NUMBER OF STATES OF 2-DIMENSIONAL CRITICAL STRING THEORY

We discuss string theory vacua which have the wrong number of spacetim e dimensions, and give a crude argument that vacua with more than four large dimensions are improbable. We then turn to two-dimensional vacu a, which naively appear to violate Bekenstein's entropy principle. A c lassical analysis shows that the naive perturbative counting of states is unjustified. All excited states of the system have strong coupling singularities which prevent us from concluding that they really exist . A speculative interpretation of the classical solutions suggests onl y a finite number of states will be found in regions bounded by a fini te area. We also argue that the vacuum degeneracy of two-dimensional c lassical string theory is removed in quantum mechanics. The system app ears to be in a Kosterlitz-Thouless phase. This leads to the conclusio n that it is also improbable to have only two large spacetime dimensio ns in string theory. However, we note that, unlike our argument for hi gh dimensions, our conclusions about the ground state have neglected t wo-dimensional quantum gravitational effects, and are at best incomple te.