Integrability in Gromov--Witten theory

我正在和斋藤合作写一本书。我们非常仔细地从Frobenius流形理论中计算出背景，该理论将在当前的提案中使用。例如，我们给出了半单Frobenius流形的所谓Painleve性质的一个自包含证明，并找到了Saito的高剩余对的一个新公式。我的主要进展是证明了一项非常重要的技术成果，该成果将在目前的提案中得到必要的使用。假设我们有一个半简单的Frobenius流形。然后我们就有了一族富氏微分方程族。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矢量。构造了以周期向量为系数的顶点算子。两个顶点算符的乘积涉及一个相位因子，该相位因子可以由沿称为相位形式的某一多值解析1形式的路径上的积分来表示。我们证明了对于给定的两个顶点算子沿其不变的判别式的闭合环，对应的相位形式的周期是2\pi的整数倍。如果我们另外假设Frobenius流形具有积分结构，那么我们的结果意味着顶点算子定义了某个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.