarithmic conformal field theory ·Zbl 1480.18014 |
| [60] | McRae, R., On the tensor structure of modules for compact orbifold vertex operator algebras, Math. Z., 296, 409-452, 2020; McRae, R., On the tensor structure of modules for compact orbifold vertex operator algebras, Math. Z., 296, 409-452, 2020 ·Zbl 1453.17017 |
| [61] | Müger, M., On the structure of modular categories, Proc. Lond. Math. Soc., 87, 291-308, 2003; Müger, M., On the structure of modular categories, Proc. Lond. Math. Soc., 87, 291-308, 2003 ·Zbl 1037.18005 |
| [62] | Müger, M., From subfactors to categories and topology. II. The quantum double of tensorcategories and subfactors, J. Pure Appl. Algebra, 180, 159-219, 2003; Müger, M., From subfactors to categories and topology. II. The quantum double of tensorcategories and subfactors, J. Pure Appl. Algebra, 180, 159-219, 2003 ·Zbl 1033.18003 |
| [63] | Ng, S.-H.; Schauenburg, P., Higher Frobenius-Schur indicators for pivotal categories, (Hopf Algebras and Generalizations. Hopf Algebras and Generalizations, Contemporary Mathematics, vol. 441, 2007, American Mathematical Society: American Mathematical Society Providence, RI), 63-90; Ng, S.-H.; Schauenburg, P., Higher Frobenius-Schur indicators for pivotal categories, (Hopf Algebras and Generalizations. Hopf Algebras and Generalizations, Contemporary Mathematics, vol. 441, 2007, American Mathematical Society: American Mathematical Society Providence, RI), 63-90 ·Zbl 1153.18008 |
| [64] | Ng, S.-H.; Schauenburg, P., Frobenius-Schur indicators and exponents of spherical categories, Adv. Math., 211, 1, 34-71, 2007; Ng, S.-H.; Schauenburg, P., Frobenius-Schur indicators and exponents of spherical categories, Adv. Math., 211, 1, 34-71, 2007 ·Zbl 1138.16017 |
| [65] | Nikulin, V. V., Integral symmetric bilinear forms and some of their applications, Math. USSR, Izv., 14, 1, 103-167, 1980; Nikulin, V. V., Integral symmetric bilinear forms and some of their applications, Math. USSR, Izv., 14, 1, 103-167, 1980 ·Zbl 0427.10014 |
| [66] | Tanabe, K., On intertwining operators and finite automorphism groups of vertex operator algebras, J. Algebra, 287, 174-198, 2005; Tanabe, K., On intertwining operators and finite automorphism groups of vertex operator algebras, J. Algebra, 287, 174-198, 2005 ·Zbl 1145.17012 |
| [67] | Xu, F., Algebraic orbifold conformal field theories, Proc. Natl. Acad. Sci. USA, 97, 14069-14073, 2000; Xu, F., Algebraic orbifold conformal field theories, Proc. Natl. Acad. Sci. USA, 97, 14069-14073, 2000 ·Zbl 1031.81038 |
| [68] | Zhu, Y., Modular invariance of characters of vertex operator algebras, J. Am. Math. Soc., 9, 237-302, 1996; Zhu, Y., Modular invariance of characters of vertex operator algebras, J. Am. Math. Soc., 9, 237-302, 1996 ·Zbl 0854.17034 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
