Integrability in Gromov--Witten theory

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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.