Landau-Ginzburg Topological Theories in the Framework of GKM and Equivalent Hierarchies

TL;DR

This paper explores the relationship between deformations of monomial solutions in the Generalized Kontsevich Model and $N=2$ Landau-Ginzburg topological theories, revealing their integrable structure and embedding into GKM.

Contribution

It establishes a connection between GKM deformations and Landau-Ginzburg theories, showing how partition functions factorize and relate to integrable hierarchies.

Findings

Partition function factorizes into a quasiclassical and a non-deformed part.

Restores explicit $p-q$ symmetry in minimal string models.

Shows embedding of supersymmetric Landau-Ginzburg models into GKM.

Abstract

We consider the deformations of ``monomial solutions'' to Generalized Kontsevich Model \cite{KMMMZ91a,KMMMZ91b} 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.