Bulk Correlation Functions in 2d Quantum Gravity

We compute the bulk three- and four-point tachyon correlators in the 2d Liouville gravity with a nonrational matter central charge $c < 1$, using and comparing two approaches. The continuous CFT approach exploits the action on the tachyons of the ground-ring generators deformed with Liouville and matter screening charges. We derive a general formula for the matter three-point OPE structure constants as a by-product. The discrete formulation of the theory is a generalization of the $ADE$ string theories, in which the target space is the semi-infinite chain of points.