Minimal Liouville gravity correlation numbers from Douglas string equation

We continue the study of (q,p) Minimal Liouville Gravity with the help of Douglas string equation. We generalize the results of http://dx.doi.org/10.1016/0550-3213(91)90548-Chttp://dx.doi.org/10.1088/1751-8113/42/30/304004, where Lee-Yang series (2,2s+1) was studied, to (3,3s+p₀) Minimal Liouville Gravity, where p₀=1,2. We demonstrate that there exist such coordinates τ_m,n on the space of the perturbed Minimal Liouville Gravity theories, in which the partition function of the theory is determined by the Douglas string equation. The coordinates τ_m,n are related in a non-linear fashion to the natural coupling constants λ_m,n of the perturbations of Minimal Lioville Gravity by the physical operators O_m,n. We find this relation from the requirement that the correlation numbers in Minimal Liouville Gravity must satisfy the conformal and fusion selection rules. After fixing this relation we compute three- and four-point correlation numbers when they are not zero. The results are in agreement with the direct calculations in Minimal Liouville Gravity available in the literature http://dx.doi.org/10.1103/PhysRevLett.66.2051http://dx.doi.org/10.1007/s11232-005-0003-3http://dx.doi.org/10.1007/s11232-006-0075-8