Mathematics

We continue the study of $(q,p)$ Minimal Liouville Gravity with the help of
Douglas string equation. We generalize the results of \cite{Moore:1991ir},
\cite{Belavin:2008kv}, 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 \cite{Goulian:1990qr}, \cite{Zamolodchikov:2005sj},
\cite{Belavin:2006ex}.