Semi-free field representation and string field theory in 2D gravity

In this paper string field theory for 2D gravity coupled to a c = 1 matter field is described. The authors use the free field representation and take into account the semi-bounded of the Liouville mode in the last step of calculations. The authors call such representation semi-free field representations. It is shown that the Witten string field theory gives a tachyon action which up to some, factor in the interaction term coincides with the Das-Jevicki-Polchinski action. Starting from string field theory with semi-free field representation we calculate string amplitudes. These amplitudes instead of energy conservation δ-functions contain energy denominators which residues coincide with the Veneziano (Shapiro-Virasoro) amplitudes for open (closed) string and exhibit very interesting property: the s, t and u poles become the poles in respect to individual external momentum. These poles are related with higher levels excitations living only with fixed momentum