Integrable systems with infinite degrees of freedom

本文主要研究了与模空间几何相关的共形场论和弦论。我们得到了以下主要结果.模函子的构造：取非交换共形场论的张量积和交换共形场论的分数次方，构造了模函子。这意味着我们可以构造新的与复单李代数相关的三重不变量.阿贝尔共形场论的重建及其与曲线退化和共形块关系的研究：利用Heisenberg代数和顶点算子代数重建阿贝尔共形场论。它是非阿贝尔共形场论的一个类似的构造。这阐明了曲线的退化与交换共形块之间的关系. KZB方程的研究：在亏格大于等于1的曲线的模空间上，描述共形块射影平坦连通的微分方程称为KZB方程。本文给出了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：Kateland和货车der Geer使用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圈类的明确描述. Painleve方程的初始条件空间：Saito等人利用初始条件空间是有理曲面这一事实，给出了Painleve方程的新的分类方法.研究超弦理论：江口和他的小组研究了朗道-金兹伯格模型，并发现了E型孤立奇点的新描述。

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.