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.

中文翻译：

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具有中心电荷164/5和236/7的顶点算子代数

本文完成了先前文献[1]中提出的分类问题，在该问题中，我们试图描述张量积所获得的最小模型和族，以及在简单模块的特征满足的条件下最小模型的简单当前扩展。三阶模微分方程，以及顶点算子代数上的温和条件。在先前的工作中，出现了几个顶点算子代数，这些代数不是最小模型。它们的五分之五被识别为众所周知的顶点算子代数，这些都是与晶格顶点算子代数的单模型相关的顶点算子代数。但是，我们无法否认简单，在我们[1]中工作的条件下，CFT和有限类型的有理顶点算子代数和中心电荷为164/5或236/7。在同一篇论文中，对具有最多两个简单模块的最小模型进行了表征。在不同的背景下，Tuite和Van（[17]）的论文中已经出现了数字164/5和236/7。但是，它们超出了我们的结论。此外，我们解决了Hampapura和Mukhi [8]提出的猜想，即$ j $函数由最小模型的字符表示。他们超出了我们的结论。此外，我们解决了Hampapura和Mukhi [8]提出的猜想，即$ j $函数由最小模型的字符表示。他们超出了我们的结论。此外，我们解决了Hampapura和Mukhi [8]提出的猜想，即$ j $函数由最小模型的字符表示。