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Quantum dimensions and quantum Galois theory


Authors: Chongying Dong, Xiangyu Jiao and Feng Xu
Journal: Trans. Amer. Math. Soc. 365 (2013), 6441-6469
MSC (2010): Primary 17B69
DOI: https://doi.org/10.1090/S0002-9947-2013-05863-1
Published electronically: August 20, 2013
MathSciNet review: 3105758
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Abstract: The quantum dimensions of modules for vertex operator algebras are defined and their properties are discussed. The possible values of the quantum dimensions are obtained for rational vertex operator algebras. A criterion for simple currents of a rational vertex operator algebra is given and a full Galois theory for rational vertex operator algebras is established using the quantum dimensions.


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  • [ABD] T. Abe, G. Buhl and C. Dong, Rationality, Regularity, and $ C_2$-cofiniteness, Trans. Amer. Math. Soc. 356 (2004), 3391-3402. MR 2052955 (2005c:17041)
  • [BPZ] A. A. Belavin, A. M. Polyakov and A. B. Zamolodchikov, Infinite conformal symmetries in two-dimensional quantum field theory, Nucl. Phys. B241 (1984), 333-380. MR 757857 (86m:81097)
  • [BH] A. E. Brouwer and W. H. Haemers, Spectra of Graphs, Universitext, Springer, 2012. MR 2882891
  • [CIZ1] A. Cappelli, C. Itzykson and J.-B. Zuber, Modular invariant partition function in two dimensions, Nucl. Phys. B280 (1987), 445-464. MR 881119 (88i:81132)
  • [CIZ2] A. Cappelli, C. Itzykson and J.-B. Zuber, The $ A$-$ D$-$ E$ classification of minimal and $ A^{(1)}_1$ conformal invariant theories, Comm. Math. Phys. 113 (1987), 1-26. MR 918402 (89b:81178)
  • [C] R. Coquereaux, Global dimensions for Lie groups at level $ k$ and their conformally exceptional quantum subgroups, Revista de la Union Matematica Argentina 51 (2010), No 2, 17-42. MR 2840162
  • [DVVV] R. Dijkgraaf, C. Vafa, E. Verlinde and H. Verlinde, The operator algebra of orbifold models, Comm. Math. Phys. 123 (1989), 485-526. MR 1003430 (91c:81132)
  • [D] C. Dong, Vertex algebras associated with even lattices, J. Alg. 161 (1993), 245-265. MR 1245855 (94j:17023)
  • [DG] C. Dong and R. Jr. Griess, Rank one lattice type vertex operator algebras and their automorphism groups, J. Alg. 208 (1998), 262-275. MR 1644007 (99h:17029)
  • [DGH] C. Dong, R. Jr. Griess and G. Hoehn, Framed vertex operator algebras, codes and the Moonshine module, Comm. Math. Phys. 193 (1998), 407-448. MR 1618135 (99g:17050)
  • [DJ] C. Dong and C. Jiang, A characterization of vertex operator algebra $ L(\frac {1}{2},0)\otimes L(\frac {1}{2},0),$ Comm. Math. Phys. 296 (2010), 69-88. MR 2606628 (2011c:17049)
  • [DL] C. Dong and J. Lepowsky, Generalized Vertex Algebras and Relative Vertex Operators, Progress in Math, Vol. 112, Birkhäuser, Boston, 1993. MR 1233387 (95b:17032)
  • [DLM1] C. Dong, H. Li and G. Mason, Compact automorphism groups of vertex operator algebras, Int. Math. Res. Not. 18 (1996), 913-921. MR 1420556 (98a:17044)
  • [DLM2] C. Dong, H. Li and G. Mason, Regularity of rational vertex operator algebras, Adv. Math. 132 (1997), 148-166. MR 1488241 (98m:17037)
  • [DLM3] C. Dong, H. Li and G. Mason, Twisted representations of vertex operator algebras, Math. Ann. 310 (1998), 571-600. MR 1615132 (99d:17030)
  • [DLM4] C. Dong, H. Li and G. Mason, Vertex operator algebras and associative algebras, J. Alg. 206 (1998), 67-96. MR 1637252 (99i:17029)
