Geometry of mirror symmetry and string theory

镜像对称几何和弦理论

• 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
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540010/
• 关键词: Calabi-Yau manifold; Mirror symmetry; Hypergeometric series; Period integral; Singularity; GKZ system; GKZ方程式系; カラピ・ヤウ多様体; カラビ・ヤウ多様体; 超幾何微分方程式; モノドロミー性質; 消滅サイクル; 導来圏; 連接層; 超幾何関数; 周期積分

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，获得了以下结果有关非紧缩的calabi-yau品种，它们的周期积分和差异化满足它们。 /z_<μ+1>，这是一个周期的积分，是由K.Saito集成的所谓原始形式。 （GKZ）作为ASASULT，我们获得了GKZ系统中原始形式的方法。它太过corresspond corresspond也太过了，周期积分。周期积分的精确定义及其在GKZ系统中的表征。例如，傅立叶在或骨上转换为时期积分的单型。

项目成果

S.Hosono, B.H.Lian, K.Oguiso, S.-T.Yau: "Kummer Structures on a K3 surface-an old question of T.Shioda"Duke Math.J.. 120. 635-687 (2003)

S.Hosono、B.H.Lian、K.Oguiso、S.-T.Yau：“K3 表面上的 Kummer 结构 - T.Shioda 的一个老问题”Duke Math.J.. 120. 635-687 (2003)

S.Hosono: "Counting BPS states via holomorphic anomaly equations"Fields Inst.Commun.. 38. 57-86 (2003)

S.Hosono：“通过全纯异常方程计算 BPS 状态”Fields Inst.Commun.. 38. 57-86 (2003)

Autoequiucilences of a K3 surface and monodrony transformations

K3 曲面的自等价性和单数变换

• 发表时间: 2004
• 期刊: Jour. Alg. Geom. 13
• 作者: S.Hosono;B.H.Lian;K.Oguiso;S.-T.Yau
• 通讯作者: S.-T.Yau

S.Hosono, B.H.Lian, K.Oguiso, S.-T.Yau: "C=2 rational toroidal conformal field theories via Gauss product"Commun.Math.Phys.. 241. 245-286 (2003)

S.Hosono、B.H.Lian、K.Oguiso、S.-T.Yau：“通过高斯积的 C=2 有理环形共形场理论”Commun.Math.Phys.. 241. 245-286 (2003)

Autoequivalences of a K3 surface and monodromy transformations

K3 曲面的自等价性和单性变换

• 发表时间: 2004
• 期刊: Jour.Alg.Geometry. 13
• 作者: S.Hosono;B.H.Lian;K.Oguiso;S.-T.Yau
• 通讯作者: S.-T.Yau