On two-dimensional quantum gravity and quasiclassical integrable hierarchies

TL;DR

This paper explores two-dimensional quantum gravity through integrable hierarchies, providing explicit computations and analyzing the relations among different string theories in various backgrounds.

Contribution

It formulates the integrable approach to minimal string theories using polynomial manipulations and residue formulas, connecting different models via the dispersionless KP hierarchy.

Findings

Explicit computations for specific models.

Relations among theories as expansions of a common solution.

Connections between different backgrounds and quantum numbers.

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 world-sheet or Liouville approach. In spherical limit the integrable side for minimal string theories is completely formulated using simple manipulations with two polynomials, based on residue formulas 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 KP hierarchy in different backgrounds, characterized by nonvanishing background values of different times, being the simplest known example of change of the quantum numbers of physical observables, when moving to a different point in the moduli space of the theory.