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

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.

14 pages, FIAN/TD-7/92 & ITEP-M-5/92, improved version

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• Towards unified theory of $2d$ gravity
• Generalized Kontsevich Model Versus Toda Hierarchy and Discrete Matrix Models
• SDiff(2) KP hierarchy
• Unification of All String Models with $c<1$

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