Riemann-Hilbert problem for Gromov-Witten invariants

每一个半单Frobenius流形都可以看作是一个经典的Riemann-Hilbert问题的解。单调数据由配备对称双线性形式的有限维复向量空间的某个子集确定。这个子集具有根系统的所有性质，除了双线性形式不一定是正定的。这个子集的元素称为反射向量，因为相对于相应的正交超平面的反射生成了Frobenius流形的单向群。问题是对对应于量子上同调的半单Frobenius流形的反射向量进行分类。众所周知，blowup运算保持了量子上同调的半简性。因此，研究爆炸操作下反射矢量的变化是很自然的。另一方面，有一个非常有趣的猜想，它给出了关于派生范畴中例外对象的反射的明确描述。我和我的学生合作开始了一个项目，目标是证明如果猜想对某个流形X成立，那么它也适用于X在有限多个点上的爆破。我们还没有完成这个项目，但我们取得了一个有趣的进展：我们证明了在膨胀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，很快就会提交到一份期刊上。

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.