Journal of the Mathematical Society of Japan

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

Yusuke ARIKE, Kiyokazu NAGATOMO, and Yuichi SAKAI

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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 $c=-114/7, 4/5$.


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

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

Received: 19 April 2016
Revised: 13 March 2017
First available in Project Euclid: 30 August 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 81T40: Two-dimensional field theories, conformal field theories, etc.
Secondary: 11F11: Holomorphic modular forms of integral weight 17B69: Vertex operators; vertex operator algebras and related structures

minimal models modular invariance modular linear differential equations 2D-chiral conformal field theory


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.

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