Defects in Jackiw-Teitelboim Quantum Gravity

TL;DR

This work classifies and analyzes defects in 2d Jackiw–Teitelboim gravity by promoting defects to deformations of the Schwarzian action via integration over Virasoro coadjoint orbits. It establishes a holographic dictionary where each orbit corresponds to a Liouville CFT sector between branes with Verlinde loop insertions, identifies explicit orbital deformations (elliptic, hyperbolic, parabolic, and exceptional), and defines two diffeomorphism-invariant geometric observables, the horizon area $\hat{\Phi}_h$ and the geodesic length $L(\gamma)$. The paper provides exact results for partition functions and correlators across the orbit families, including a solvable complex SYK example with a $U(1)$ symmetry and a non-abelian generalization, and it interprets defects as chemical-potential twists in a particle-on-a-group picture. Together, these results yield a unified 2d CFT/BFJT framework for JT defects with concrete holographic and SYK-like applications and open avenues for studying exceptional Virasoro orbits and higher-rank extensions.

Abstract

We classify and study defects in 2d Jackiw-Teitelboim gravity. We show these are holographically described by a deformation of the Schwarzian theory where the reparametrization mode is integrated over different coadjoint orbits of the Virasoro group. We show that the quantization of each coadjoint orbit is connected to 2d Liouville CFT between branes with insertions of Verlinde loop operators. We also propose an interpretation for the exceptional orbits. We use this perspective to solve these deformations of the Schwarzian theory, computing their partition function and correlators. In the process, we define two geometric observables: the horizon area operator $Φ_h$ and the geodesic length operator $L(γ)$. We show this procedure is structurally related to the deformation of the particle-on-a-group quantum mechanics by the addition of a chemical potential. As an example, we solve the low-energy theory of complex SYK with a U(1) symmetry and generalize to the non-abelian case.

Figures (15)

• Figure 1: Left: AdS$_3$ global spacetime with a defect along the time direction (green), measured by the holonomy integrals $\oint A_\phi$ and $\oint \bar{A}_\phi$. Right: Defect in 2d Jackiw-Teitelboim gravity (green), measured by the holonomy integral $\oint A_0$.
• Figure 2: Scheme of all Virasoro highest weight representations at large $c$. Hyperbolic, parabolic and elliptic orbits span all representations. Both unitary and non-unitary representations are required.
• Figure 3: Schematic situation of the action $S[f]$ in functional space. The elliptic ($0\leq \theta \leq 1$) and hyperbolic orbits have a unique and stable saddle solution. The exceptional elliptic ($\theta>1$), including the higher $\theta=n$ orbits, have a unique saddle solution, but it has unstable directions. The exceptional hyperbolic and parabolic orbits have no extrema at all and are unbounded in action both from above and below.
• Figure 4: Inserting Verlinde loop operators (red) between Liouville primary insertions (blue crosses) gives defect operators in the JT bulk.
• Figure 5: Left: Chiral WZW model ($\chi$WZW) and its dimensional reductions along both axes. The quantum particle on the group manifold ($\partial_t = 0$) appears in the dual channel of the original chiral WZW model, which by itself reduces to the character $\chi$ of the group $G$ ($\partial_\phi = 0$). See Appendix \ref{['app:compact']} for the equations. Right: Liouville between ZZ-branes leads to a torus surface (grey) with mirrored operator insertions. The holographic bulk is found as the exterior of the torus. In the Schwarzian limit, the torus degenerates, the bulk becomes independent of the angular coordinate (rotating around the grey tube) and can be viewed as a disk (lightblue) bounded by the degenerate torus. Primary operator insertions (blue) lead to bilocal lines in the disk. Verlinde loop operators (red) are topologically supported. Deforming them into the bulk, they are Chern-Simons Wilson lines puncturing the BF bulk disk at an arbitrary point (green).