## Reverse orbifold construction and uniqueness of holomorphic vertex operator algebras

HTML articles powered by AMS MathViewer

- by Ching Hung Lam and Hiroki Shimakura PDF
- Trans. Amer. Math. Soc.
**372**(2019), 7001-7024 Request permission

## Abstract:

In this article, we develop a general technique for proving the uniqueness of holomorphic vertex operator algebras based on the orbifold construction and its “reverse” process. 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 $1$ Lie algebra if the Lie algebra has the type $E_{6,3}G_{2,1}^3$, $A_{2,3}^6$, or $A_{5,3}D_{4,3}A_{1,1}^3$.## References

- Toshiyuki Abe, Geoffrey Buhl, and Chongying Dong,
*Rationality, regularity, and $C_2$-cofiniteness*, Trans. Amer. Math. Soc.**356**(2004), no. 8, 3391–3402. MR**2052955**, DOI 10.1090/S0002-9947-03-03413-5 - 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**, DOI 10.1007/s00220-017-2901-2 - Richard E. Borcherds,
*Vertex algebras, Kac-Moody algebras, and the Monster*, Proc. Nat. Acad. Sci. U.S.A.**83**(1986), no. 10, 3068–3071. MR**843307**, DOI 10.1073/pnas.83.10.3068 - S. Carnahan and M. Miyamoto,
*Regularity of fixed-point vertex operator subalgebras*, arXiv:1603.05645 (2016). - J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson,
*$\Bbb {ATLAS}$ of finite groups*, Oxford University Press, Eynsham, 1985. Maximal subgroups and ordinary characters for simple groups; With computational assistance from J. G. Thackray. MR**827219** - J. H. Conway and N. J. A. Sloane,
*Sphere packings, lattices and groups*, 3rd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 290, Springer-Verlag, New York, 1999. With additional contributions by E. Bannai, R. E. Borcherds, J. Leech, S. P. Norton, A. M. Odlyzko, R. A. Parker, L. Queen and B. B. Venkov. MR**1662447**, DOI 10.1007/978-1-4757-6568-7 - 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**, DOI 10.1007/BF02103716 - 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**, DOI 10.1007/s002200000242 - Chongying Dong and Geoffrey Mason,
*Holomorphic vertex operator algebras of small central charge*, Pacific J. Math.**213**(2004), no. 2, 253–266. MR**2036919**, DOI 10.2140/pjm.2004.213.253 - Chongying Dong and Geoffrey Mason,
*Rational vertex operator algebras and the effective central charge*, Int. Math. Res. Not.**56**(2004), 2989–3008. MR**2097833**, DOI 10.1155/S1073792804140968 - Chongying Dong and Geoffrey Mason,
*Integrability of $C_2$-cofinite vertex operator algebras*, Int. Math. Res. Not. , posted on (2006), Art. ID 80468, 15. MR**2219226**, DOI 10.1155/IMRN/2006/80468 - Chongying Dong and Kiyokazu Nagatomo,
*Automorphism groups and twisted modules for lattice vertex operator algebras*, Recent developments in quantum affine algebras and related topics (Raleigh, NC, 1998) Contemp. Math., vol. 248, Amer. Math. Soc., Providence, RI, 1999, pp. 117–133. MR**1745258**, DOI 10.1090/conm/248/03821 - Chongying Dong, Li Ren, and Feng Xu,
*On orbifold theory*, Adv. Math.**321**(2017), 1–30. MR**3715704**, DOI 10.1016/j.aim.2017.09.032 - J. van Ekeren, S. Möller, and N. Scheithauer,
*Construction and classification of holomorphic vertex operator algebras*, arXiv:1507.08142 (2015). J. Reine Angew. Math. (to appear). - J. van Ekeren, S. Möller, and N. Scheithauer,
*Dimension formulae in genus zero and uniqueness of vertex operator algebras*, arXiv:1704.00478 (2017). Int. Math. Res. Not. IMRN (to appear). - Igor B. Frenkel, Yi-Zhi Huang, and James Lepowsky,
*On axiomatic approaches to vertex operator algebras and modules*, Mem. Amer. Math. Soc.**104**(1993), no. 494, viii+64. MR**1142494**, DOI 10.1090/memo/0494 - 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** - 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**, DOI 10.1215/S0012-7094-92-06604-X - 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**, DOI 10.1007/s00220-015-2292-1 - James E. Humphreys,
*Introduction to Lie algebras and representation theory*, Graduate Texts in Mathematics, Vol. 9, Springer-Verlag, New York-Berlin, 1972. MR**0323842**, DOI 10.1007/978-1-4612-6398-2 - Victor G. Kac,
*Infinite-dimensional Lie algebras*, 3rd ed., Cambridge University Press, Cambridge, 1990. MR**1104219**, DOI 10.1017/CBO9780511626234 - Kazuya Kawasetsu, Ching Hung Lam, and Xingjun Lin,
