Study of the AdS$_2$/CFT$_1$ Correspondence with the Contribution from the Weyl Anomaly

We consider the Almheiri–Polchinski model of the AdS$$_2$$ back-reaction coupled with a Liouville field, which is necessary for quantum consistency. In this model, the Liouville field is determined classically by a bulk conformal transformation. The boundary time is also reparametrized by this transformation. It is shown that the on-shell action on the boundary for the gravity sector is given by a bulk integral containing the Liouville field. This integral stems from Weyl anomaly and is SL(2,R) invariant. A prescription is given for computing correlation functions of the operators dual to massless scalars. The generating function of the correlation functions of these operators is given by a sum of matter action and the bulk integral containing the Liouville field. The latter integral leads to extra contributions to $$n(\geq 6)$$ point functions.