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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.

本文主要研究了模空间几何中的共形场理论和弦理论。我们得到了以下主要结果：模函子的构造：利用非阿贝尔共形场论的张量积与阿贝尔共形场论的分数次幂构造了模函子。这意味着我们可以构造与复简单李代数相关的新的三倍不变量。阿贝尔共形场论的重构及其与曲线和共形块退化关系的研究：利用Heisenberg代数和顶点算子代数重构了阿贝尔共形场论。它是非阿贝尔共形场论的一个类似构造。这阐明了曲线退化与阿贝尔共形块之间的关系。KZB方程的研究：描述保形块在大于等于1的曲线模空间上的投影平面连接的微分方程称为KZB方程。本文对KZB方程给出了新的简单描述，并对其性质进行了研究。阿贝尔曲面和K3曲面的模空间研究：Katsura和van der Geer利用Artin-Mazur形式群给出了阿贝尔曲面和K3曲面的模空间分层。他们给出了与超奇异曲面相对应的轨迹的环类的明确描述。5 . Painleve方程的初始条件空间：Saito等人利用Painleve方程的初始条件空间是有理曲面这一事实，给出了Painleve方程分类的新方法。超弦理论的研究：Eguchi和他的团队研究了Landau-Ginzburg模型，并发现了E.Saito和他的团队研究了有理椭圆表面的镜像对称性。