Wiggling boundary and corner edge modes in JT gravity with defects

TL;DR

The work investigates how wiggling boundaries and defects in JT gravity give rise to gravitational and gauge edge modes (GrEMs and GaEMs). By analyzing conical and wormhole defects, it derives boundary actions that are Schwarzian or deformed Schwarzian, with the boundary dynamics controlled by horizon data and topological constraints. A detailed corner-analysis reveals a consistent SL$(2,\mathbb{R})$-type algebra governing corner degrees of freedom, and a Maurer–Cartan packaging of extrinsic data yields gauge-invariant edge modes tied to the bulk via gluing conditions. The results illuminate how boundary information encodes physical degrees of freedom, linking bulk topology, boundary dynamics, and corner symmetries, and suggest avenues for computing correlation functions and exploring higher symmetry structures.

Abstract

We study the gravitational edge modes (GrEMs) and gauge edge modes (GaEMs) in Jackiw-Teitelboim (JT) gravity on a wiggling boundary. The wiggling effect manifests as a series of spacetime topological and bulk constraints for both conical and wormhole defect solutions. For the conical defect solution, we employ the generalized Fefferman-Graham (F-G) gauge to extend the boundary action, allowing for non-constant temperature and horizon position. We find that the infrared behavior of this boundary action is determined by the local dynamics of the temperature and horizon. For the wormhole defect solution, the boundary action can, in special cases, be described by a field with variable mass subject to a constant external force. We classify this corner system as a first-class constrained system influenced by field decomposition, confirming that the physical degrees of freedom are determined by constraints from the wiggling boundary information. We find that GrEMs and GaEMs can be linked at the corners by imposing additional constraints. Additionally, we show that the ``parallelogram'' composed of corner variables exhibits discreteness under a unitary representation. Finally, we explore that information from extrinsic vectors can be packaged into the GaEMs via a Maurer-Cartan form, revealing the boundary degrees of freedom as two copies of the $\mathfrak{sl}(2,\mathbb{R})$ algebra. By separating pure gauge transformations, we identify the gluing condition for gauge invariance and the corresponding integrable charges.

Figures (1)

• Figure 1: In the left panel, the normal and tangent vectors at the asymptotic boundary are marked with red lines with arrows, labeled $\bar{y}$ (which is one of $\bar{y}_1$ or $\bar{y}_2$) and $\bar{n}$ (which corresponds to the chosen $\bar{y}$), respectively. The conical defect angle is $\alpha^c$. Due to this angle, there are two sets of tangent and normal vectors, which intersect at the corners ${\cal S}_1$ and ${\cal S}_2$. The wiggling boundary corresponds to the curved boundary and is described by a coordinate transformation from base coordinates to target coordinates. In the right panel, we consider only a single corner, where a similar coordinate transformation exists. The normal and tangent vectors are labeled $\tilde{n}$ and $\tilde{y}$, respectively.