Geometry on String Theory and Moduli spaces

弦论和模空间的几何

• 批准号: 12440008
• 负责人: SAITO Masa-hiko
• 金额: $ 10.3万
• 依托单位: Kobe University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2000
• 资助国家: 日本
• 起止时间: 2000 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-12440008/
• 关键词: BPS invariants; Gopakumar-Vafa conjecture; Homological Mirror Symmetry; Calabi-Yau manifolds; Gromov-Witten invariants; Okamoto-Painleve pairs; Painleve equations; Moduli spaces of Vector Bundles; グロモフ・ウイッテン不変量; 弦双対性; Verlinde公式; フレアーコホモロジー; ラ

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.

项目期间，我们对以下课题进行了调查，并取得了以下成果。 (i) Calabi-Yau 流形 - Gopakumar-Vafa 猜想的 Gromov-Witten 不变量和 BPS 不变量，(ii) 同调镜像对称和派生类别的几何，(iii) Painleve 方程初始条件空间的代数几何，(iv) Painleve 方程及其推广的李理论研究，(v) 向量模空间的对称性束，（vi）不变理论的新发展。对于(i)，我们提出了 Calabi-Yau 3 倍的 BPS 不变量的数学定义，并验证了它们对于假设 Gopakumar-Vafa 猜想的 Gromov-Witten 不变量的已知计算的兼容性。在(iii)中，我们引入了Okamoto-Painleve对的概念，并证明了可以通过Okamoto-Painleve对的变形理论推导Painleve方程。对于Painleve方程(iv)的研究，我们课题组一直在发展Lie理论方法和代数几何方法，阐明Painleve方程、Affine Weyl群的对称性和有理面几何之间的关系。

项目成果

M.Noumi: "Backlund transformations and the manifolds of Painleve systems"Funkcial. Ekvac.. 45. 237-258 (2002)

M.Noumi：“Backlund 变换和 Painleve 系统的流形”Funkcial。

M. -H. Saito(ed): "Proceedings of the workshop, Algebraic Geometry and Integrable Systems related to String Theory in: RIMS, Kokyuroku, No. 1232"RIMS. 173 (2001)

M.-H。

M.-H.Saito (編集): "Proceedings of the workshop, Algebraic Geometry and Integrable Systems related to String Theory, (RIMS, Kokyuroku, No. 1232)"RIMS. 173 (2001)

M.-H. Saito（编辑）：“研讨会论文集，与弦理论相关的代数几何和可积系统，（RIMS，Kokyuroku，No. 1232）”RIMS 173（2001）。

K.Fukaya: "Lagrangian Intersection Floer Theory, -Anomaly and Obstruction-"AMS. (to appear). (2001)

K.Fukaya：“拉格朗日交点弗洛尔理论，-异常与障碍-”AMS。

M.-H.Saito: "Painleve equations and deformations of rational surfaces with rational double points"Proceedings of the Nagoya Int. Workshop, Physics and combinatorics. 320-365 (2000)

M.-H.Saito：“Painleve 方程和有理双点有理曲面的变形”名古屋国际学院学报。