Abstract
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 AdS3/CFT2 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.
Highlights
The present study 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
It is given by the nonlocal Polyakov action [14], which can alternatively be written as the action of a Liouville theory
The primary goal of the present work is to review the emergence of the Polyakov and Liouville actions within AdS3/CFT2 by direct application of the GKPW dictionary (1.3), and to compare it with the Hamiltonian reduction of three-dimensional gravity to flat Liouville theory in the tradition initiated by Coussaert, Henneaux and van Driel [2]
Summary
In any euclidean two-dimensional conformal field theory, the generating functional W [gij]. Note that any nonzero holomorphic function f (z) is associated with a vanishing source μ = 0 as can be seen from (2.21) the stress tensor expectation manifestly depends on f (z) This extra freedom relates to the existence of inequivalent but conformally related flat geometries naturally covered by the complex coordinate w = f (z). Where μ(z, z) is the only gravitational degree of freedom left after gauge fixing This directly follows from the preceding discussion, and amounts to a re-interpretion of the Polyakov action (2.5) as describing dynamical gravity in two dimensions rather than generating the stress tensor correlations of a CFT with the metric acting as a non-dynamical background source.
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