Two-dimensional quantum gravity and quasiclassical integrable hierarchies**Based on talk presented at the conference Liouville Field Theory and Statistical Models (dedicated to the memory of Alexei B Zamolodchikov) in Moscow in June 2008.

Abstract

The main results for the two-dimensional quantum gravity, conjectured from the matrix model or integrable approach, are presented in the form to be compared with the worldsheet or Liouville approach. In a spherical limit, the integrable side for minimal string theories is completely formulated using simple manipulations with two polynomials, based on residue formulae from quasiclassical hierarchies. Explicit computations for particular models are performed and certain delicate issues of nontrivial relations among them are discussed. They concern the connections between different theories, obtained as expansions of basically the same stringy solution to dispersionless Kadomtsev–Petviashvili hierarchy in different backgrounds, characterized by nonvanishing background values of different times, being the simplest known example of the change of the quantum numbers of physical observables, when moving to a different point in the moduli space of the theory.