Entanglement entropy in Jackiw-Teitelboim Gravity

TL;DR

Using the BF formulation of JT gravity with gauge group $SL(2,\mathbb{R})$, the black hole entropy of an $AdS_2$ wormhole is shown to coincide with a logarithmic edge term arising from entanglement across a boundary in a gauge-theoretic Hilbert space. The replica-trick calculation in the Hartle–Hawking state yields $S_{EE}=\int dk\, p_{k,\beta}[ -\log p_{k,\beta}+\log(k\sinh 2\pi k) ]$, where $d\mu(k)=k\sinh(2\pi k)\,dk$ is the Plancherel measure for $SL(2,\mathbb{R})$ and $p_{k,\beta}$ is the normalized eigen-representation distribution. In the classical (large $\alpha$) limit the edge term evaluated at the peak $k_{max}$ reproduces the Bekenstein–Hawking contribution $S_{BH}=\langle\phi_h\rangle/(4G_N)$, establishing the JT analogue of the edge/area correspondence. The work thus supplies a concrete realization of the RT=bulk-EE-with-edge-modes idea in an emergent gauge-gravity system, and suggests that black hole entropy counts degrees of freedom associated with continuous $SL(2,\mathbb{R})$ representations rather than local IR gauge-invariant bulk operators. These results motivate further exploration of edge-mode entanglement in higher-dimensional gravity and its connections to holography, the SYK model, and the microstate problem.

Abstract

I show that the black hole entropy associated to an $AdS_2$ wormhole is an entanglement edge term related to a natural measure on the gauge group in the $SL(2)$ gauge theory formulation of $1+1d$ Jackiw-Teitelboim gravity. I comment on what the entropy appears to be counting.