We study the context of AdS \(_3\) Einstein gravity and its boundary description in the CFT \(_2\) framework. This framework describes gravity dynamics in AdS \(_3\) using gauge theories. In AdS/CFT correspondence, AdS space is related to a CFT on its boundary. The boundary description does not have modular symmetry, and it is not covariant. The breaking of scale invariance due to quantum corrections in CFT causes the Weyl anomaly. It induces the appearance of the Liouville theory in CFT \(_2\) in general when having the covariance. We apply the Weyl transformation to the boundary theory, and it reproduces the Liouville theory on different manifolds such as torus and cylinder. The Liouville theory characteristics are accurately reproduced with the introduction of an extra boundary term via the transformation. Combining techniques from CFT \(_2\) and the one-loop exact torus partition function provides a way to solve the Rényi-2 mutual information between two disconnected intervals, a measure of entanglement in quantum systems. We consider all bulk manifolds with the AdS \(_3\) boundary condition. The non-perturbative effects in the theory result in smoothing out or altering the behavior of first-order phase transition in the mutual information of two intervals’ separation length.