Integrability in Gromov--Witten theory

格罗莫夫--维滕理论中的可积性

基本信息

批准号：
22K03265
负责人：
MILANOV Todor
金额：
$ 2.58万
依托单位：
The 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：
2022
资助国家：
日本
起止时间：
2022-04-01 至 2027-03-31
项目状态：
未结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-22K03265/
关键词：
Frobenius manifolds quantum cohomology vertex operators Gromov-Witten invariants

项目摘要

I am writing a book in collaboration with K. Saito. We worked out very carefully the background from the theory of Frobenius manifolds that will be used in the current proposal. For example, we gave a self-contained proof of the so-called Painleve property of a semi-simple Frobenius manifold and we found a new formula for Saito's higher-residue pairing. My main progress is in proving a very important technical result which will be used in an essential way in the current proposal. Suppose that we have a semi-simple Frobenius manifold. Then we have a certain isomonodromic family of Fuchsian differential equations. The corresponding solutions can be viewed as generalization of the period integrals of analytic hypersurfaces. That is why we call them period vectors. We construct vertex operators whose coefficients are the period vectors. The product of two vertex operators involves a phase factor that can be represented by an integral along the path of a certain multivalued analytic 1-form called the phase form. We prove that for a given closed loop around the discriminant along which the two vertex operators are invariant, the corresponding periods of the phase form are integer multiples of 2\pi i. If we assume in addition that the Frobenius manifold has an integral structure, then our result implies that the vertex operators define a twisted representation of a certain lattice vertex algebra.

我正在与K. Saito合作写一本书。我们非常仔细地研究了Frobenius歧管理论的背景，该理论将在当前的提案中使用。例如，我们给出了半简单的Frobenius歧管的所谓的painleve属性的独立证明，我们找到了锡托高级救助配对的新公式。我的主要进步是证明一个非常重要的技术结果，该结果将在当前建议中以必不可少的方式使用。假设我们有一个半简单的frobenius歧管。然后，我们拥有一定的紫红色微分方程的等词性家族。相应的解决方案可以看作是分析性超曲面周期积分的概括。这就是为什么我们称它们为时期向量。我们构建了其系数为时期向量的顶点操作员。两个顶点算子的乘积涉及一个相位因子，该相位因子可以由沿特定多值分析1形式的路径的积分表示，称为相形式。我们证明，对于给定的闭合环，两个顶点操作员是不变的，相应的时期是2 \ pi i的整数倍数。如果我们还假设Frobenius歧管具有积分结构，那么我们的结果意味着顶点操作员定义了某个晶格顶点代数的扭曲表示。