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arxiv: 2604.21014 · v1 · submitted 2026-04-22 · ✦ hep-th

Recognition: unknown

3D near-de Sitter gravity and the soft mode of DSSYK

Herman Verlinde, Tommaso Marini, Xiao-Liang Qi

Authors on Pith no claims yet

Pith reviewed 2026-05-09 23:27 UTC · model grok-4.3

classification ✦ hep-th
keywords double-scaled SYKde Sitter gravityIsrael junction conditionssoft modesreparametrization modeholographic dualitydS3entropy
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The pith

3D de Sitter gravity with a dS2 energy slice reproduces the soft reparametrization dynamics of double-scaled SYK.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that the complex reparametrization mode governing soft dynamics in double-scaled SYK with time-dependent coupling has a direct dual in 2+1 dimensional Einstein-de Sitter gravity. An energy distribution localized on a dS2 slice inside dS3 produces effective equations of motion that match the SYK equations through Israel junction conditions. The one-dimensional effective action for this soft mode equals the action obtained from the 3D gravitational theory with conformal boundary conditions on past and future infinity. These boundaries split into hyperbolic k=-1 slices whose intersection hosts the holographic screen. The construction also reproduces the semiclassical DSSYK entropy and equates de Sitter boundary-to-boundary Green functions to the square of SYK two-point functions.

Core claim

The effective SYK equations of motion take the form of the Israel junction conditions across the dS2 slice. The 1D effective action of the SYK soft mode coincides with the effective action derived from 3D Einstein-de Sitter gravity with conformal boundary conditions on I±. The boundary conditions split I± into two hyperbolic k=-1 slices, and the holographic screen is placed at the intersection. An adapted Gibbons-Hawking calculation reproduces the semiclassical DSSYK entropy, while the boundary-to-boundary Green functions in 3D de Sitter equal the square of DSSYK two-point functions.

What carries the argument

The Israel junction conditions across a dS2 energy slice embedded in dS3, which enforce matching between the SYK soft-mode equations and the gravitational constraints under the split k=-1 conformal boundary conditions.

If this is right

  • The semiclassical DSSYK entropy is recovered from a Gibbons-Hawking calculation adapted to the k=-1 boundary conditions.
  • Boundary-to-boundary propagators in 3D de Sitter space equal the square of the DSSYK two-point functions.
  • The time-dependent Maldacena-Qi coupling in SYK is realized geometrically by the dynamics of the dS2 energy distribution.
  • An alternative dual description of the same SYK soft mode exists in 3D AdS gravity with two time directions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Higher-point correlation functions in DSSYK could be computed from bulk de Sitter diagrams using the same junction geometry.
  • The construction offers a geometric mechanism for incorporating time-dependent boundary couplings into de Sitter holography via interior slices.
  • The duality may extend to other SYK variants by varying the energy distribution on the dS2 slice.

Load-bearing premise

The dual gravity system takes the form of 2+1-dimensional Einstein-de Sitter gravity with an energy distribution localized on a dS2 slice within dS3, and that the boundary conditions on I± split into two hyperbolic k=-1 slices with the holographic screen at their intersection.

What would settle it

A direct computation showing that the gravitational effective action on the k=-1 slices fails to reproduce the 1D SYK soft-mode action for the complex reparametrization mode.

Figures

Figures reproduced from arXiv: 2604.21014 by Herman Verlinde, Tommaso Marini, Xiao-Liang Qi.