  • [DLM5] C. Dong, H. Li and G. Mason, Modular-invariance of trace functions in orbifold theory and generalized moonshine, Comm. Math. Phys. 214 (2000), 1-56. MR 1794264 (2001k:17043)
  • [DM1] C. Dong and G. Mason, On quantum Galois theory, Duke Math. J. 86 (1997), 305-321. MR 1430435 (97k:17042)
  • [DM2] C. Dong and G. Mason, Quantum Galois theory for compact Lie groups, J. Alg. 214 (1999), 92-102. MR 1684904 (2000g:17043b)
  • [DM3] C. Dong and G. Mason, Rational vertex operator algebras and the effective central charge, Int. Math. Res. Not. 56 (2004), 2989-3008. MR 2097833 (2005k:17034)
  • [DMZ] C. Dong, G. Mason and Y. Zhu, Discrete series of the Virasoro algebra and the moonshine module, Proc. Symp. Pure. Math, Amer. Math. Soc. 56 II, (1994), 295-316. MR 1278737 (95c:17043)
  • [DY] C. Dong and N. Yu, $ Z$-graded weak modules and regularity, Comm. Math. Phys. 316 (2012), 269-277. MR 2989460
  • [ENO] P. Etingof, D. Nikshych and V. Ostrik, On fusion categories, Ann. Math. (2) 162 (2005), 581-642. MR 2183279 (2006m:16051)
  • [FMS] P. D. Francesco, P. Mathieu and D. Snchal, Conformal field theory, Springer-Verlag, 1997. MR 1424041 (97g:81062)
  • [FHL] I. Frenkel, Y. Huang and J. Lepowsky, On axiomatic approaches to vertex operator algebras and modules, Mem. Amer. Math. Soc. 104 1993. MR 1142494 (94a:17007)
  • [FLM] I. Frenkel, J. Lepowsky and A. Meurman, Vertex Operator Algebras and the Monster, Pure and Applied Math., Vol. 134, Academic Press, Boston, 1988. MR 996026 (90h:17026)
  • [FZ] I. Frenkel and Y. Zhu, Vertex operator algebras associated to representations of affine and Virasoro algebras, Duke Math. J. 66 (1992), 123-168. MR 1159433 (93g:17045)
  • [GL] D. Guido, and R. Longo, An algebraic spin and statistics theorem, Comm. Math. Phys. 172 (1995), 517-533. MR 1354258 (97i:81068)
  • [HMT] A. Hanaki, M. Miyamoto, and D. Tambara, Quantum Galois theory for finite groups, Duke Math. J. 97 (1999), 541-544. MR 1682988 (2000g:17043a)
  • [H] Y.-Z. Huang, Vertex operator algebras and the Verlinde Conjecture, Comm. Contemp. Math. 10 (2008), 103-154. MR 2387861 (2009e:17056)
  • [HL1] Y.-Z. Huang and J. Lepowsky, Toward a theory of tensor product for representations of a vertex operator algebra, in Proc. 20th Intl. Conference on Diff. Geom. Methods in Theoretical Physics, New York, 1991, ed. S. Catto and A. Rocha, World Scientific, Singapore, 1992, Vol. 1, 344-354. MR 1225125 (94k:17045)
  • [HL2] Y.-Z. Huang and J. Lepowsky, Tensor products of modules for a vertex operator algebra and vertex tensor categories, Lie Theory and Geometry, Birkhauser, Boston, 1994, 349-383. MR 1327541 (96e:17061)
  • [HL3] Y.-Z. Huang and J. Lepowsky, A theory of tensor products for module categories for a vertex operator algebra, I, Sel. Math. 1 (1995), 699-756. MR 1383584 (98a:17047)
  • [HL4] Y.-Z. Huang and J. Lepowsky, A theory of tensor products for module categories for a vertex operator algebra, III, J. Pure Appl. Alg. 100 (1995), 141-171. MR 1344848 (98a:17049)
  • [IZ] C. Itzykson and J.-B. Zuber , Two-dimensional conformal invariant theories on a torus, Nucl. Phys. B27, (1986), 580-616. MR 865230 (88f:81111)
  • [J] V.F.R. Jones, Index for subfactors, Inv. Math. 72 (1983), 1-25. MR 696688 (84d:46097)
  • [K] V. G. Kac, Infinite-dimensional Lie algebras, Third edition, Cambridge University Press, Cambridge, 1990. MR 1104219 (92k:17038)
