Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

$ \mathbb{Z}_2$-orbifold construction associated with $ (-1)$-isometry and uniqueness of holomorphic vertex operator algebras of central charge 24


Authors: Kazuya Kawasetsu, Ching Hung Lam and Xingjun Lin
Journal: Proc. Amer. Math. Soc. 146 (2018), 1937-1950
MSC (2010): Primary 17B69
DOI: https://doi.org/10.1090/proc/13881
Published electronically: December 11, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The vertex operator algebra structure of a strongly regular holomorphic vertex operator algebra $ V$ of central charge $ 24$ is proved to be uniquely determined by the Lie algebra structure of its weight one space $ V_1$ if $ V_1$ is a Lie algebra of the type $ A_{1,4}^{12}$, $ B_{2,2}^6$, $ B_{3,2}^4$, $ B_{4,2}^3$, $ B_{6,2}^2$, $ B_{12,2}$, $ D_{4,2}^2B_{2,1}^4$, $ D_{8,2}B_{4,1}^2$, $ A_{3,2}^4A_{1,1}^4$, $ D_{5,2}^2A_{3,1}^2$, $ D_{9,2}A_{7,1}$, $ C_{4,1}^4$, or $ D_{6,2}B_{3,1}^2C_{4,1}$.


References [Enhancements On Off] (What's this?)

  • [ACKL] Tomoyuki Arakawa, Thomas Creutzig, Kazuya Kawasetsu, and Andrew R. Linshaw, Orbifolds and Cosets of Minimal $ \mathcal{W}$-Algebras, Comm. Math. Phys. 355 (2017), no. 1, 339-372. MR 3670736, https://doi.org/10.1007/s00220-017-2901-2
  • [Bou] Nicolas Bourbaki, Lie groups and Lie algebras. Chapters 4-6, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2002. Translated from the 1968 French original by Andrew Pressley. MR 1890629
  • [DGM] L. Dolan, P. Goddard, and P. Montague, Conformal field theories, representations and lattice constructions, Comm. Math. Phys. 179 (1996), no. 1, 61-120. MR 1395218
  • [DGH] Chongying Dong, Robert L. Griess Jr., and Gerald Höhn, Framed vertex operator algebras, codes and the Moonshine module, Comm. Math. Phys. 193 (1998), no. 2, 407-448. MR 1618135, https://doi.org/10.1007/s002200050335
  • [DLM00] Chongying Dong, Haisheng Li, and Geoffrey Mason, Modular-invariance of trace functions in orbifold theory and generalized Moonshine, Comm. Math. Phys. 214 (2000), no. 1, 1-56. MR 1794264, https://doi.org/10.1007/s002200000242
  • [DM04a] Chongying Dong and Geoffrey Mason, Holomorphic vertex operator algebras of small central charge, Pacific J. Math. 213 (2004), no. 2, 253-266. MR 2036919, https://doi.org/10.2140/pjm.2004.213.253
  • [DM04b] Chongying Dong and Geoffrey Mason, Rational vertex operator algebras and the effective central charge, Int. Math. Res. Not. 56 (2004), 2989-3008. MR 2097833, https://doi.org/10.1155/S1073792804140968
  • [EMS] J. van Ekeren, S. Möller, and N. Scheithauer, Construction and classification of holomorphic vertex operator algebras. arXiv:1507.08142 (2015)
  • [FLM] Igor Frenkel, James Lepowsky, and Arne Meurman, Vertex operator algebras and the Monster, Pure and Applied Mathematics, vol. 134, Academic Press, Inc., Boston, MA, 1988. MR 996026
  • [FZ92] Igor B. Frenkel and Yongchang Zhu, Vertex operator algebras associated to representations of affine and Virasoro algebras, Duke Math. J. 66 (1992), no. 1, 123-168. MR 1159433, https://doi.org/10.1215/S0012-7094-92-06604-X
  • [Hel78] Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, vol. 80, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 514561
  • [HKL] Yi-Zhi Huang, Alexander Kirillov Jr., and James Lepowsky, Braided tensor categories and extensions of vertex operator algebras, Comm. Math. Phys. 337 (2015), no. 3, 1143-1159. MR 3339173, https://doi.org/10.1007/s00220-015-2292-1
  • [Hum90] James E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. MR 1066460
  • [Ka] Victor G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. MR 1104219
  • [KM15] Matthew Krauel and Masahiko Miyamoto, A modular invariance property of multivariable trace functions for regular vertex operator algebras, J. Algebra 444 (2015), 124-142. MR 3406171, https://doi.org/10.1016/j.jalgebra.2015.07.013
  • [LLin] C. Lam and X. Lin, Holomorphic vertex operator algebra of central charge 24 with Lie algebra $ F_{4,6}A_{2,2}$. arXiv:1612.08123 (2016)
  • [LS1] Ching Hung Lam and Hiroki Shimakura, Classification of holomorphic framed vertex operator algebras of central charge 24, Amer. J. Math. 137 (2015), no. 1, 111-137. MR 3318088, https://doi.org/10.1353/ajm.2015.0001
  • [LS16] Ching Hung Lam and Hiroki Shimakura, Orbifold construction of holomorphic vertex operator algebras associated to inner automorphisms, Comm. Math. Phys. 342 (2016), no. 3, 803-841. MR 3465432, https://doi.org/10.1007/s00220-015-2484-8
  • [LS3] Ching Hung Lam and Hiroki Shimakura, A holomorphic vertex operator algebra of central charge 24 whose weight one Lie algebra has type $ A_{6,7}$, Lett. Math. Phys. 106 (2016), no. 11, 1575-1585. MR 3555415, https://doi.org/10.1007/s11005-016-0883-1
  • [LS4] C. Lam, H. Shimakura, Reverse orbifold construction and uniqueness of holomorphic vertex operator algebras. arXiv:1606.08979 (2016)
  • [Li96] Hai-Sheng Li, Local systems of twisted vertex operators, vertex operator superalgebras and twisted modules, Moonshine, the Monster, and related topics (South Hadley, MA, 1994) Contemp. Math., vol. 193, Amer. Math. Soc., Providence, RI, 1996, pp. 203-236. MR 1372724, https://doi.org/10.1090/conm/193/02373
  • [Mo] P. S. Montague, Orbifold constructions and the classification of self-dual $ c=24$ conformal field theories, Nuclear Phys. B 428 (1994), no. 1-2, 233-258. MR 1299260, https://doi.org/10.1016/0550-3213(94)90201-1
  • [SS] Daisuke Sagaki and Hiroki Shimakura, Application of a $ \mathbb{Z}_3$-orbifold construction to the lattice vertex operator algebras associated to Niemeier lattices, Trans. Amer. Math. Soc. 368 (2016), no. 3, 1621-1646. MR 3449220, https://doi.org/10.1090/tran/6382
  • [Sch] A. N. Schellekens, Meromorphic $ c=24$ conformal field theories, Comm. Math. Phys. 153 (1993), no. 1, 159-185. MR 1213740

