Abstract
We present a revisitation of the Almheiri-Polchinski dilaton gravity model from a two-dimensional (2D) bulk perspective. We describe a peculiar feature of the model, namely the pattern of conformal symmetry breaking using bulk Killing vectors, a covariant definition of mass and the flow between different vacua of the theory. We show that the effect of the symmetry breaking is both the generation of an infrared scale (a mass gap) and to make local the Goldstone modes associated with the asymptotic symmetries of the 2D spacetime. In this way a non vanishing central charge is generated in the dual conformal theory, which accounts for the microscopic entropy of the 2D black hole. The use of covariant mass allows to compare energetically the two different vacua of the theory and to show that at zero temperature the vacuum with a constant dilaton is energetically preferred. We also translate in the bulk language several features of the dual CFT discussed by Maldacena et al. The uplifting of the 2D model to $(d+2)-$dimensional theories exhibiting hyperscaling violation is briefly discussed.
Highlights
Two-dimensional (2D) dilaton gravity models have a long history
If one uses MADM as the mass of the solution the three vacua become completely degenerate. They all have MADM 1⁄4 0, as it is evident from Eq (5), because the constant dilaton vacuum (CDV) is characterized by η0 1⁄4 0, whereas the linear dilaton vacuum (LDV) and the interpolating linear dilaton vacuum (ILDV) have a 1⁄4 0
Using a covariant definition of the mass, bulk Killing vectors and the flow between the two different vacua of the theory characterized, respectively, by a constant and linear varying dilaton, we have given a description of the pattern of conformal symmetry breaking, which is complementary to that emerging in the dual CFT [15]
Summary
Two-dimensional (2D) dilaton gravity models have a long history (see Ref. [1] for a review). [16,17,18] and the flow between a “symmetry-respecting” vacuum and a “symmetry-violating” vacuum In this way we can understand, from a purely 2D bulk gravitational perspective, the generation of an IR scale (the mass gap/scale of conformal symmetry breaking in the conformal correlators) and the appearance of local Goldstone modes. We will show that whereas at nonvanishing temperature the LDV is always energetically preferred, at T 1⁄4 0 the situation is reversed and the CDV is favorite This signals a T 1⁄4 0 quantum phase transition which, from an higher-dimensional perspective, can be thought as a spontaneous dimensional reduction from a d þ 2 to d 1⁄4 2 dimensions spacetime, whose possible role in quantum gravity has been emphasized in Ref. In the Appendix we discuss the uplifting of a 2D model to a (d þ 2)–dimensions model exhibiting hyperscaling violation in the ultraviolet (UV)
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