Geometry on String Theory and Moduli spaces

在项目期间，我们对以下主题进行了调查，并获得了以下结果。(i)Calabi-Yau流形-Gopakumar-Vafa猜想的Gromov-Witten不变量和BPS不变量，（ii）同调镜像对称和导出范畴的几何，（iii）Painleve方程初始条件空间的代数几何，（iv）Painleve方程及其推广的Lie理论研究，（v）向量丛模空间的对称性，（vi）不变量理论的新发展。对于（i），我们给出了Calabi-Yau三重BPS不变量的数学定义，并验证了它们与已知的Gopakumar-Vafa猜想下Gromov-Witten不变量计算的相容性。在（iii）中，我们引入了Okamoto-Painleve对的概念，并证明了可以通过Okamoto-Painleve对的形变理论导出Painleve方程。关于Painleve方程的研究（iv），我们的研究小组发展了Lie理论方法和代数几何方法，阐明了Painleve方程、仿射Weyl群的对称性和有理曲面几何之间的关系。

During the period of the project, we have investigated the following subjects and obtained the following results. (i) Gromov-Witten invariants and BPS invariants for Calabi-Yau manifolds-Gopakumar-Vafa conjecture, (ii) Homological Mirror Symmetry and Geometry of derived categories, (iii) Algebraic Geometry of spaces of initial conditions of Painleve equations, (iv) Lie theoretic studies for Painleve equations and its generalizations, (v) Symmetries of the moduli spaces of vector bundles, (vi) New development of invariant theory. As for (i), we proposed a mathematical definition of BPS invariants for Calabi-Yau 3-fold and verified their compatibility for the known calculation of Gromov-Witten invariants assuming Gopakumar-Vafa conjecture. In (iii), we introduce the notion of Okamoto-Painleve pairs and proved that one can derive Painleve equations through deformation theory of Okamoto-Painleve pairs. As for the studies of Painleve equations (iv), our group have been developing Lie theoretic approach and algebro-geometric approach, which clarify the relations among Painleve equations, the symmetry of Affine Weyl groups and the geometry of rational surfaces.