NUMBER OF DEGREES OF FREEDOM, DENSITY OF STATES AND TACHYONS IN STRING THEORY AND CFT

We analyze the asymptotic density of states in two-dimensional conformal field theories as a measure of their number of degrees of freedom. In theories where some of the states contribute - 1 to the torus amplitude - "space-time fermions" - it is natural to study the asymptotic behavior of the number of bosons minus fermions. This quantity is relevant when the (CFT is coupled to two-dimensional gravity. The combined gravity and matter theory is stable only when the asymptotic density of states of the world-sheet theory (space-time bosons minus fermions) is that of a quantum mechanical system of one free degree of freedom i.e. one with no net field theoretic degrees of freedom. From a space-time point of view, the asymptotic density of states of the bosons and the fermions can be exponentially large but the difference between them is that of a two-dimensional field theory.