Menagerie of AdS$\boldsymbol{_2}$ boundary conditions

TL;DR

This work provides the most general AdS$_2$ boundary conditions for Jackiw–Teitelboim gravity in the linear dilaton sector, allowing leading-order dilaton fluctuations at the boundary. By formulating JT as a Poisson sigma model and implementing a novel dilaton kinetic counterterm, the authors obtain a well-defined variational principle and an on-shell Schwarzian action, while revealing a spectrum of asymptotic symmetry algebras (centerless $rak{sl}(2)$ current, Virasoro, warped conformal, and $$(1)) across boundary-condition classes. The analysis shows how off-shell algebras can be infinite-dimensional but may reduce on-shell, echoing SYK-like conformal breaking, and connects these boundary symmetries to half of the corresponding 3D AdS$_3$ gravity algebras. Thermodynamically, the entropy is captured by Wald and Cardy-like formulas, with a central charge $c=6kar{Y}/$ governing the off-shell Virasoro structure and a Schwarzian action governing the boundary dynamics, providing a coherent holographic picture for AdS$_2$ black holes.

Abstract

We consider different sets of AdS$_2$ boundary conditions for the Jackiw-Teitelboim model in the linear dilaton sector where the dilaton is allowed to fluctuate to leading order at the boundary of the Poincaré disk. The most general set of boundary condtions is easily motivated in the gauge theoretic formulation as a Poisson sigma model and has an $\mathfrak{sl}(2)$ current algebra as asymptotic symmetries. Consistency of the variational principle requires a novel boundary counterterm in the holographically renormalized action, namely a kinetic term for the dilaton. The on-shell action can be naturally reformulated as a Schwarzian boundary action. While there can be at most three canonical boundary charges on an equal-time slice, we consider all Fourier modes of these charges with respect to the Euclidean boundary time and study their associated algebras. Besides the (centerless) $\mathfrak{sl}(2)$ current algebra we find for stricter boundary conditions a Virasoro algebra, a warped conformal algebra and a $\mathfrak{u}(1)$ current algebra. In each of these cases we get one half of a corresponding symmetry algebra in three-dimensional Einstein gravity with negative cosmological constant and analogous boundary conditions. However, on-shell some of these algebras reduce to finite-dimensional ones, reminiscent of the on-shell breaking of conformal invariance in SYK. We conclude with a discussion of thermodynamical aspects, in particular the entropy and some Cardyology.