Integrability in Gromov--Witten theory

我正在和齐藤合作写一本书。我们非常仔细地研究了Frobenius流形理论的背景，这将在当前的提案中使用。例如，我们给出了半简单Frobenius流形的Painleve性质的自包含证明，并找到了Saito高残数对的新公式。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以必要的方式使用。假设我们有一个半简单的Frobenius流形。然后我们就得到了一类等同的傅氏微分方程。相应的解可以看作是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。构造其系数为周期向量的顶点算子。两个顶点算子的乘积涉及一个相位因子，该相位因子可以用一个沿一定的多值解析1形式（称为相位形式）路径的积分来表示。我们证明了对于一个给定的围绕判别式的闭环，其中两个顶点算子是不变的，相形式的相应周期是2\pi i的整数倍。如果我们另外假设Frobenius流形具有积分结构，那么我们的结果表明顶点算子定义了一个特定晶格顶点代数的扭曲表示。

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.