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

TL;DR

This paper explores various boundary conditions in AdS$_2$ for the Jackiw-Teitelboim model, revealing diverse asymptotic symmetries and their implications for holography, thermodynamics, and connections to 3D gravity.

Contribution

It introduces a comprehensive set of AdS$_2$ boundary conditions with novel boundary counterterms, analyzing their asymptotic symmetries and relation to 3D gravity algebras.

Findings

Discovery of multiple boundary algebras including Virasoro and warped conformal

Identification of a boundary counterterm as a kinetic term for the dilaton

Connection between AdS$_2$ boundary symmetries and 3D Einstein gravity symmetries

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 Poincar\'e 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.