## Journal of the Mathematical Society of Japan

### Vertex operator algebras, minimal models, and modular linear differential equations of order 4

#### Abstract

In this paper we classify vertex operator algebras with three conditions which arise from Virasoro minimal models: (A) the central charge and conformal weights are rational numbers, (B) the space spanned by characters of all simple modules of a vertex operator algebra coincides with the space of solutions of a modular linear differential equation of order $4$ and (C) the dimensions of first three weight subspaces of a VOA are $1, 0$ and $1$, respectively. It is shown that vertex operator algebras which we concern have central charges $c=-46/3, -3/5, -114/7, 4/5$, and are isomorphic to minimal models for $c=-46/3, -3/5$ and ${\mathbb{Z}}_2$-graded simple current extensions of minimal models for $c=-114/7, 4/5$.

#### Note

The first author was supported by JSPS KAKENHI Grant Number 25800003. The second author was partially supported by JSPS KAKENHI Grant Number 17K04171, International Center of Theoretical Physics, Italy, and Max Planck institute for Mathematics, Germany. The third author was partially supported by JSPS KAKENHI Grant Numbers 15K13428 and 16H06336.

#### Article information

Source
J. Math. Soc. Japan, Volume 70, Number 4 (2018), 1347-1373.

Dates
Revised: 13 March 2017
First available in Project Euclid: 30 August 2018

https://projecteuclid.org/euclid.jmsj/1535616221

Digital Object Identifier
doi:10.2969/jmsj/74957495

Mathematical Reviews number (MathSciNet)
MR3868210

Zentralblatt MATH identifier
07009705

#### Citation

ARIKE, Yusuke; NAGATOMO, Kiyokazu; SAKAI, Yuichi. Vertex operator algebras, minimal models, and modular linear differential equations of order 4. J. Math. Soc. Japan 70 (2018), no. 4, 1347--1373. doi:10.2969/jmsj/74957495. https://projecteuclid.org/euclid.jmsj/1535616221

#### References

• Y. Arike, M. Kaneko, K. Nagatomo and Y. Sakai, Affine vertex operator algebras and modular linear differential equations, Lett. Math. Phys., 106 (2016), 693–718.
• Y. Arike, K. Nagatomo and Y. Sakai, Characterization of the simple Virasoro vertex operator algebras with $2$ and $3$-dimensional space of characters, In: Lie algebras, vertex operator algebras, and related topics, Contemp. Math., 695, Amer. Math. Soc., Providence, 2017, 175–204.
• Y. Arike and K. Nagatomo, Central charges $164/5$ and $236/7$, preprint.
• C. Dong, H. Li and G. Mason, Modular-invariance of trace functions in orbifold theory and generalized Moonshine, Commun. Math. Phys., 214 (2000), 1–56.
• C. Dong and G. Mason, Shifted vertex operator algebras, Math. Proc. Camb. Phil. Soc., 141 (2006), 67–80.
• B. Feigin and D. Fuchs, Representations of the Virasoro algebra, In: Representation theory of Lie groups and related topics, Adv. Stud. Contemp. Math., 7, Gordon and Breach, New York, 1990, 465–554.
• B. Feigin and D. Fuchs, Verma modules over the Virasoro algebra, In: Topology (Leningrad, 1982), Lect. Notes in Math., 1984, Springer, Berlin, 1982, 230–245.
• I. Frenkel, Y. Huang and J. Lepowsky, On axiomatic approaches to vertex operator algebras and modules, Mem. Amer. Math. Soc., 104 (1993), no. 494.
• I. Frenkel and Y. Zhu, Vertex operator algebras associated to representations of affine and Virasoro algebras, Duke Math. J., 66 (1992), 123–168.
• H. R. Hampapura and S. Mukhi, Two-dimensional RCFT's without Kac–Moody symmetry, J. High Energ. Phys., 2016 (2016), No. 7, Article 138.
• E. L. Ince, Ordinary differential equations, Dover Publications, New York, 1944.
• K. Iohara and Y. Koga, Representation theory of the Virasoro algebra, Springer Monographs in Math., Springer-Verlag London, Ltd., London, 2011.
• M. Kaneko and M. Koike, On extremal quasimodular forms, Kyushu J. Math., 60 (2006), 457–470.
• C. H. Lam and H. Yamauchi, On the structure of framed vertex operator algebras and their pointwise frame stabilizers, Commun. Math. Phys., 277 (2008), 237–285.
• J. Lepowsky and H.-S. Li, Introduction to vertex operator algebras and their representations, Prog. Math., 227, Birkhäuser Boston, Inc., Boston, 2004.
• C. Marks, Irreducible vector-valued modular forms of dimension less than six, Illinois J. Math., 55 (2011), 1267–1297.
• G. Mason, Vector-valued modular forms and linear differential operators, Int. J. Number Theory, 3 (2007), 377–390.
• A. Matsuo and K. Nagatomo, Axioms for a vertex algebra and the locality of quantum fields, MSJ Memoirs, 4, Mathematical Society of Japan, 1999.
• S. D. Mathur, S. Mukhi and A. Sen, On the classification of rational conformal field theories, Phys. Letter B, 213 (1988), 303–308.
• A. Milas, Ramanujan's “Lost Notebook" and the Virasoro algebra, Commun. Math. Phys., 251 (2004), 567–588.
• A. Milas, Virasoro algebra, Dedekind $\eta$-function and specialized MacDonald identities, Transformation Groups, 9 (2004), 273–288.
• K. Nagatomo and Y. Sakai, Characterization of minimal models and 6th order modular linear differential equations, in preparation.
• Y. Sakai, The generalized modular linear differential equations, in preparation.
• M. P. Tuite and H. D. Van, On exceptional vertex operator (super) algebras, In: Developments and retrospectives in Lie theory, Dev. Math., 38, Springer, Cham, 2014, 351–385.
• W. Wang, Rationality of Virasoro vertex operator algebras, Internat. Math. Res. Notices, 1993 (1993), 197–211.
• H. Yamauchi, Module categories of simple current extensions of vertex operator algebras, J. Pure Appl. Algebra, 189 (2004), 315–328.
• Y. Zhu, Modular invariance of characters of vertex operator algebras, J. Amer. Math. Soc., 9 (1996), 237–302.