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The gauge formulation of Einstein gravity in AdS$_3$ background leads to aboundary theory that breaks modular symmetry and loses the covariant form. Weexamine the Weyl anomaly for the cylinder and torus manifolds. The divergentterm is the same as the Liouville theory when transforming from the cylinder tothe sphere. The general Weyl transformation on the torus also reproduces theLiouville theory. The Weyl transformation introduces an additional boundaryterm for reproducing the Liouville theory, which allows the use of CFTtechniques to analyze the theory. The torus partition function in this boundarytheory is one-loop exact, and an analytical solution to disjoint two-intervalR\'enyi-2 mutual information can be obtained. We also discuss a first-orderphase transition for the separation length of two intervals, which occurs atthe classical level but is smoothed out by non-perturbative effects captured byaveraging over a modular group in the boundary theory.

Modular Average and Weyl Anomaly in Two-Dimensional Schwarzian Theory