D=2D=2 String Theory in Target Space/World-Sheet Light-Cone Gauge

$D=2$ free string in linear dilaton background is considered in so called target space/world-sheet light cone gauge $X^{+}=0,~g_{++}=0,~g_{+-}=1$. After gauge fixing the theory has the residual Virasoro and $U(1)$ current symmetries. The physical spectrum related to $SL_2$ invariant vacuum is found to be trivial. We find that the theory has a nontrivial spectrum if the states in different non-equivalent representations ("pictures") of CFT algebra of matter fields are considered.Comment: LATEX, 10 page

Divergences of Discrete States Amplitudes and Effective Lagrangian in 2D String Theory

Scattering amplitudes for discrete states in 2D string theory are considered. Pole divergences of tree-level amplitudes are extracted and residues are interpreted as renormalized amplitudes for discrete states. An effective Lagrangian generating renormalized amplitudes for open string is written and corresponding Ward identities are presented. A relation of this Lagrangian with homotopy Lie algebra is discussed