*$\Bbb {Z}_2$-orbifold construction associated with $(-1)$-isometry and uniqueness of holomorphic vertex operator algebras of central charge 24*, Proc. Amer. Math. Soc.**146**(2018), no. 5, 1937–1950. MR**3767347**, DOI 10.1090/proc/13881 - 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**, DOI 10.1016/j.jalgebra.2015.07.013 - Ching Hung Lam,
*On the constructions of holomorphic vertex operator algebras of central charge 24*, Comm. Math. Phys.**305**(2011), no. 1, 153–198. MR**2802303**, DOI 10.1007/s00220-011-1212-2 - C. H. Lam and X. Lin,
*A holomorphic vertex operator algebra of central charge $24$ with weight one Lie algebra $F_{4,6}A_{2,2}$*, arXiv:1612.08123 (2016). - Ching Hung Lam and Hiroki Shimakura,
*Quadratic spaces and holomorphic framed vertex operator algebras of central charge 24*, Proc. Lond. Math. Soc. (3)**104**(2012), no. 3, 540–576. MR**2900236**, DOI 10.1112/plms/pdr041 - 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**, DOI 10.1353/ajm.2015.0001 - 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**, DOI 10.1007/s00220-015-2484-8 - 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**, DOI 10.1007/s11005-016-0883-1 - C. H. Lam and H. Shimakura,
*On orbifold constructions associated with the Leech lattice vertex operator algebra*, arXiv:1705.01281 (2017). Math. Proc. Cambridge Philos. Soc. (to appear). - C. H. Lam and H. Shimakura,
*Inertia groups and uniqueness of holomorphic vertex operator algebras*, arXiv:1804.02521 (2018). - Ching Hung Lam and Hiroshi Yamauchi,
*A characterization of the moonshine vertex operator algebra by means of Virasoro frames*, Int. Math. Res. Not. IMRN**1**(2007), Art. ID rnm003, 10. MR**2331901**, DOI 10.1093/imrn/rnm003 - J. Lepowsky,
*Calculus of twisted vertex operators*, Proc. Nat. Acad. Sci. U.S.A.**82**(1985), no. 24, 8295–8299. MR**820716**, DOI 10.1073/pnas.82.24.8295 - Hai Sheng Li,
*Symmetric invariant bilinear forms on vertex operator algebras*, J. Pure Appl. Algebra**96**(1994), no. 3, 279–297. MR**1303287**, DOI 10.1016/0022-4049(94)90104-X - 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**, DOI 10.1090/conm/193/02373 - Masahiko Miyamoto,
*A $\Bbb Z_3$-orbifold theory of lattice vertex operator algebra and $\Bbb Z_3$-orbifold constructions*, Symmetries, integrable systems and representations, Springer Proc. Math. Stat., vol. 40, Springer, Heidelberg, 2013, pp. 319–344. MR**3077690**, DOI 10.1007/978-1-4471-4863-0_{1}3 - Masahiko Miyamoto,
*$C_2$-cofiniteness of cyclic-orbifold models*, Comm. Math. Phys.**335**(2015), no. 3, 1279–1286. MR**3320313**, DOI 10.1007/s00220-014-2252-1 - Masahiko Miyamoto and Kenichiro Tanabe,
*Uniform product of $A_{g,n}(V)$ for an orbifold model $V$ and $G$-twisted Zhu algebra*, J. Algebra**274**(2004), no. 1, 80–96. MR**2040864**, DOI 10.1016/j.jalgebra.2003.11.017 - S. Möller,
*A cyclic orbifold theory for holomorphic vertex operator algebras and applications*, arXiv:1611.09843 (2016). Thesis (Ph.D.)–Technische Universität Darmstadt. - 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**, DOI 10.1016/0550-3213(94)90201-1 - Daisuke Sagaki and Hiroki Shimakura,
*Application of a $\Bbb {Z}_3$-orbifold construction to the lattice vertex operator algebras associated to Niemeier lattices*, Trans. Amer. Math. Soc.**368**(2016), no. 3, 1621–1646. MR**3449220**, DOI 10.1090/tran/6382 - A. N. Schellekens,
*Meromorphic $c=24$ conformal field theories*, Comm. Math. Phys.**153**(1993), no. 1, 159–185. MR**1213740**, DOI 10.1007/BF02099044

## Additional Information

**Ching Hung Lam**- Affiliation: Institute of Mathematics, Academia Sinica, Taipei 10617, Taiwan and National Center for Theoretical Sciences of Taiwan, Taipei, Taiwan
- MR Author ID: 363106
- Email: chlam@math.sinica.edu.tw
**Hiroki Shimakura**- Affiliation: Graduate School of Information Sciences, Tohoku University, Sendai 980-8579, Japan
- MR Author ID: 688879
- Email: shimakura@tohoku.ac.jp
- Received by editor(s): March 5, 2017
- Received by editor(s) in revised form: November 19, 2018
- Published electronically: June 28, 2019
- Additional Notes: The first author was partially supported by MoST Grant Number 104-2115-M-001-004-MY3 of Taiwan

The second author was partially supported by JSPS KAKENHI Grant Numbers 26800001 and 17K05154

The authors were partially supported by the 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”. - © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**372**(2019), 7001-7024 - MSC (2010): Primary 17B69
- DOI: https://doi.org/10.1090/tran/7887
- MathSciNet review: 4024545