Geometry of mirror symmetry and string theory

本研究项目对非紧的Calabi-Yau簇及其周期积分和微分方程得到了如下结果：首先，当给定一个Calabi-Yau簇作为奇点C^2/Z_<μ+1>的c_1 = 0分解时，发现其周期积分非常接近于K. Saito引入的所谓的原始形式;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Gelfand-Kapranov-Zelvinski（GKZ）系统的微分方程组。结果，我们得到了一种用GKZ系统重新表述原始形式理论的方法。已知A_n根系的Weyl群给出本原型的单值性。其次，我们研究了三维奇异性C^3/G（G ∈ SL（3，C）：有限交换群）及其c_1 = 0分解的情形。我们得到了周期积分的精确定义及其在GKZ系统上的刻画。我们观察到，monodromy的周期积分是密切相关的McKay对应连接的表示理论，代数几何。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，在相干层的导出范畴上的Fourier-Mukai变换表现为周期积分的单值性。我们验证了这种“镜像单值关系”在明确的例子。这种单值性已经被精确地作为一种数学猜想，在某些超几何级数的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.