Equivalence between light-cone and conformal gauge in the two-dimensional string

Aiming towards understanding the question of the discrete states in the light-cone gauge in the theory of two-dimensional strings with a linear background charge term, we study the path-integral formulation of the theory. In particular, by gauge fixing Polyakov's path-integral expression for the 2-d strings, we show that the light-cone gauge-fixed generating functional is the same as the conformal gauge-fixed one and is critical for the same value of the background charge (Q=2 $\sqrt 2 $). Since the equivalence is shown at the generating functional level, one expects that the spectra of the two theories are the same. The zero modes of the ratio of the determinants are briefly analyzed and it is shown that only the constant mode survives in this formulation. This is an indication that the discrete states may lie in these zero modes. This result is not particular to the light-cone gauge, but it holds for the conformal gauge as well.