$D=2$ 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.