Communications in Number Theory and Physics

Volume 14 (2020)

Number 3

Vertex operator algebras with central charges 164/5 and 236/7

Pages: 487 – 509



Yusuke Arike (Research Field in Education, Law, Economics and the Humanities Area Research and Education Assembly, Kagoshima University, Kagoshima, Japan)

Kiyokazu Nagatomo (Graduate School of Information Science and Technology, Osaka University, Suita, Osaka, Japan)


This paper completes the classification problem which was proposed in the previous paper [1] in which we attempted to characterize the minimal models and families obtained by the tensor products and the simple current extensions of minimal models under the condition that the characters of simple modules satisfy modular differential equations of the third order, and a mild condition on vertex operator algebras. In the previous work, several vertex operator algebras which are not the minimal models appeared. Five elevenths of them are identified to well-known vertex operator algebras which are all vertex operator algebras related with orbifold models of lattice vertex operator algebras. However, we were not able to deny the existence of simple, rational vertex operator algebras of CFT and finite type with central charges either 164/5 or 236/7 under the condition on which we worked in [1]. The characterization of minimal models with at most two simple modules was achieved in the same paper.

The numbers 164/5 and 236/7 were already appeared in the paper of Tuite and Van ([17]) in the different context. However, they were out of reach of our conclusion. Moreover, we solve the conjecture, which was proposed by Hampapura and Mukhi [8], that the $j$‑function is expressed by characters of the minimal models.


vertex operator algebras, modular linear differential equations, quantum dimensions, global dimensions

2010 Mathematics Subject Classification


The first author is supported by JSPS KAKENHI Grant Number 19K03406.

The second author was supported in part by JSPS KAKENHI Grant Number 17K04171, International Center of Theoretical Physics, Italy, and Max Planck Institute for Mathematics, Germany.

Received 11 October 2017

Accepted 6 February 2020

Published 13 July 2020