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 17B69

Retrieve articles in all journals with MSC (2010): 17B69


Additional Information

Kazuya Kawasetsu
Affiliation: Institute of Mathematics, Academia Sinica, Taipei 10617, Taiwan
Address at time of publication: School of Mathematics and Statistics, Faculty of Science, The University of Melbourne, Victoria 3052, Australia
Email: kazuya.kawasetsu@unimelb.edu.au

Ching Hung Lam
Affiliation: Institute of Mathematics, Academia Sinica, Taipei 10617, Taiwan
Email: chlam@math.sinica.edu.tw

Xingjun Lin
Affiliation: Institute of Mathematics, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan
Address at time of publication: Collaborative Innovation Centre of Mathematics, School of Mathematics and Statistics, Wuhan University, Luojiashan, Wuhan, Hubei 430072, People’s Republic of China
Email: linxingjun88@126.com

DOI: https://doi.org/10.1090/proc/13881
Keywords: Vertex operator algebra, orbifold construction, Niemeier lattices
Received by editor(s): January 3, 2017
Received by editor(s) in revised form: July 7, 2017
Published electronically: December 11, 2017
Additional Notes: The second author was partially supported by MoST grant 104-2115-M-001-004-MY3 of Taiwan
The third author is an “Overseas researchers under Postdoctoral Fellowship of Japan X1Society for the Promotion of Science” and is supported by JSPS Grant No. 16F16020.
Communicated by: Kailash C. Misra
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society