模朱代数理论与维拉索罗顶点代数
Modular Zhu algebra theory and Virasoro vertex algebras
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中文总结 AI 辅助
该研究利用高阶朱代数和模转换代数发展顶点代数表示理论，给出正特征域上莫比乌斯顶点代数合理性条件，证明特定离散系列维拉索罗顶点算子代数的合理性，还研究了特征$p>2$时简单商的全纯性情况。
中文摘要 AI 辅助
我们利用高阶朱代数和模转换代数发展了任意环上顶点代数的表示理论。给出了正特征域上莫比乌斯顶点代数合理性的几个等价条件，推广了达米奥利尼、吉布尼和克拉申的工作。作为应用，证明了对于特征0的域$\mathbb{F}$及互质整数$r,s>1$，离散系列维拉索罗顶点算子代数$L_{\operatorname{Vir}}(c_{r,s},0)_{\mathbb{F}}$是合理的。还研究了特征$p>2$时的情况，如$p = 3,5$以及$p = 7$时的相关结果。
英文摘要
We develop the representation theory of vertex algebras over arbitrary commutative rings as part of a relative theory, suitable for various applications such as the study of modular forms, modular tensor categories, and sheaves of coinvariants and conformal blocks. Towards this end, we construct Zhu algebras (and higher analogues) and mode transition algebras over arbitrary rings via the universal enveloping algebra. We prove that these constructions are compatible with base change and give rationality criteria for Möbius vertex algebras over arbitrary fields satisfying mild assumptions. We apply this framework to study Virasoro vertex operator algebras over arbitrary fields. Using integral forms and base change, we extend the rationality of the discrete series Virasoro VOAs from $\mathbb{C}$ to arbitrary fields of characteristic zero. In positive characteristic, the Virasoro theory exhibits surprising new phenomena: the so-called restricted Virasoro VOA is $C_2$-cofinite and has a semisimple Zhu algebra, but it is not rational. We also show that this VOA is not simple because it admits nontrivial singular vectors in positive characteristic for every central charge. We calculate the singular vector of degree $2p$ in characteristics $p=5,7$ using Mathematica.