  • [KR] V. G. Kac and A. Raina, Highest Weight Representations of Infinite Dimensional Lie Algebras, World Scientific, Adv. Ser. In Math. Phys. Singapore, 1987. MR 1021978 (90k:17013)
  • [Ka] M. Kaku, Strings, Conformal Fields, and Topology, Grad. Texts in Contemp. Phys. Springer-Verlag, New York, 1991. MR 1102894 (92i:81288)
  • [KLM] Y. Kawahigashi, R. Longo and M. Müger, Multi-interval subfactors and modularity of representations in conformal field theory, Comm. Math. Phys. 219 (2001), 631-669. MR 1838752 (2002g:81059)
  • [LS] P. W. H. Lemmens and J. J. Seidel, Equiangular lines, J. Alg. 24 (1973), 494-512. MR 0307969 (46:7084)
  • [Li1] H. Li, Local systems of vertex operators, vertex superalgebras and modules, J. Pure Appl. Alg. 109 (1996), 143-195. MR 1387738 (97d:17016)
  • [Li2] H. Li, An analogue of the Hom functor and a generalized nuclear democracy theorem, Duke Math. J. 93 (1998), 73-114. MR 1620083 (99d:17031)
  • [M] A. Milas, Fusion rings for degenerate minimal models, J. Alg. 254 (2002), 300-335. MR 1933872 (2003k:17037)
  • [MS] G. Moore and N. Seiberg, Classical and quantum conformal field theory, Comm. Math. Phys. 123 (1989), 177-254. MR 1002038 (90e:81216)
  • [PP] M. Pimsner and S. Popa, Entropy and index for subfactors, Ann. Sci. Ec. Norm. Sup. 19 (1986), 57-106. MR 860811 (87m:46120)
  • [S] J. H. Smith, Some properties of the spectrum of a graph, Combinatorial Structures and their Applications (Proc. Calgary Internat. Conf., Calgary, Alta.), pp. 403-406, New York: Gordon and Breach, 1970. MR 0266799 (42:1702)
  • [V] E. Verlinde, Fusion rules and modular transformation in 2D conformal field theory, Nucl. Phys. B300 (1988), 360-376. MR 954762 (89h:81238)
  • [W] W. Wang, Rationality of Virasoro vertex operator algebras, Duke Math. J. 71, (1993), 197-211. MR 1230296 (94i:17034)
  • [Wa] A. Wassermann, Operator algebras and conformal field theory III: Fusion of positive energy representations of $ LSU(N)$ using bounded operators, Inv. Math. 133 (1998), 467-538. MR 1645078 (99j:81101)
  • [X] F. Xu, Algebraic orbifold conformal field theories, Proc. Natl. Acad. Sci. USA 97 (2000), 14069-14073. MR 1806798 (2002a:81156)
  • [Xu] X. Xu, Introduction to Vertex Operator Superalgebras and Their Modules, Mathematics and its Applications, Vol. 456, Kluwer Academic Publishers, Dordrecht, 1998. MR 1656671 (2000h:17019)
  • [Z] Y. Zhu, Modular invariance of characters of vertex operator algebras, J. Amer, Math. Soc. 9 (1996), 237-302. MR 1317233 (96c:17042)

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Additional Information

Chongying Dong
Affiliation: Department of Mathematics, University of California, Santa Cruz, California 95064

Xiangyu Jiao
Affiliation: Department of Mathematics, University of California, Santa Cruz, California 95064

Feng Xu
Affiliation: Department of Mathematics, University of California, Riverside, California 92521

DOI: https://doi.org/10.1090/S0002-9947-2013-05863-1
Received by editor(s): January 12, 2012
Received by editor(s) in revised form: April 24, 2012
Published electronically: August 20, 2013
Additional Notes: The first author was supported by NSF grants and a faculty research fund from the University of California at Santa Cruz
The third author was supported by an NSF grant and a faculty research fund from the University of California at Riverside.
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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