Flat coordinates of topological conformal field theory and solutions of the Gauss–Manin system

Abstract

It was shown many years ago by Dijkgraaf, Velinde, and Verlinde for two-dimensional topological conformal field theory and more recently for the non-critical String theory that some models of these two types can be solved using their connection to the special case of Frobenius manifolds—the so-called Saito Frobenius manifolds connected to a deformed singularity. The crucial point for obtaining an explicit expression for the correlators is finding the flat coordinates of Saito Frobenius manifolds as functions of the parameters of the deformed singularity. We suggest a direct way to find the flat coordinates, using the integral representation for the solutions of the Gauss–Manin system connected to the corresponding Saito Frobenius manifold for the singularity.