Moduli spaces and symmetries in string theory

以“通过椭圆曲面接近Seiberg-Witten理论”为主题，清水与名古屋大学的a .土屋、日本大学的A.Kato等人进行了联合研究。斋藤（东京大学）、阿瓦塔（东京大学）等其中，构造了标记有理椭圆曲面的模空间及其周期映射，研究了椭圆曲面的反演问题、与简单椭圆奇点的关系，构造了椭圆Lie代数和椭圆Weyl群，实现了Seiberg-Witten可积系统。2002年，为了扩大我们的研究和寻找经典周期的量子类比，我们引入了椭圆K3曲面的周期、模和相干束的派生范畴。sueno和Shimizu研究了以阿贝尔电流为对称的CFT的共形块的退化行为，并出版了一本关于模理论的书。上野用CFT构造了一个模函子。Saito从复杂结构变形理论的角度重新考虑了painlevel方程的初值空间，引入了okamoto - painlevel对的概念，对其进行了分类，并对微分方程进行了改写。suzuki研究了弱距离正则有向图和带连杆不变量的II型矩阵。森本将球上的解析泛函表示为热函数的初值。Yamakawa研究了可解李群的余维1局部自由作用

On the theme of Seiberg-Witten theory approached through elliptic surfaces, Shimizu did the joint research with A.Tsuchiya (Nagoya Univ.), A.Kato (Unlv. Tokyo) Y.Saito (Univ. Tokyo), H.Awata (Univ. Tokyo) etc. Among others, we constructed the moduli space of marked rational elliptic surfaces and their period map, studied the inversion problem, the relation with simply elliptic singularities, constructed the elliptic Lie algebra and the elliptic Weyl group, and carried out the realization of a Seiberg-Witten integrable system. In 2002, we introduced the periods of elliptic K3 surfaces, the moduli and the derived category of coherent sheaves in order to enlarge our study and to look for the quantum analog of classical periodsUeno and Shimizu studied the degeneration behavior of the conformal blocks of the CFT with abelian currents as symmetries, and published a book on moduli theory. Ueno constructed a modular functor using CFT. From the viewpoint of deformation theory of complex structure, Saito reconsidered the space of initial values of the Painleve equations, introduced the notion of Okamoto-Painleve pairs, classified them and rewrited the differential equationsSuzuki studied weakly distance-regular digraphs and type II matrices associated with link invariants. Morimoto expressed the analytic functionals on the sphere as initial values of the heat functions. Yamakawa studied codimension one locally free action of solvable Lie groups