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Moonlight顶点算子代数，它是为了解释Monster单有限群(最大零星有限单群)与经典椭圆模函数之间的一种神秘关系(Moonlight猜想)而构造的。我们这个项目的目的是阐明VOAS的模不变性，并将其推广到多变量中。(1)利用伊辛模型，我们发现了月光顶点算子代数的一种新结构，它提供了一种新的多元模不变性。与原来的建设相比，我们的建设是简单的，我们的建设可以适用于许多其他的VoA。(2)证明了C2-条件足以得到模不变性。经典的合理性(模的完全可约性)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)对于任意复中心电荷c，我们构造了一个以Euclidian Jordan代数为Griess代数的无穷多VoA，从而构造了一个Siegel模不变性的候选者。(4)给出了确定VOAS自同构群的序公式。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在全自同构群中得到关于对合集中心子的信息。问题是我们是否能从中确定自同构群。

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.