Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-26T08:44:04.416Z Has data issue: false hasContentIssue false
  • English
  • Français

On orbifold constructions associated with the Leech lattice vertex operator algebra

Published online by Cambridge University Press:  05 September 2018

CHING HUNG LAM  and
HIROKI SHIMAKURA
CHING HUNG LAM
Affiliation:
Institute of Mathematics, Academia Sinica and National Center for Theoretical Sciences of Taiwan, Taipei 10617, Taiwan. e-mail: chlam@math.sinica.edu.tw
HIROKI SHIMAKURA
Affiliation:
Graduate School of Information Sciences, Tohoku University, Sendai 980-8579, Japan. e-mail: shimakura@tohoku.ac.jp
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we study orbifold constructions associated with the Leech lattice vertex operator algebra. As an application, we prove that the structure of a strongly regular holomorphic vertex operator algebra of central charge 24 is uniquely determined by its weight one Lie algebra if the Lie algebra has the type A3,43A1,2, A4,52, D4,12A2,6, A6,7, A7,4A1,13, D5,8A1,2 or D6,5A1,12 by using the reverse orbifold construction. Our result also provides alternative constructions of these vertex operator algebras (except for the case A6,7) from the Leech lattice vertex operator algebra.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 168 , Issue 2 , March 2020 , pp. 261 - 285
Copyright
Copyright © Cambridge Philosophical Society 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

†

Partially supported by MoST grant 104-2115-M-001-004-MY3 of Taiwan.

‡

Partially supported by JSPS KAKENHI Grant Numbers JP26800001 and JP17K05154.

§

Both authors were partially supported by JSPS Program for Advancing Strategic International Networks to Accelerate the Circulation of Talented Researchers “Development of Concentrated Mathematical Center Linking to Wisdom of the Next Generation”.

