Discrete State Moduli of String Theory from c=1 Matrix Model

We propose a new formulation of the space-time interpretation of the $c=1$ matrix model. Our formulation uses the well-known leg-pole factor that relates the matrix model amplitudes to that of the 2-dimensional string theory, but includes fluctuations around the fermi vacuum on {\sl both sides} of the inverted harmonic oscillator potential of the double-scaled model, even when the fluctuations are small and confined entirely within the asymptotes in the phase plane. We argue that including fluctuations on both sides of the potential is essential for a consistent interpretation of the leg-pole transformed theory as a theory of space-time gravity. We reproduce the known results for the string theory tree level scattering amplitudes for flat space and linear dilaton background as a special case. We show that the generic case corresponds to more general space-time backgrounds. In particular, we identify the parameter corresponding to background metric perturbation in string theory (black hole mass) in terms of the matrix model variables. Possible implications of our work for a consistent nonperturbative definition of string theory as well as for quantized gravity and black-hole physics are discussed.