Geometry of mirror symmetry and string theory

镜像对称几何和弦理论

基本信息

批准号：
15540010
负责人：
HOSONO Shinobu
金额：
$ 1.86万
依托单位：
Graduate School of Mathematical Sciences, University of Tokyo
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2005
项目状态：
已结题

项目摘要

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.

在该研究项目中，获得了以下结果，这些结果是关于非紧缩的calabi-yau品种，它们的周期积分和微分方程所满足它们的。由于该周期积分在某些圆环动作下是不变的，因此我们发现它们满足了一个称为Gel'Fand-Kapranov-Zelvinski（GKZ）系统的微分方程系统。结果，我们获得了一种根据GKZ系统来重塑原始形式理论的方法。众所周知，原始形式的单片由A_N根系的Weyl群给出。在我们的情况下，发现这种Weyl组的作用也扩展了相应的仿生Weyl组在周期积分上的作用。第二，我们研究了三维奇异性C^3/g（g⊂sl（3，c）：有限的abelian群体）及其C_1 = 0分辨率的情况。我们在GKZ系统方面获得了周期积分及其表征的精确定义。我们观察到，该时期积分的单片与将表示理论与代数几何形状联系起来的McKay对应关系密切相关。也就是说，在镜像对称性下，McKay对应关系转化为先验周期的理论，例如，在衍生的相干滑轮类别上的傅立叶 - 木叶变换似乎是周期积分的单片。我们在明确的例子中验证了这种“镜像单片关系”。根据某些超小几幅序列，将其作为数学猜想确定为数学猜想。