Abstract
We construct the holographic dictionary for both running and constant dilaton solutions of the two dimensional Einstein-Maxwell-Dilaton theory that is obtained by a circle reduction from Einstein-Hilbert gravity with negative cosmological constant in three dimensions. This specific model ensures that the dual theory has a well defined ultraviolet completion in terms of a two dimensional conformal field theory, but our results apply qualitatively to a wider class of two dimensional dilaton gravity theories. For each type of solutions we perform holographic renormalization, compute the exact renormalized one-point functions in the presence of arbitrary sources, and derive the asymptotic symmetries and the corresponding conserved charges. In both cases we find that the scalar operator dual to the dilaton plays a crucial role in the description of the dynamics. Its source gives rise to a matter conformal anomaly for the running dilaton solutions, while its expectation value is the only non trivial observable for constant dilaton solutions. The role of this operator has been largely overlooked in the literature. We further show that the only non trivial conserved charges for running dilaton solutions are the mass and the electric charge, while for constant dilaton solutions only the electric charge is non zero. However, by uplifting the solutions to three dimensions we show that constant dilaton solutions can support non trivial extended symmetry algebras, including the one found by Compère, Song and Strominger [1], in agreement with the results of Castro and Song [2]. Finally, we demonstrate that any solution of this specific dilaton gravity model can be uplifted to a family of asymptotically AdS2 × S 2 or conformally AdS2 × S 2 solutions of the STU model in four dimensions, including non extremal black holes. The four dimensional solutions obtained by uplifting the running dilaton solutions coincide with the so called ‘subtracted geometries’, while those obtained from the uplift of the constant dilaton ones are new.
Highlights
Of a good candidate for the holographic dual, the focus has been on attempts to describe the effects of the strong gravitational backreaction on AdS2 by finite energy excitations
We further show that the only non trivial conserved charges for running dilaton solutions are the mass and the electric charge, while for constant dilaton solutions only the electric charge is non zero
By uplifting the solutions to three dimensions we show that constant dilaton solutions can support non trivial extended symmetry algebras, including the one found by Compere, Song and Strominger [1], in agreement with the results of Castro and Song [2]
Summary
We obtain the most general solutions of the 2D action (1.1), including the general solution with constant dilaton found earlier in [14], and discuss their connection. As we show in the subsection, there is an interpolating flow between the extremal elements of each family of solutions, but not for the non extremal ones Another property of the constant dilaton solutions (2.11) that is worth pointing out is that the asymptotic form of the gauge field At is rather unlike that of gauge fields in AdSd+1 with d > 2, or the solution (2.4) in the case of running dilaton, since the mode that asymptotically dominates is not μ(t), but rather α(t). It corresponds to the one-point function of the scalar operator dual to the dilaton ψ, which is an irrelevant operator of dimension 2 relative to the theory dual to constant dilaton solutions (2.11) This mode has been discussed before, e.g. in [14, 61], but its holographic interpretation was different due to the use of different boundary counterterms.
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