Toward to a generalization of modular invariance of vertex operator algebras into Hilbert type and Siegel type.

顶点算子代数（简称VOA）的概念起源于moonshine顶点算子代数，它是为了解释Monster简单有限群（最大的偶发有限群）与经典椭圆模函数之间的神秘关系（moonshine猜想）而构造的。我们这个项目的目的是澄清voa的模不变性，并在多变量中扩展它。(1)利用Ising模型构造了moonshine顶点算子代数，提供了一种新的多变量模不变性。与原来的施工相比，我们的施工简单，可以应用于许多其他的voc。(2)我们证明了c2条件足以得到模不变性。传统上，合理性（模块的完全可约性）被认为比c2条件更重要，但我们的研究表明，我们不需要合理性。(3)用欧几里得Jordan代数作为Griess代数构造了任意复杂中心电荷c的无限多voa，从而构造了一个候选的Siegel模不变性。(4)建立了一个确定voa自同构群的顺序公式。在我们的构造中，宫本对合起着至关重要的作用，因此我们可以很容易地得到关于对合在全自同构群中的中心化的信息。问题是我们能否从它确定自同构群。

A concept of vertex operator algebras (VOA shortly) has originated from the moonshine vertex operator algebra, which was constructed in order to explain a mysterious relation (the moonshine conjecture) between Monster simple finite group (the largest sporadic finite simple group) and the classical elliptic modular function. Our purpose of this project is to clarify the modular invariance property of VOAs and extend it in multivarables. (1)We found a new construction of the moonshine vertex operator algebra by using Ising models, which offers a new modular invariance in multivariables. Compared with the original construction, our construction is easy and we can apply our construction for many other VOAs. (2)We have shown that C2-condition is enough to get, a modular invariance. Classically, the rationality (completely reducibility of modules) was considered to be more important than C2-condition, but our research has shown that we don't need rationality. (3)We construct an infinitely many VOAs with Euclidian Jordan Algebras as Griess algebras for any complex central charge c. So we construct a candidate of Siegel modular invariance. (4)We found an order formula to determine the automorphism group of VOAs. In our construction, Miyamoto involution plays an essential role and so we can easily get information about the centralizer of Involution in the full automorphism group. The question is if we can determine the automorphism group from it.