Integrals over moduli spaces, ground ring, and four-point function in minimal Liouville gravity

Directly evaluating the correlation functions in 2D minimal gravity requires integrating over the moduli space. For degenerate fields, the higher equations of motion of the Liouville field theory allow converting the integrand to a derivative, which reduces the integral to boundary terms and the so-called curvature contribution. The latter is directly related to the vacuum expectation value of the corresponding ground-ring element. The action of this element on the cohomology related to a generic matter primary field is evaluated directly in terms of the operator product expansions of the degenerate fields. This allows constructing the ground-ring algebra and evaluating the curvature term in the four-point function. We also analyze the operator product expansions of the Liouville "logarithmic primaries" and calculate the relevant logarithmic terms. Based on this, we obtain an explicit expression for the four-point correlation number of one degenerate and three generic matter fields. We compare this integral with the numbers obtained from the matrix models of 2D gravity and discuss some related problems and ambiguities.