Fine Structure of Jackiw-Teitelboim Quantum Gravity

TL;DR

The paper argues that Jackiw-Teitelboim gravity is precisely captured by an SL^+(2,R) BF theory with coset boundary constraints, and that the ubiquitous sinh(2π√E) density of states is the SL^+(2,R) Plancherel measure. By analyzing compact and noncompact group BF theories, it develops a coherent holographic picture where one sees Schwarzian dynamics on asymptotic boundaries and frozen SL^+(2,R) edge modes on horizons, which account for black hole entropy and enable factorization. The work further shows that two-boundary configurations map onto Liouville CFT on the torus in a Schwarzian double-scaling limit, and that correlation functions with crossing Wilson lines in JT gravity align with this Liouville/Virasoro structure. These results illuminate how edge states and coset structures resolve the factorization puzzle in JT and provide a bridge to higher-dimensional gravity via connections to 3d gravity and quantum groups.

Abstract

We investigate structural aspects of JT gravity through its BF description. In particular, we provide evidence that JT gravity should be thought of as (a coset of) the noncompact subsemigroup SL$^+$(2,R) BF theory. We highlight physical implications, including the famous sinh Plancherel measure. Exploiting this perspective, we investigate JT gravity on more generic manifolds with emphasis on the edge degrees of freedom on entangling surfaces and factorization. It is found that the one-sided JT gravity degrees of freedom are described not just by a Schwarzian on the asymptotic boundary, but also include frozen SL$^+$(2,R) degrees of freedom on the horizon, identifiable as JT gravity black hole states. Configurations with two asymptotic boundaries are linked to 2d Liouville CFT on the torus surface.

Figures (18)

• Figure 1: Left: defect channel slicing of the amplitude where $gh^{-1} = U$. Middle: Angular slicing of the amplitude. Right: Circular slicing connecting the inner boundary and the outer.
• Figure 2: Left: Angular slicing of the amplitude. Middle and Right: Circular slicing and annular region connecting the inner boundary and the outer, the latter projecting on the invariant indices. From hereon coset boundaries will always we depicted in red.
• Figure 3: Left: Configuration of crossing Wilson lines that enclose a bulk region. The coset projection (denoted by the $0$-symbol and the red-colored boundary) is not felt in the deep interior, but it is crucial to use the $\sinh 2\pi \sqrt{E}$ measure to agree with a semi-classical shockwave computation of the same topology (Right).
• Figure 4: Quantizing BF theory on a circle gives a complete basis by Peter-Weyl as the set of all characters of all unitary irreps. Gluing proceeds by using this basis.
• Figure 5: We evolve a set of oriented Cauchy slices (black) through the disk. In this way, an orientation is associated to each of the boundaries of the smaller disks (blue) that allows for an $\text{SL}^+(2,\mathbb{R})$ BF calculation in each of these disks. The black dot represents the horizon.