Integrable systems with infinite degrees of freedom

具有无限自由度的可积系统

基本信息

批准号：
09304002
负责人：
UENO Kenji
金额：
$ 21.38万
依托单位：
KYOTO UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (A).
财政年份：
1997
资助国家：
日本
起止时间：
1997 至 2000
项目状态：
已结题

项目摘要

In the present research we studied mainly conformal filed theory and string theory related to geometry of moduli spaces. We obtained the following main results.1. Construction of modular functor :Taking the tensor product of non-abelian conformal field theory and a fractional power of abelian conformal field theory we constructed modularfunctor. This implies that we can construct new invariants of threefolds associated with complex simple Lie algebras.2. Reconstruction of abelian conformal field theory and study of relationship with degeneration of curves and conformal blocks :Using Heisenberg algebra and vertex operator algebra we reconstruct abelian conformal field theory. It is a similar construction of non-abelian conformal field theory. This clarifies relationship between degeneration of curves and abelian conformal blocks.3. Study of KZB equation :The differential equations describing projectively flat connection of conformal blocks over the moduli space of curves of genus greater than or equal to one is called KZB equation. In the present study we gave new simple description of KZB equation and studied its properties.4. Study of the moduli spaces of abelian surfaces and K3 surfaces :Katsura and van der Geer gave stratification of the moduli spaces of abelian surfaces and K3 surfaces using the Artin-Mazur formal groups. They gave explicit description of cycle classes of the loci corresponding to supersingular surfaces.5. The spaces of initial conditions of Painleve equations :Saito and his group gave new method of classification of Painleve equations by using the fact the spaces of initial conditions are rational surfaces.6. Study of superstring theory :Eguchi and his group studied Landau-Ginzburg models and found a new description of isolated singularities of type E.Saito and his group studied mirror symmetry of rational elliptic surfaces.

在目前的研究中，我们研究了与模量空间几何相关的保形理论和弦理论。我们获得了以下主要结果。1。模块化函子的构建：采用非阿布尔形成性场理论的张量和我们构建模块化器的Abelian共形场理论的分数。这意味着我们可以构建与复杂的简单谎言代数相关的三倍的新不变式。2。重建亚伯式保形场理论以及与曲线变性和共形块的关系的研究：使用Heisenberg代数和顶点操作员代数，我们重建了Abelian的保形场理论。这是非亚伯共同场理论的类似结构。这种阐明了曲线变性与阿贝尔形式的块之间的关系。3。 KZB方程的研究：在大于或等于一个的曲线的模量曲线上，描述共形块的较大层连接的微分方程称为KZB方程。在本研究中，我们对KZB方程式提供了新的简单描述，并研究了其属性4。研究Abelian表面和K3表面的模量空间：Katsura和van der Geer使用Artin-Mazur正式组对Abelian表面和K3表面的模量分层进行了分层。他们给出了对应于超高表面的基因座的周期类别的明确描述。5。潘leve方程的初始条件的空间：Saito及其小组通过使用最初条件的空间为理性表面，从而提供了新的Painleve方程分类方法。6。 SuperString理论的研究：Eguchi和他的小组研究了Landau-Ginzburg模型，并发现了E.Saito型孤立奇点的新描述，他的小组研究了理性椭圆表面的镜像对称性。