On pq-duality and explicit solutions in $c \le 1$ $2d$ gravity models

TL;DR

This paper explores integral representations of solutions in nonperturbative $c \,\leq\, 1$ 2D gravity models, revealing $p-q$ duality and providing explicit solutions for various models.

Contribution

It derives an integral model for arbitrary $(p,q)$ minimal models coupled to gravity, explicitly demonstrating $p-q$ duality and solving string equations.

Findings

Demonstrates $p-q$ duality in minimal models

Provides explicit integral solutions for various models

Analyzes exact solutions to string equations and reduction conditions

Abstract

We study the integral representation for the exact solution to nonperturbative $c le 1$ string theory. A generic solution is determined by two functions $W(x)$ and $Q(x)$ which behaive at infinity like $x^p$ and $x^q$ respectively. The integral model for arbitrary $(p,q)$ models is derived which explicitely demonstrates $p-q$ duality of minimal models coupled to gravity. We discuss also the exact solutions to string equation and reduction condition and present several explicit examples.