Riemann-Hilbert problem for Gromov-Witten invariants

每一个半简单的Frobenius流形都可以看作是经典黎曼-希尔伯特问题的解。一元数据由具有对称双线性形式的有限维复向量空间的某个子集确定。除了双线性形式不一定是正定的以外，这个子集具有根的所有性质。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产生了Frobenius流形的单群。问题是对量子上同调下的半简单Frobenius流形对应的反射向量进行分类。已知爆破运算保留了量子上同调的半简单性。因此，研究在爆破操作下反射矢量如何变化是很自然的。另一方面，有一个非常有趣的猜想，它给出了一个明确的描述反射的例外对象在派生范畴。我和我的学生合作开始了一个项目，目标是证明如果这个猜想对某个流形X成立，那么它对X在有限多个点上的膨胀也成立。我们还没有完成这个项目，但我们取得了一个有趣的进展：我们证明了在放大的特殊除数上支持的某些特殊物体是反射向量。我们写了一篇论文，现在可以在arXiv上看到，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.