AdS$_2$ Holographic Dictionary

TL;DR

This work clarifies AdS2 holography for a specific 2D dilaton gravity model obtained from 3D AdS gravity by circle reduction, revealing two distinct holographic doctrines: running-dilaton and constant-dilaton sectors. It delivers exact holographic renormalization, computes the full set of renormalized one-point functions with arbitrary sources, and derives the corresponding Ward identities and anomalies, notably a Schwarzian effective action in the running case. The constant-dilaton sector exhibits a Coulomb-branch structure with vanishing stress tensor and an irrelevant dilaton operator whose VEV parameterizes degenerate vacua, and it admits extended asymptotic symmetries when embedded in 3D gravity (CSS boundary conditions). Both branches uplift to 4D STU subtracted geometries, with running-dilaton uplifts yielding known subtracted geometries and constant-dilaton uplifts producing new AdS2×S2 solutions, highlighting a rich interplay between 2D holography and higher-dimensional uplifts. Together, these results advance a precise, UV-complete AdS2/CFT1 dictionary and illuminate the role of the dilaton operator in 2D holography, including its influence on anomalies and conserved charges.

Abstract

We construct the holographic dictionary for both running and constant dilaton solutions of the 2D Einstein-Maxwell-Dilaton theory obtained by a circle reduction from 3D gravity with negative cosmological constant. This specific model ensures that the dual theory has a well defined ultraviolet completion in terms of a 2D CFT, but our results apply qualitatively to a wider class of 2D 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 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, in agreement with results by Castro and Song. Finally, we demonstrate that any solution of this specific dilaton gravity model can be uplifted to a family of asymptotically AdS$_2\times S^2$ or conformally AdS$_2\times S^2$ solutions of the STU model in four dimensions. 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.

Figures (2)

• Figure 1: This diagram shows how the EMD model \ref{['action']} arises as the low energy effective theory of non extremal asymptotically conformally AdS$_2\times S^2$ subtracted geometries in four dimensions, in the parameterization introduced in An:2016fzu. For non rotating black holes the two routes to \ref{['action']}, a direct $S^2$ reduction (black arrow) and the more general procedure through the uplift to five dimensions (blue arrows), coincide. However, only the latter is available for rotating black holes. The relevant Kaluza-Klein Ansätze are given in \ref{['KK-relations']} and \ref{['KK-Ansatz']}. A similar diagram applies for subtracted geometries in five dimensions, replacing $S^2$ with $S^3$.
• Figure 2: The compact part of the solutions \ref{['classII-uplift']} for increasing values of the ratio $\lambda B/\sqrt{LQ}$: 0 for plot (a), 1 for plot (b), and 3 for plot (c). Solutions with $\lambda=0$ are static, but when $\lambda\neq 0$ the solutions acquire a non zero angular velocity, which goes to infinity at the AdS$_2$ boundary.