Flat coordinates for Saito Frobenius manifolds and String theory

Abstract

It was shown in \cite{DVV} for $2d$ topological Conformal field theory (TCFT) \cite{EY,W} and more recently in \cite{BSZ}-\cite{BB2} for the non-critical String theory \cite{P}-\cite{BAlZ} that a number of models of these two types can be exactly solved using their connection with the Frobenius manifold (FM) structure introduced by Dubrovin\cite{Dub}. More precisely these models are connected with a special case of FMs, so called Saito Frobenius manifolds (SFM)\cite{Saito} (originally called Flat structure together with the Flat coordinate system), which arise on the space of the versal deformations of the isolated Singularities after choosing of a suitabe so-called Primitive form, and which also arises on the quotient spaces by reflection groups. In this paper we explore the connection of the models of TCFT and non-critical String theory with SFM. The crucial point for obtaining an explicit expression for the correlators is finding the flat coordinates of SFMs 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 Gauss-Manin system connected with the corresponding SFM for a simple singularity. Also, we address the possible generalization of our approach for the models investigated in \cite{Gep} which are ${SU(N)_k}/({SU(N-1)_{k+1} \times U(1)})$ Kazama-Suzuki theories \cite{KS}.