Abstract
We consider different sets of AdS2 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 conditions is easily motivated in the gauge theoretic formulation as a Poisson sigma model and has an mathfrak{s}mathfrak{l}(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{s}mathfrak{l}(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.
Highlights
The study of dilaton gravity in two dimensions began in the 1980s with the introduction of the Jackiw-Teitelboim (JT) model [1, 2], and has been punctuated by periods of increased interest in the community
Besides the sl(2) current algebra we find for stricter boundary conditions a Virasoro algebra, a warped conformal algebra and a u(1) current algebra
They have to satisfy the equations of motion that follow from varying Aρ in the action (X ±)′ ± (L±X +(ε0)′ − (X 0) − L0X ±) = 0 (X 0)′ + 2(L+X − − L−X +) = 0
Summary
The study of dilaton gravity in two dimensions began in the 1980s with the introduction of the Jackiw-Teitelboim (JT) model [1, 2], and has been punctuated by periods of increased interest in the community. They must transform the dilaton in a non-trivial way If this is allowed by the boundary conditions the AdS2 isometries remain part of the asymptotic symmetries. In the present work we do not impose such conditions and instead let the dilaton fluctuate to leading order at the boundary. We present the loosest set of boundary conditions, (3.1)–(3.13), for the JT model (in first order formulation), leading to a generalized Fefferman-Graham expansion of metric (3.18) and dilaton (3.19). We show that naive Cardyology works, not just in the Virasoro case and for warped conformal boundary conditions, in the sense that the chiral Cardy formula (and its warped conformal generalization) lead to a result for entropy compatible with the macroscopic Wald entropy. Appendix B discusses toy models that amount to Poisson sigma models in Casimir-Darboux coordinates, which elucidates some of the subtle issues encountered in the main text and paves the way towards generic models of two-dimensional dilaton gravity
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