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当前提案中将使用的弗罗贝尼乌斯流形理论的背景。例如，我们给出了半简单 Frobenius 流形的所谓 Painleve 性质的独立证明，并且我们发现了 Saito 高留数配对的新公式。我的主要进展是证明了一项非常重要的技术成果，该成果将在当前提案中得到重要使用。假设我们有一个半单弗罗贝尼乌斯流形。那么我们就有了一定的等单性 Fuchsian 微分方程族。相应的解可以被视为解析超曲面周期积分的推广。这就是为什么我们称它们为周期向量。我们构造顶点算子，其系数是周期向量。两个顶点算子的乘积涉及一个相位因子，该相位因子可以用沿着某个多值解析 1-形式（称为相位形式）的路径的积分来表示。我们证明，对于围绕判别式的给定闭环（两个顶点算子沿该闭环不变），相位形式的相应周期是 2\pi i 的整数倍。如果我们另外假设弗罗贝尼乌斯流形具有积分结构，那么我们的结果意味着顶点算子定义了某个晶格顶点代数的扭曲表示。