Figure 1
Figure 1. Figure 1: The near-dS2 geometry Σ is embedded as a gluing surface inside 3D de Sitter spacetime with conformal boundary conditions at I ˘. The shape of the gluing surface Σ and the holographic screen C ˘ are determined by the 3D Einstein equations, which reduce to the Israel junction conditions on Σ. The resulting equations of motion match the Lensky-Qi equations that govern the SYK boundary dynamics. 1.2 2+1-D near… view at source ↗
Figure 2
Figure 2. Figure 2: The Einstein-Hilbert action of 2+1D near-de Sitter gravity splits up into the usual bulk term, two Gibbons-Hawking-York boundary terms on Σ ˘ and I ˘, and Hayward corner terms on C ˘. The normalization of the boundary terms at I ˘ is fixed by our choice of conformal boundary conditions. Israel junction conditions (here rKabs denotes the discontinuity of Kab at Σ) rKabs ´ rKshab “ 8πGN Tab. (7) Our hypothes… view at source ↗
Figure 3
Figure 3. Figure 3: Schematic depiction of the AdS1`2 holographic perspective with two time directions. The soft mode ψptq traces a 1D path (red) on the 2D boundary of the emergent 3D geometry. The 2D gluing surface Σ spans between the 1D paths. The bulk light cone centered at p is oriented towards the 2D boundary. The 1D curves lie at spacelike future and past infinity relative to the bulk lightcone. 1.3 dS2`1 or AdS1`2 ? In… view at source ↗
Figure 4
Figure 4. Figure 4: The DSSYK spectrum and energy as a function of the variable θ. We see that θ P r0, π{2s, v P r0, 1s, corresponds to positive temperature (ρ increases as E increases), while θ P rπ{2, πs, v P r´1, 0s, corresponds to positive temperature (ρ decreases as E decreases). non-linear semiclassical regime where λ Ñ 0 but in which energy and temperature can be of order 1{λ. This semiclassical high energy theory [12,… view at source ↗
Figure 5
Figure 5. Figure 5: On the left, in blue, the contour used to compute expectation values in the TFD state. The horizontal and vertical directions are Euclidean and Lorentzian time, respectively. The red points denote the operator locations in gRLpu1, u2q and the wiggly red line represents the MQ coupling µpuq. On the right, the domain of u1, u2. We can use reflection symmetry to restrict to the shaded region u1 ě u2. The µpuq… view at source ↗
Figure 6
Figure 6. Figure 6: The future region of the spacetime obtained by cutting and gluing global 3D de Sitter along hypersurfaces Σ, Σ ˚ given by the function ψpuq “ xpuq ˘ iypuq. We remove the dark gray region and glue together its boundaries. The vertical direction is the global time τ . momentum tensor on the gluing surface Σ Tab “ ˜ T00 0 0 0 ¸ . (40) Here T00 is delta-function supported on Σ and may depend on the spatial coo… view at source ↗
Figure 7
Figure 7. Figure 7: The gluing surfaces Σ and Σ ˚ for the TFD solution (left) and their intersection with a constant τ slice (right). The conical angle at the cusps is equal to 2θ. Depending on the sign of the temperature, i.e. on whether θ ă π 2 or θ ą π 2 , either the interior/exterior of the wedge is physical/removed or vice versa. 12 To plot the gluing membrane, we projected the spatial sphere to the coordinates y1,2 ” co… view at source ↗
Figure 8
Figure 8. Figure 8: The gluing surfaces Σ and Σ ˚ for the shockwave solution with µˆ ą 0. Delta function source with µˆ ă 0. We can also choose the sign of ˆµ such that it corresponds to a particle with positive mass as seen from the de Sitter region outside of the wedge. On the SYK side, this regime corresponds to the negative temperature regime ´1 ă v ă 0. The green trajectory creates a conical deficit on the outside and a … view at source ↗
Figure 9
Figure 9. Figure 9: The gluing surfaces Σ and Σ ˚ for the shockwave solution with µˆ ă 0 [PITH_FULL_IMAGE:figures/full_fig_p023_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The gluing membranes Σ and Σ ˚ for the solution with constant µˆ. 3.4 Schwarzian limit The low energy limit of the SYK model is known to be dual to JT gravity on an AdS2 spacetime. Here we show how to recover this limit from our setup. We set µ “ 0 for simplicity. Consider the v Ñ 1 (β Ñ 8) limit of the thermal solution in equation (29). In this limit, the y “ Imψ location of the surface Σ goes to zero, a… view at source ↗
Figure 11
Figure 11. Figure 11: In the low energy limit, the two sides of the gluing surface Σ “ Σ ˚ coincide. We identify the surface Σ with the AdS2 geometry of JT gravity, after changing the overall sign of the metric. The intersection between the AdS2 slice and the cutoff (54) determines the boundary curve of the AdS2 spacetime14. Fluctuations of the Σ embedding translate into fluctuations of the boundary curve. In terms of the cano… view at source ↗
Figure 12
Figure 12. Figure 12: Schematic depiction of the standard Euclidean gravity path integral of the Schwarzschild-de Sitter spacetime. The left panel shows the overlap of two Hartle-Hawking wave functions prepared by the tan (Euclidean) and blue (Lorentzian) regions B3 and dS3. The Lorentzian parts cancel in the overlap, leaving a three sphere obtained by gluing together two three balls B3 with a small tube around the conical def… view at source ↗
Figure 13
Figure 13. Figure 13: Contours C and γ used in evaluating the action contour ¸ pdϕ, see [17] Finding the solution to these equations is straightforward [17] ϕpηq “ ´ logˆ sin θ cos η ˙ η “ iu sin θ ´ πv 2 , (86) ppηq “ 1 2 logˆ z ` e 2iθ 1 ` ze2iθ ˙ , z “ e 2iη . (87) Here η is a rescaled Euclidean 1D time coordinate. Note that the solution is periodic in η with period π. This looks promising, as it appears to support our assu… view at source ↗
Figure 14
Figure 14. Figure 14: The Euclidean path integral of near-de Sitter gravity. The two Hartle-Hawking wave functions are glued together via the k “ ´1 conformal boundary conditions on I ˘. This requires a Hayward term at the cusp C. The Lorentzian contributions cancel, but the Euclidean Hayward term survives. The action integrals over the volume and the surface of the green tube combine to contribute the entropy SSYK, while the … view at source ↗
Figure 15
Figure 15. Figure 15: The constant temperature solution corresponds to a Schwarzschild-de Sitter spacetime with a conical defect. Translation d du along the holographic screen C ` can be extended to a Killing vector that is space-like in the future patch but time-like in the static patch. On the observer worldline, the Killing vector is proportional to static patch time translation, with c “ cos πv 2 . Euclidean Hayward term s… view at source ↗
Figure 16
Figure 16. Figure 16: The two types of geodesics that connect points on the holographic screen correspond to the two types of SYK two-point functions gRR and gRL. The lengths of the geodesics are computed in the standard de Sitter geometry. The geodesic that reconnects with I ` is complex. corresponding SYK two-point functions21 ℓpX ` 1 , X`˚ 2 q “ 2T ´ gRRpu1, u2q ` gRRpu1, u2q 2 , ℓpX ` 1 , X´ 3 q “ 2T ´ gRLpu1, u3q ` gRLpu1… view at source ↗
Figure 17
Figure 17. Figure 17: Illustration of two paths in ψ plane ψptq and ψ˜ptq which have the same end points and are tangential at the end points, so that ψ and p 1 are the same for the two paths at the end points. a more local time dependence than the most generic situation: the two-point function G∆pt1, t2q only depends on ψptq and its derivative ψ 1 ptq at t “ t1, t2. Consider two trajectories ψptq and ψ˜ptq corresponding to di… view at source ↗
read the original abstract

