Geometry of mirror symmetry and string theory

在本研究项目中，对于非紧的Calabi-Yau变量，得到了其周期积分和微分方程满足的结果。首先，在奇异点C^2/Z_<μ+1>的c_1 = 0分辨率下给出Calabi-Yau变分时，发现周期积分非常接近于K.Saito引入的所谓原始形式。由于周期积分在某些环面作用下是不变的，我们发现它们满足一个称为Gel'fand-Kapranov-Zelvinski （GKZ）系统的微分方程组。因此，我们获得了一种用GKZ系统重新表述原始形式理论的方法。已知原始形式的单性是由A_n根的Weyl群给出的。在我们的例子中，我们发现这个Weyl群作用扩展到周期积分上相应的仿射Weyl群作用。其次，研究了三维奇点C^3/G （G∧SL(3，C)：有限阿贝尔群）及其c_1 = 0分辨率的情况。我们得到了周期积分的精确定义及其在GKZ系统中的表征。我们观察到周期积分的单性与麦凯对应密切相关，麦凯对应将表示理论与代数几何联系起来。也就是说，在镜像对称下，McKay对应被转化为超越循环理论，例如，在相干束的派生范畴上的Fourier-Mukai变换表现为周期积分的一元。我们用明确的例子验证了这种“镜像单一性关系”。这一单性已被精确地作为一个数学猜想，以某些超几何级数在相关上同调群中取其值。

In this research project, the following results are obtained regarding non-compact Calabi-Yau varieties, their period integrals and differential equations satisfied them.Firstly, when a Calabi-Yau variety is given as the c_1 = 0 resolution of the singularity C^2/Z_<μ+1>, it is found that the period integrals are very close to the so-called primitive forms introduced by K.Saito. Since the period integrals are invariant under certain torus actions, we found that they satisfy a system of differential equations called Gel'fand-Kapranov-Zelvinski (GKZ) system. As a result we obtained a way to rephrase the theory of the primitive forms in terms of the GKZ system. It is known that the monodromy of the primitive forms is given by the Weyl group of the A_n root system. In our case, it is found that this Weyl group action is extended too the corresponding affine Weyl group actions on the period integrals.Secondly, we studied the cases of three dimensional singularity C^3/G (G⊂SL(3,C) : a finite abelian group) and its c_1 = 0 resolutions. We obtained a precise definition of the period integrals and their characterization in terms of the GKZ systems. We observed that the monodromy of the period integrals is closely related to the McKay correspondence which connects the representation theory to algebraic geometry. Namely, under mirror symmetry, the McKay correspondence is transformed to the theory of transcendental cycles, and for example, Fourier-Mukai transforms on the derived category of coherent sheaves appear as the monodromy of the period integrals. We verified this 'mirror monodromy relations' in explicit examples. This monodromy property has been made precise as a mathematical conjecture in terms of certain hypergeometric series taking its values in the relevant cohomology group.