New boundary conditions for AdS$_2$

TL;DR

<3-5 sentence high-level summary>We formulate new AdS$_2$ boundary conditions in JT gravity using Bondi gauge, which enlarges the asymptotic symmetry to a warped Diff(S^1)\ltimes C^∞(S^1) and yields a boundary theory that generalizes the Schwarzian via a coadjoint action of the warped Virasoro group. This boundary dynamics reproduces the low-energy action of the complex SYK model and allows a complete Euclidean path integral that matches a refined random matrix ensemble (SSS) with boundary charges Q and energy shifts; a flat-space analogue via the \widehat{CGHS} model shows a similar ensemble structure. The construction provides a finer probe of near-extremal black holes, including deformations of RN and embedding into near-extreme Kerr, and it links to warped CFTs and Kerr/CFT ideas. Together, the results unify boundary dynamics, symmetry structure, and holographic duality across AdS$_2$ and flat geometries, highlighting ensemble averages as a recurring theme in two-dimensional gravity.

Abstract

We describe new boundary conditions for AdS$_2$ in Jackiw-Teitelboim gravity. The asymptotic symmetry group is enhanced to $\r{Diff}(S^1)\ltimes C^\infty(S^1)$ whose breaking to $\r{SL}(2,\R)\times Ů(1)$ controls the near-AdS$_2$ dynamics. The action reduces to a boundary term which is a generalization of the Schwarzian theory and can be interpreted as the coadjoint action of the warped Virasoro group. This theory reproduces the low-energy effective action of the complex SYK model. We compute the Euclidean path integral and derive its relation to the random matrix ensemble of Saad, Shenker and Stanford. We study the flat space version of this action, and show that the corresponding path integral also gives an ensemble average, but of a much simpler nature. We explore some applications to near-extremal black holes.

Figures (4)

• Figure 1: Cartoon of the geometry that computes $Z_{g,n}(\{\beta_i\},\{Q_i\})$. At the end of each trumpet $i$, we specify the boundary conditions $\beta_i$ and $Q_i$, which corresponds to the insertion of $\mathrm{Tr}\,e^{-\beta_i \left( H + {Q_i^2/ (2\gamma)} \right)}$ in the matrix ensemble.
• Figure 2: The trumpet geometry: we specify the boundary conditions $\beta$ and $Q$ at the end of the asymptotic boundary. The small end is labeled by the length $b$ of the geodesic boundary.
• Figure 3: Flat space boundary particles in Lorentzian signature. On the right, we have the thermal state which is analogous to a Schwarzschild black hole. On the left, we have the vacuum Minkowski state which is similar to the global AdS$_2$ solution of JT gravity and should be interpreted as an eternal traversable wormhole Maldacena:2018lmt.
• Figure 4: The cylinder geometry. We specify two boundary conditions $\beta_1$ and $\beta_2$ at each end. The only modulus is the circumference $b$.