Landau–Ginzburg topological theories in the framework of GKM and equivalent hierarchies

We consider the deformations of “monomial solutions” to Generalized Kontsevich Model [1,2] and establish the relation between the flows generated by these deformations with those of $N=2$ Landau–Ginzburg topological theories. We prove that the partition function of a generic Generalized Kontsevich Model can be presented as a product of some “quasiclassical” factor and non-deformed partition function which depends only on the sum of Miwa transformed and flat times. This result is important for the restoration of explicit $p-q$ symmetry in the interpolation pattern between all the $(p,q)$-minimal string models with $c<1$ and for revealing its integrable structure in $p$-direction, determined by deformations of the potential. It also implies the way in which supersymmetric Landau–Ginzburg models are embedded into the general context of GKM. From the point of view of integrable theory these deformations present a particular case of what is called equivalent hierarchies.