Subcritical closed string field theory in less than 26 dimensions

In this paper we construct the second-quantized action for subcritical closed string field theory with zero cosmological constant in dimensions 2≤D<26, generalizing the nonpolynomial closed string field theory action proposed by the author and the Kyoto and MIT groups for D=26. The proof of gauge invariance is considerably complicated by the presence of the Liouville field φ and the nonpolynomial nature of the action. However, we explicitly show that the polyhedral vertex functions obey BRST invariance to all orders. By point-splitting methods we calculate the anomaly contribution due to the Liouville field, and show in detail that it cancels only if D-26+1+3Q2=0, in both the bosonized and unbosonized polyhedral vertex functions. We also show explicitly that the four-point function generated by this action reproduces the shifted Shapiro-Virasoro ampltiude found from c=1 matrix models and Liouville theory in two dimensions. This calculation is nontrivial because the conformal transformation from the z to the ρ plane requires rather complicated third elliptic integrals and is hence much more involved than the ones found in the usual polynomial theories