Quantization of string theory for $c \leq 1$

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 $τ$-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.