Quantization of string theory for $c \leq 1$

TL;DR

This paper explores the canonical quantization of $c \, \leq \, 1$ string theories, comparing it with matrix model results, and discusses solutions, challenges, and connections to mirror symmetry and background independence.

Contribution

It provides an explicit trivial solution for the quantization scheme and examines the potential for non-trivial solutions via a dressing operator, highlighting analytic continuation issues.

Findings

Derived a trivial topological solution for $c \, \leq \, 1$ string theories.

Identified challenges in explicitly computing non-trivial solutions due to analytic continuation problems.

Discussed parallels with mirror symmetry and background independence in string theory.

Abstract

We consider the canonical quantization scheme for $c \leq 1$ ($(p,q)$ -) string theories and compare it with what is known from matrix model approach. We derive explicitly a trivial ($\equiv $ topological) solution. We discuss a ``dressing" operator which in principle allows one to obtain a non-trivial solution, but an explicit computation runs into a problem of analytic continuation of the formal expressions for $\tau $-functions. We discuss also the application of proposed scheme to the case of discrete matrix model and consider some parallels with mirror symmetry and background independence in string theory.