Non-Rational 2D Quantum Gravity: I. World Sheet CFT

We address the problem of computing the tachyon correlation functions in Liouville gravity with generic (non-rational) matter central charge c<1. We consider two variants of the theory. The first is the conventional one in which the effective matter interaction is given by the two matter screening charges. In the second variant the interaction is defined by the Liouville dressings of the non-trivial vertex operator of zero dimension. This particular deformation, referred to as "diagonal'', is motivated by the comparison with the discrete approach, which is the subject of a subsequent paper. In both theories we determine the ground ring of ghost zero physical operators by computing its OPE action on the tachyons and derive recurrence relations for the tachyon bulk correlation functions. We find 3- and 4-point solutions to these functional equations for various matter spectra. In particular, we find a closed expression for the 4-point function of order operators in the diagonal theory.