References

REFERENCES

[Bo86] Borcherds, R. E. Vertex algebras, Kac-Moody algebras and the Monster. Proc. Nat'l. Acad. Sci. U.S.A. 83 (1986), 3068–3071.Google Scholar
[Bo92] Borcherds, R. E. Monstrous moonshine and monstrous Lie superalgebras. Invent. Math. 109 (1992), 405–444.Google Scholar
[BCP97] Bosma, W., Cannon, J. and Playoust, C. The Magma algebra system I: the user language. J. Symbolic Comput. 24 (1997), 235–265.Google Scholar
[CM] Carnahan, S. and Miyamoto, M. Regularity of fixed-point vertex operator subalgebras. arXiv:1603.05645.Google Scholar
[DGM96] Dolan, L., Goddard, P. and Montague, P. Conformal field theories, representations and lattice constructions. Comm. Math. Phys. 179 (1996), 61–120.Google Scholar
[DL96] Dong, C. and Lepowsky, J. The algebraic structure of relative twisted vertex operators. J. Pure Appl. Algebra 110 (1996), 259–295.Google Scholar
[DLM00] Dong, C., Li, H., and Mason, G. Modular-invariance of trace functions in orbifold theory and generalised moonshine. Comm. Math. Phys. 214 (2000), 1–56.Google Scholar
[DM04a] Dong, C. and Mason, G. Holomorphic vertex operator algebras of small central charge. Pacific J. Math. 213 (2004), 253–266.Google Scholar
[DM04b] Dong, C. and Mason, G. Rational vertex operator algebras and the effective central charge. Int. Math. Res. Not. (2004), 2989–3008.Google Scholar
[DM06] Dong, C. and Mason, G. Integrability of C 2-cofinite vertex operator algebras. Int. Math. Res. Not. (2006), Art. ID 80468, 15 pp.Google Scholar
[DN99] Dong, C. and Nagatomo, K. Automorphism groups and twisted modules for lattice vertex operator algebras, in Recent developments in quantum affine algebras and related topics (Raleigh, NC, 1998), 117–133, Contemp. Math. 248 (Amer. Math. Soc., Providence, RI, 1999).Google Scholar
[EMS1] Van Ekeren, J., Möller, S. and Scheithauer, N. Construction and classification of holomorphic vertex operator algebras. J. Reine Angew. Math. (published online).Google Scholar
[EMS2] Van Ekeren, J., Möller, S. and Scheithauer, N. Dimension formulae in genus zero and uniqueness of vertex operator algebras. Internat. Math. Res. Notices (published online).Google Scholar
[FHL93] Frenkel, I. B., Huang, Y. and Lepowsky, J. On axiomatic approaches to vertex operator algebras and modules. Mem. Amer. Math. Soc. 104 (1993), viii+64 pp.Google Scholar
[FLM88] Frenkel, I. B., Lepowsky, J. and Meurman, A. Vertex operator algebras and the monster. Pure and Appl. Math. vol. 134 (Academic Press, Boston, 1988).Google Scholar
[FZ92] Frenkel, I. and Zhu, Y. Vertex operator algebras associated to representations of affine and Virasoro algebras. Duke Math. J. 66 (1992), 123–168.Google Scholar
[HL90] Harada, K. and Lang, M. L. On some sublattices of the Leech lattice. Hokkaido Math. J. 19 (1990), 435–446.Google Scholar
[HM16] Höhn, G. and Mason, G. The 290 fixed-point sublattices of the Leech lattice. J. Algebra 448 (2016), 618–637.Google Scholar
[Ka90] Kac, V. G. Infinite-Dimensional Lie Algebras, third edition (Cambridge University Press, Cambridge, 1990).Google Scholar
[KLL18] Kawasetsu, K., Lam, C. H. and Lin, X. ℤ2-orbifold construction associated with (-1)-isometry and uniqueness of holomorphic vertex operator algebras of central charge 24. Proc. Amer. Math. Soc. 146 (2018), 1937–1950.Google Scholar
[La11] Lam, C. H. On the constructions of holomorphic vertex operator algebras of central charge 24. Comm. Math. Phys. 305 (2011), 153–198Google Scholar
[LL] Lam, C. H. and Lin, X. A Holomorphic vertex operator algebra of central charge 24 with weight one Lie algebra F 4,6A 2,2. arXiv:1612.08123.Google Scholar
[LS12] Lam, C. H. and Shimakura, H. Quadratic spaces and holomorphic framed vertex operator algebras of central charge 24. Proc. Lond. Math. Soc. 104 (2012), 540–576.Google Scholar
[LS15] Lam, C. H. and Shimakura, H. Classification of holomorphic framed vertex operator algebras of central charge 24. Amer. J. Math. 137 (2015), 111–137.Google Scholar
[LS16a] Lam, C. H. and Shimakura, H. Orbifold construction of holomorphic vertex operator algebras associated to inner automorphisms. Comm. Math. Phys. 342 (2016), 803–841.Google Scholar
[LS16b] Lam, C. H. and Shimakura, H. A holomorphic vertex operator algebra of central charge 24 whose weight one Lie algebra has the type A 6,7. Lett. Math. Phys. 106 (2016), 1575–1585.Google Scholar
[LS] Lam, C. H. and Shimakura, H. Reverse orbifold construction and uniqueness of holomorphic vertex operator algebras. arXiv:1606.08979.Google Scholar
[Le85] Lepowsky, J. Calculus of twisted vertex operators. Proc. Natl. Acad. Sci. USA 82 (1985), 8295–8299.Google Scholar
[Li94] Li, H. Symmetric invariant bilinear forms on vertex operator algebras. J. Pure Appl. Algebra, 96 (1994), 279–297.Google Scholar
[Li96] Li, H. Local systems of twisted vertex operators, vertex operator superalgebras and twisted modules, in Moonshine, the Monster, and related topics. Contemp. Math. 193, (Amer. Math. Soc., Providence, RI, 1996), 203–236.Google Scholar
[Li01] Li, H. Certain extensions of vertex operator algebras of affine type. Comm. Math. Phys. 217 (2001), 653–696.Google Scholar
[Mi13] Miyamoto, M. A ℤ3-orbifold theory of lattice vertex operator algebra and ℤ3-orbifold constructions, in Symmetries, integrable systems and representations. Springer Proc. Math. Stat. 40 (Springer, Heidelberg, 2013), 319–344.Google Scholar
[Mi15] Miyamoto, M. C 2-cofiniteness of cyclic-orbifold models. Comm. Math. Phys. 335 (2015), 1279–1286.Google Scholar
[Mö16] Möller, S. A Cyclic Orbifold Theory for holomorphic vertex operator algebras and applications, dissertation (Darmstadt, 2016). arXiv:1611.09843.Google Scholar
[Mo94] Montague, P. S. Orbifold constructions and the classification of self-dual c = 24 conformal field theories. Nuclear Phys. B 428 (1994), 233–258.Google Scholar
[SS16] Sagaki, D. and Shimakura, H. Application of a ℤ3-orbifold construction to the lattice vertex operator algebras associated to Niemeier lattices. Trans. Amer. Math. Soc. 368 (2016), 1621–1646.Google Scholar
[Sc93] Schellekens, A. N. Meromorphic c = 24 conformal field theories. Comm. Math. Phys. 153 (1993), 159–185.Google Scholar
[Wi83] Wilson, R. A. The maximal subgroups of Conway's group Co 1. J. Algebra 85 (1983), 144–165.Google Scholar