Generalization of A.M.Polyakov's 2d gravity with the simplest non-invariant states

In the 1980s A.M.Polyakov, V.G.Knizhnik and A.B.Zamolodchikov developed a model of two-dimensional quantum gravity. The starting point was the formulation by A.Polyakov of Liouville theory of one scalar field as a theory of 2d gravity induced by conformal matter. Further development of the Polyakov-Liouville theory led to numerous results such as exact correlation functions and applications to matrix models. However, in those investigations, QFT was always considered over Poincaré invariant states. The purpose of our work is to modify Polyakov-Liouville theory of 2d gravity for the simplest cases of Poincaré-non-invariant states. Namely, the insertion of primary state of conformal field theory and the thermal state of conformal matter will be discussed.