Nambu-Goto string as a higher-derivative Liouville theory

I propose a generalization of the Liouville action which corresponds to the Nambu-Goto string like the usual Liouville action corresponds to the Polyakov string. The two differ by higher-derivative terms which are negligible classically but revive quantumly. An equivalence with the four-derivative action suggests that the Nambu-Goto string in four dimensions can be described by the (4,3) minimal model analogously to the critical Ising model on a dynamical lattice. While critical indices are the same as in the usual Liouville theory, the domain of applicability becomes broader.