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

## TL;DR

This paper explores the AdS$_2$/CFT$_1$ correspondence by incorporating the Weyl anomaly through the Liouville field in the Almheiri-Polchinski model, revealing new insights into boundary correlation functions and quantum consistency.

## Contribution

It introduces a novel boundary on-shell action involving a bulk integral with the Liouville field, stemming from Weyl anomaly, and provides a method to compute correlation functions in this framework.

## Key findings

- The boundary on-shell action includes a Weyl anomaly-induced bulk integral.
- Correlation functions are computed using a new prescription involving the Liouville field.
- Extra contributions to higher-point functions are identified from the bulk integral.

## Abstract

In this paper we will consider the Almheiri-Polchinski model of the AdS$_2$ back reaction coupled with 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.