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 [1, 2], where Lee–Yang series $(2, 2s + 1)$ was studied, to $(3, 3s + p_0)$ Minimal Liouville Gravity, where $p_0 = 1, 2$. We demonstrate that there exist such coordinates $\tau_{m,n}$ 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 $\tau_{m,n}$ are related in a non-linear fashion to the natural coupling constants $\lambda_{m,n}$ 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 [3–5].