Riemann-Hilbert problem for Gromov-Witten invariants

Gromov-Witten 不变量的黎曼-希尔伯特问题

基本信息

批准号：
17K05193
负责人：
MILANOV Todor
金额：
$ 2.25万
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：
2017
资助国家：
日本
起止时间：
2017-04-01 至 2024-03-31
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-17K05193/
关键词：
Frobenius manifolds quantum cohomology vertex operators frobenius manifolds matrix model FJRW invariants matrix models integrable hierarchies Gromov-Witten invariants period integrals Frobenius structures Frobenius Manifolds

项目摘要

Every semi-simple Frobenius manifold can be viewed as a solution of a classical Riemann-Hilbert problem. The monodromy data is determined by a certain subset of a finite dimensional complex vector space equipped with a symmetric bilinear form. This subset has all the properties of a root system except that the bilinear form is not necessarily positive definite. The elements of this subset are called reflection vectors because the reflections with respect to the corresponding orthogonal hyperplanes generate the monodromy group of the Frobenius manifold. The problem is to classify the reflection vectors corresponding the semi-simple Frobenius manifolds that underly quantum cohomology. It is known that the blowup operation preserves semi-simplicity of quantum cohomology. Therefore, it is natural to investigate how does the reflection vectors change under the blow up operation. On the other hand, there is a very interesting conjecture that gives an explicit description of the reflections in terms of exceptional objects in the derived category. I have started a project in collaboration with my student in which the goal is to prove that if the conjecture holds for some manifold X, then it holds for the blowup of X at finitely many points. We did not complete the project yet but we made an interesting progress: we proved that certain exceptional objects supported on the exceptional divisor of the blowup are reflection vectors. We wrote a paper which is now available on the arXiv and it will be submitted to a journal soon.

每个半简单的Frobenius歧管都可以看作是经典的Riemann-Hilbert问题的解决方案。单肌数据由配备对称双线性形式的有限尺寸复合载体空间的一定子集确定。该子集具有根系的所有属性，但双线性形式不一定是正定的。该子集的元素称为反射载体，因为相对于相应的正交超平面的反射会产生Frobenius歧管的单型组。问题是要对反射矢量进行分类，这些反射矢量与量子共同体相关的半简单frobenius歧管。众所周知，爆炸操作保留了量子同谋的半动作。因此，很自然地研究反射向量如何在爆破操作下发生变化。另一方面，有一个非常有趣的猜想，可以根据派生类别中的特殊对象对反射进行明确描述。我已经与我的学生合作启动了一个项目，其目标是证明，如果猜想对某些歧视X持有，那么它将在有限的许多方面爆炸。我们还没有完成该项目，但我们取得了一个有趣的进步：我们证明了爆炸的特殊除数支持的某些特殊对象是反射矢量。我们撰写了一篇论文，该论文现在可以在Arxiv上使用，并将很快提交日记。

项目成果

期刊论文数量（0）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

Primitive forms and Frobenius structures on the Hurwiotz spaces

Hurwiotz 空间上的原始形式和 Frobenius 结构

DOI：
发表时间：
2019
期刊：
影响因子：
0
作者：
Milanov Todor;Tonita Valentin;Todor Milanov;Todor Milanov;Todor Milanov;Todor Milanov;Todor Milanov;Todor Milanov;Todor Milanov;Todor Milanov
通讯作者：
Todor Milanov

The 2-Component BKP Grassmannian and Simple Singularities of Type D

DOI：
10.1093/imrn/rnz325
发表时间：
2018-04
期刊：
International Mathematics Research Notices
影响因子：
1
作者：
Jipeng Cheng;T. Milanov
通讯作者：
Jipeng Cheng;T. Milanov

Gromov--Witten invariants of Fano orbifold lines of type D and integrable hierarchies

Gromov--D 型法诺环折线和可积层次的维滕不变量

DOI：
发表时间：
2019
期刊：
影响因子：
0
作者：
Milanov Todor;Tonita Valentin;Todor Milanov;Todor Milanov;Todor Milanov;Todor Milanov;Todor Milanov;Todor Milanov;Todor Milanov;Todor Milanov;Todor Milanov;Todor Milanov
通讯作者：
Todor Milanov

Fano orbifold lines of type D and integrable hierarchies

D 型 Fano Orbifold 线和可积层次结构