Integrability in Gromov--Witten theory

我正在和K合作写一本书。齐藤我们非常仔细地从弗罗贝纽斯流形的理论背景，将用于目前的建议。例如，我们给出了半单Frobenius流形的所谓Painleve性质的自包含证明，并且我们发现了Saito的高残数配对的新公式。我的主要进展是证明了一个非常重要的技术成果，它将在目前的提案中以重要的方式使用。假设我们有一个半单Frobenius流形。这样我们就得到了一类Fuchsian微分方程的等单径族。相应的解可以看作是解析超曲面的周期积分的推广。这就是为什么我们称之为周期向量。我们构造了顶点算子，其系数是周期向量。两个顶点算子的乘积包含一个相位因子，该相位因子可以表示为沿着称为相位形式的某个多值解析1-形式的路径的积分沿着。我们证明了，对于一个给定的闭环周围的判别沿着的两个顶点操作是不变的，相应的周期的相位形式是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.