Riemann-Hilbert problem for Gromov-Witten invariants

每个半单Frobenius流形都可以看作是一个经典Riemann-Hilbert问题的解。单值数据由有限维复向量空间的某个子集确定，该子集具有对称双线性形式。这个子集具有根系的所有性质，除了双线性形式不一定是正定的。这个子集的元素被称为反射向量，因为相对于相应的正交超平面的反射生成弗罗贝纽斯流形的单值群。问题的关键是对基于量子上同调的半单Frobenius流形的反射向量进行分类。已知爆破操作保持量子上同调的半简单性。因此，研究在爆破操作下反射向量如何变化是很自然的。另一方面，有一个非常有趣的猜想，给出了一个明确的描述反射的例外对象在派生类别。我已经开始了一个项目，在合作与我的学生，其目标是证明，如果猜想成立的一些流形X，那么它成立的爆破X在许多点。我们还没有完成这个项目，但我们取得了一个有趣的进展：我们证明了在爆破的例外因子上支持的某些例外对象是反射向量。我们写了一篇论文，现在可以在arXiv上找到，很快就会提交给期刊。

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.