  The generating functional of stress tensor correlation functions in
two-dimensional conformal field theory is the nonlocal Polyakov action, or
equivalently, the Liouville or Alekseev-Shatashvili action. I review its
holographic derivation within the AdS$_3$/CFT$_2$ correspondence, both in
metric and Chern-Simons formulations. I also provide a detailed comparison with
the well-known Hamiltonian reduction of three-dimensional gravity to a flat
Liouville theory, and conclude that the two results are unrelated. In
particular, the flat Liouville action is still off-shell with respect to bulk
equations of motion, and simply vanishes in case the latter are imposed. The
present study also suggests an interesting re-interpretation of the computation
of black hole spectral statistics recently performed by Cotler and Jensen as
that of an explicit averaging of the partition function over the boundary
source geometry, thereby providing potential justification for its agreement
with the predictions of a random matrix ensemble.
