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Holographic duals of JT gravity at finite cutoff: the classical case

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Andreas Belaey, Thomas G. Mertens, Lorenzo Russo, Alex Tarana

High Energy Physics - Theory · Submitted 29 Oct 2026 · 86 references

Pith's verdict: A careful BF phase-space reduction that delivers a genuinely new closed Schwarzian action at finite cutoff, conditional on the hyperbolic patch covering the configurations it claims.

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arXiv ID2610.40391 (version 1)
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CitationAlex Tarana, Andreas Belaey, Lorenzo Russo, Thomas G. Mertens. "Holographic duals of JT gravity at finite cutoff: the classical case." arXiv:2610.40391 (2026). https://pith.science/paper/PO57PEJM
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The paper

The paper claims

JT gravity with a holographic screen at finite distance has an exact classical boundary dual, a phase-space action whose variables are the embedding coordinates of the screen curve in AdS2.

The reading

JT gravity is the universal low-energy model of near-extremal black holes, and its holographic description is standardly taken at the asymptotic boundary of AdS$_2$, where it becomes Schwarzian quantum mechanics. This paper argues that when the holographic screen is moved inward to a finite distance — a finite cutoff — the classical theory still has an exact boundary dual, written as a phase-space action rather than an action for a single reparametrization. The claim is that the four variables $(f, p_f, \ell, p_\ell)$ are the coordinates of the finite-cutoff curve in AdS$_2$ together with their conjugate momenta, and that the resulting Hamiltonian reproduces the known finite-cutoff energy on shell. A sympathetic reader would care because this is a closed-form, off-shell description of the finite-cutoff Schwarzian model, giving a concrete setting to probe bulk locality and, in the companion work, to quantize the system.

What carries the argument

The engine is symplectic reduction of the BF theory on a strip. Bulk Gauss-law constraints reduce the local fields to the cotangent bundle $T^*G$ of the $\widetilde{\mathrm{SL}}(2,\mathbb{R})$ group, with the open holonomy $g$ and boundary values of the $B$-field as canonical data. Each gravitational boundary condition fixes one component of the moment map — the dilaton value $\Phi_b$ at a finite-cutoff boundary, the tension $\mu$ at an EOW brane — and quotienting the stabilizer flow leaves the physical phase space. The surviving configuration coordinate is chosen as the proper length $L$ of the geodesic connecting the boundaries, with conjugate momentum $p_L$; in the one-sided case the configuration space becomes the coset $H_+ \setminus \widetilde{\mathrm{SL}}(2,\mathbb{R}) \simeq \mathrm{AdS}_2$, parametrized by $g = e^{\eta P_2} e^{-\ell H} e^{f F}$. This patched parametrization is the load-bearing identity: it converts the group-theoretic data into the embedding coordinates of the boundary curve and produces the symplectic potential and Hamiltonian of Eq. (5.20).

Extended reading notes

Core claim

The paper's central claim is that the first-order (BF) formulation of JT gravity, reduced with finite-cutoff and end-of-the-world boundary conditions, yields an exact phase-space description of the boundary dynamics. For the one-sided system the result is the action $S = \int d\tau\,(p_\ell \dot\ell + p_f \dot f - \Phi_b + \sqrt{(e^{-\ell} p_f - \Phi_b)^2 - p_\ell^2})$, with boundary Hamiltonian $H_\partial = \Phi_b - \sqrt{(e^{-\ell} p_f - \Phi_b)^2 - p_\ell^2}$ that equals on shell the known energy $E = \Phi_b - \sqrt{\Phi_b^2 - \Phi_h^2}$. The coordinates $(f(\tau), \ell(\tau))$ describe the embedding of the finite-cutoff curve in AdS$_2$ in Poincaré coordinates, $z = e^{-\ell}$, and the Hamilton equations imply $e^{2\ell}\dot f^2 - \dot\ell^2 = 1$, so $\tau$ is proper time along the curve. For two boundaries the paper claims the system reduces to a single canonical pair $(L, p_L)$, the geodesic length between the boundaries and its conjugate, governed by Pöschl–Teller potentials for equal cutoffs and by a brane-plus-cutoff potential whose known limits are recovered. It also derives an $\mathrm{SL}^+(2,\mathbb{R})$ positive-semigroup structure as the local algebraic content of the gravity phase space, emerging from positivity of infinitesimal spatial length.

Load-bearing premise

The whole reduction picks the double-coset representative and patch $g = e^{\eta P_2} e^{-\ell H} e^{f F}$ (and the two-boundary analogue), which covers only hyperbolic holonomies with $\mathrm{Tr}(g) > 2$; if the physical finite-cutoff sector includes elliptic or parabolic holonomies, Eq. (5.20) is a patch-level action rather than the exact dual.

