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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.

在本研究中，我们主要研究了与模空间几何有关的共形场理论和弦理论。我们得到了以下主要结果：1.模函子的构造：取非阿贝尔共形场理论的张量积和阿贝尔共形场理论的分数次幂构造模函子。这意味着我们可以构造与复单李代数相关的三重不变量。阿贝尔共形场理论的重建及其与曲线和共形块退化的关系研究：利用海森堡代数和顶点算符代数，我们重建了阿贝尔共形场理论。它是非阿贝尔共形场理论的类似构造。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：描述亏格大于或等于1的曲线的模空间上共形块的射影平坦连接的微分方程称为KZB方程。在本研究中，我们给出了KZB方程的新的简单描述，并研究了它的性质。阿贝尔曲面和K3曲面的模空间的研究：Katsura和van der Geer利用Artin-Mazur形式群给出了阿贝尔曲面和K3曲面的模空间的分层。给出了超奇异曲面对应轨迹的循环类的显式刻画。Painleve方程的初始条件空间：Saito和他的团队利用初条件空间是有理曲面的事实，给出了Painleve方程的新的分类方法。超弦理论的研究：江口和他的团队研究了Landau-Ginzburg模型，发现了E.Saito和他的团队研究有理椭圆曲面的镜像对称性的孤立奇点的新描述。