Some Properties of ( Non ) Critical Strings

We review some recent developments in string theory, emphasizing the importance of vacuum instabilities, their relation to the density of states, and the role of space-time fermions in non-critical string theory. We also discuss the classical dynamics of two dimensional string theory. In the last few years many people have been studying simple models of string theory , which correspond to string propagation in two dimensional space-time (and simple generalizations thereof). Of course, for most applications [1], [2] one needs to consider much more complicated models, however many important issues in string theory are still not understood, and the hope is that the two dimensional theory will serve as a useful toy model, in which some of these issues may be addressed. Due to the low dimension of space-time, the number of degrees of freedom in the theory is vastly smaller than in the twenty six (or ten) dimensional case. The physical on shell states include one field theoretic degree of freedom, the " tachyon " center of mass of the string (which is actually massless in two dimensions), and a discrete infinite set of massive states – the remnants of the tower of oscillator states in D > 2 (where the number of field theoretic degrees of freedom with mass ≤ m 0 diverges exponentially with m 0). Due to the absense of transverse excitations, one might expect the dynamics to be simpler in two dimensions. Indeed, following ideas of [3] [4] it has been shown [5] [6] using large N matrix model techniques, that many properties of two dimensional (2d) strings are exactly calculable to all orders in the topological (genus) expansion. The simplicity of these models is closely related to an underlying free fermion structure [3] – [7]. There are two main potential domains of application of the results of [5], [6]. One is two dimensional (world sheet) quantum gravity. The implications of matrix models for quantum gravity have been recently discussed in [8], [9], [10]. The second application, which will be of main interest to us here, is to critical (unified) string theory. The two are of course closely connected, and we will mention some aspects of this relation as we go along. The approach we will take is to try and understand the matrix model results in the continuum path integral formalism [1], and use this understanding to shed light on the structure of the …