Editorial extensions

If this is right

  • The finite-cutoff boundary action (5.20) is exact at the classical level on phase space, so the energy, boundary proper time, and embedding curve are all determined off shell, not only on the classical solution.
  • At infinite cutoff the action reduces to the standard Schwarzian phase-space actions: the Poincaré-frame form (5.24), the thermal Rindler form (5.30), and the global-frame form (5.37).
  • The two-sided systems close: equal cutoffs give a Pöschl–Teller Hamiltonian (4.18) matching the box model of [20], and a cutoff with an EOW brane gives a potential (4.48) whose large-cutoff limit is the known Morse model of [23].
  • Positivity of infinitesimal spatial length — the 1+1-dimensional analogue of $\sqrt{h} > 0$ — selects an $\mathrm{SL}^+(2,\mathbb{R})$ semigroup as the local algebraic structure of the gravity phase space, replacing the full gauge group.
  • On shell the Hamiltonian reproduces the standard finite-cutoff energy, but off shell it is not a $\bar{T}T$-type square-root deformation of the Schwarzian; the two descriptions agree on solutions while differing as phase-space functions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A direct test of the framework is quantum: imposing the inequalities (5.18)–(5.19) as operator constraints in the companion quantization should select principal-series $\mathfrak{sl}(2,\mathbb{R})$ representations with an upper Casimir bound, a specific checkable prediction.
  • The hyperbolic-patch limitation suggests the honest dual may need a continuation or gluing into elliptic and parabolic sectors; one could test whether analytically continuing (5.20) across the boundary of the region (5.18)–(5.19) reproduces the horizonless global-AdS$_2$ sector the paper interprets as $L < L_0$.
  • Since $p_f$ cannot be integrated out in closed form, any Euclidean path-integral comparison with existing finite-cutoff results will need a contour prescription for the phase-space inequalities, analogous to the lapse-contour problem in canonical gravity; the authors flag this as open.
  • The same reduction applied to de Sitter JT yields a trigonometric counterpart with opposite kinetic sign, and the paper leaves open whether a de Sitter analogue of the $\mathrm{SL}^+(2,\mathbb{R})$ semigroup organizes the gluing of regions there.
Original abstract

We formulate JT gravity with finite-cutoff boundaries and end-of-the-world branes in the first-order BF formulation. We dynamically reduce the BF model into a one-dimensional system living at the boundaries, generalizing earlier constructions. We obtain in particular a geometrical understanding of the phase space degrees of freedom associated to each of these boundaries. We leverage these results to probe bulk locality and the underlying structure of spacetime. Within our framework, we are able to derive a closed-form expression for the holographic Schwarzian description of JT gravity at finite-cutoff in phase space. The resulting degrees of freedom describe the embedding of the finite-cutoff curve in AdS$_2$. A quantization of these systems is presented in a companion paper.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-30: 1 request, 5,416 input tokens.

What the abstract itself states, as probabilities: claims first 0.09 · states limitation 0.39 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.07 · compares with prior results 0.36 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We formulate JT gravity with finite-cutoff boundaries and end-of-the-world branes in the first-order BF formulation. We dynamically reduce the BF model into a one-dimensional system living at the boundaries, generalizing earlier constructions. We obtain in particular a geometrical understanding of the phase space degrees of freedom associated to each of these boundaries. We leverage these results to probe bulk locality and the underlying structure of spacetime. Within our framework, we are able to derive a closed-form expression for the holographic Schwarzian description of JT gravity at finite-cutoff in phase space. The resulting degrees of freedom describe the embedding of the finite-cutoff curve in AdS$_2$. A quantization of these systems is presented in a companion paper.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We formulate JT gravity with finite-cutoff boundaries and end-of-the-world branes in the first-order BF formulation. contribution 0.71 0.95 0.06 asserted 1.00
s1 We dynamically reduce the BF model into a one-dimensional system living at the boundaries, generalizing earlier constructions. method 0.72 0.94 0.12 asserted 1.00
s2 We obtain in particular a geometrical understanding of the phase space degrees of freedom associated to each of these boundaries. result 0.81 0.94 0.14 asserted 1.00
s3 We leverage these results to probe bulk locality and the underlying structure of spacetime. contribution 0.45 0.81 0.08 asserted 0.78
s4 Within our framework, we are able to derive a closed-form expression for the holographic Schwarzian description of JT gravity at finite-cutoff in phase space. result 0.95 0.96 0.57 asserted 0.97
s5 The resulting degrees of freedom describe the embedding of the finite-cutoff curve in AdS$_2$. interpretation 0.71 0.85 0.20 asserted 1.00
s6 A quantization of these systems is presented in a companion paper. future work 0.80 0.33 0.03 asserted 0.98

