Topological versus Non--Topological Theories and $p-q$ Duality in $c \le 1$ 2d Gravity Models

TL;DR

This paper explores the non-perturbative formulation of $c \, \leq \, 1$ string theory, presenting a unified field theory approach for topological and non-topological models, and deriving integral representations that respect $p-q$ duality.

Contribution

It introduces a field theory-like formulation for $c \, \leq \, 1$ models and derives integral representations for $(p,q)$ solutions that explicitly demonstrate $p-q$ duality.

Findings

Derived integral representations for $(p,q)$ solutions.

Explicitly demonstrated $p-q$ duality in these models.

Provided exact solutions to the string equations.

Abstract

We discuss the non--perturbative formulation for $c \leq 1$ string theory. The field theory like formulation of topological and non--topological models is presented. The integral representation for arbitrary $(p,q)$ solutions is derived which explicitly obeys $p-q$ duality of these theories. The exact solutions to string equation and various examples are also discussed.