What kind of paper the classifier reports this is: theoretical result 0.99 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read
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not examined
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  • figures and tables
  • references and citing sentences
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not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. The Penrose diagram for JT gravity in the presence of an EOW brane boundary (green curve) and a finite-cutoff boundary (red curve). The spacetime is the shaded region between the two boundaries. 2.2 Review: First-order formulation It is well understood that the classical bulk JT action can be rewritten as a topological BF gauge theory with an underlying sl(2, R) algebra [7, 8, 9, 10]. Here we derive the appropriate first-order analogues of the boundary JT actions of (2.1). In Lorentzian signature, the bulk manifold M is spanned by a timelike coordinate ν and a radial spacelike…
Figure 2. Adapted local frame on each of the boundary brane segments. brane geodesic is now given by4 K = e µ a∂µn a + e µ aω a µbn b = e µ //ωµ. (2.33) 4We have used that ω 0 µ 1 = −ωµ from ϵ 0 1 = −1 since ϵ01 = 1 and 0 is timelike. 11
Figure 3. Local Lorentz frame adapted to the bulk geodesic curve γ. of the spin connection measures the curvature and since γ is a geodesic γ ∗ω = 0 by (2.33). Therefore the pullback of A reduces to γ ∗A = H ds. The Wilson line, hence, admits the representative gγ = P exp − Z γ A  = e −LH, L ≡ Z γ ds. (3.23) Equation (3.23) holds in the gauge adapted to the geodesic, so it does not describe a generic Wilson line g on T ∗G. Residual gauge transformations preserving the bound￾ary conditions, induce physical symmetries at the endpoints (3.5). A generic element in the equivalence clas
Figure 4. Potential of the dynamical system with two finite-cutoff boundaries as a function of bulk geodesic length L, with zero at L0 and maximum at Lmax. where Vcutoff(Lmax) = (Φ− b ) 2 . This means that the barrier cannot be overcome as long as we consider configurations with “energy” variable Φ 2 h below this threshold: Φ 2 h < (Φ− b ) 2 < (Φ+ b ) 2 . Our goal is now to provide some geometric interpretation for the different classes of classical solutions. For the present discussion, it is convenient to work in Schwarzschild gauge in the bulk: ds 2 = −(r 2 − Φ 2 h ) dt 2
Figure 6. Spacetime between two nearby finite cutoff boundaries (here in Poincaré coordinates) allows us to probe emergence of bulk locality. The bulk geodesic length between both adjacent boundaries is positive L > 0 for any choice of Φ − b and Φ + b . is given by (4.1): g 7→ e β(E+F) e −LHe γ(E+F) (4.36) =   e −L/2 cosh β cosh γ + e L/2 sinh β sinh γ e−L/2 cosh β sinh γ + e L/2 sinh β cosh γ e −L/2 sinh β cosh γ + e L/2 cosh β sinh γ e−L/2 sinh β sinh γ + e L/2 cosh β cosh γ   , with L > 0. We observe now that there exists a conjugation matrix C = 1 √ 2
Figure 7. Potential VEOW(L) for the two cases: Φ 2 b − µ 2 > 0 (left) and Φ 2 b − µ 2 < 0 (right). At the top we denote the various EOW brane (green) and finite-cutoff boundary (red) configurations depending on L. potential. It is clear that some features of the potential depend on the hierarchy between the boundary data µ and Φb, suggesting to consider two separate regimes: • Φb > µ (
Figure 8. Left: example of Euclidean boundary curve (f(τ ), ℓ(τ )) parametrized in proper time τ , that is not monotonic in f. Right: example of a boundary curve that self-intersects. TT¯-flow of symmetry generators. The finite-cutoff system has a set of sl(2, R) symme￾try generators (5.39), where only the last one is different from the infinite cutoff generators. Can we determine directly the flow of these symmetry generators by integrating the T T¯-flow equations [60]? 45
Figure 9. Diff-invariant construction of the dilaton field using two-sided gravitational information (L) and the geodesic length s along this geodesic. be interesting. The relation with one-sided dressing constructions as in [62, 63, 64] would also be interesting to understand. The dilaton field usually plays an inverse role as follows. The value of any scalar field, such as Φ, can be used to label bulk points in terms of the value of the field itself [65]. Taking the dilaton field Φ on a fixed Cauchy slice then naturally defines a radial location in the bulk. Timelike boundaries in de Sitter…
Figure 10. The on-shell geometry for JT gravity with two different Dirichlet boundaries, with Φ − b and Φ + b corresponding to the left and right red curves, respectively. On the left panel, the boundaries lie on different sides of the black hole. On the right panel, the boundary curves both lie to the right of the horizon. where we used the boundary condition (2.4). Hence, we conclude that EOW boundaries carry no energy and do not contribute to the total Hamiltonian. Our strategy consists of foliating the geometry (2.9) with spacelike boundary-to-boundary Cauchy slices on which we specify…

Showing 9 of the paper's 10 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:2610.40391 by the authors.

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