We present a dual gravity interpretation of the complex reparametrization mode $\psi(u)$ that governs the soft dynamics of double-scaled SYK in the presence of a time-dependent Maldacena-Qi coupling. We find that the dual gravity system takes the form of 2+1-dimensional Einstein-de Sitter gravity with an energy distribution localized on a dS$_2$ slice within dS$_3$. The effective SYK equations of motion take the form of the Israel junction conditions across the dS$_2$ slice. We study the 1D effective action of the SYK soft mode and show that it coincides with the effective action derived from 3D Einstein-de Sitter gravity with conformal boundary conditions on $\mathscr{I}^\pm$. The boundary conditions split $\mathscr{I}^\pm$ into two hyperbolic $k=-1$ slices, and the holographic screen is placed at the intersection. We adapt the Gibbons-Hawking calculation of the Schwarzschild-de Sitter entropy to the case with $k=-1$ boundary conditions and find that it reproduces the semiclassical DSSYK entropy. The boundary-to-boundary Green functions in 3D de Sitter are equal to the square of DSSYK two-point functions. We give an alternative holographic interpretation of our results in terms of 3D AdS gravity with two time directions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 3 minor

Summary. The manuscript proposes a holographic duality between the complex reparametrization soft mode ψ(u) of double-scaled SYK (DSSYK) with time-dependent Maldacena-Qi coupling and 3D Einstein-de Sitter gravity. It identifies the effective SYK equations of motion with Israel junction conditions across an embedded dS₂ slice in dS₃, shows that the 1D SYK effective action coincides with the gravitational effective action under conformal boundary conditions on ℐ± (split into two k=-1 hyperbolic slices with the holographic screen at their intersection), adapts the Gibbons-Hawking entropy calculation to reproduce the semiclassical DSSYK entropy, and demonstrates that boundary-to-boundary Green functions in 3D de Sitter equal the square of DSSYK two-point functions. An alternative interpretation in terms of 3D AdS gravity with two time directions is also presented.

Significance. If the claimed exact coincidences hold, the work establishes a concrete dictionary linking SYK soft-mode dynamics to near-de Sitter gravitational degrees of freedom, which could advance holographic approaches to dS gravity and clarify the role of reparametrization modes. The explicit matching of equations of motion to junction conditions, effective actions, entropy, and Green functions (with the square relation) constitutes a strength, providing a falsifiable and parameter-free correspondence in the semiclassical regime. This is particularly notable given the direct derivation of gravitational features from the SYK side without post-hoc fitting.

major comments (1)
  1. The identification of the SYK equations of motion with Israel junction conditions (abstract and corresponding section) is load-bearing for the duality. The explicit form of the localized energy distribution on the dS₂ slice and its precise matching to the time-dependent MQ coupling should be shown without implicit assumptions on the stress-energy tensor; a derivation gap here would affect the central claim that the EOM take the form of junction conditions.
minor comments (3)
  1. The splitting of ℐ± into k=-1 hyperbolic slices and placement of the holographic screen at their intersection would benefit from an accompanying diagram to clarify the geometry.
  2. Notation for the conformal boundary conditions and the adapted Gibbons-Hawking entropy calculation could be made more explicit by including the precise boundary term used in the action.
  3. A summary table mapping SYK quantities (ψ(u), two-point functions, entropy) to their gravitational counterparts would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading, positive assessment, and constructive suggestion. We address the major comment below and have incorporated revisions to strengthen the explicitness of the derivation.

read point-by-point responses

  1. Referee: The identification of the SYK equations of motion with Israel junction conditions (abstract and corresponding section) is load-bearing for the duality. The explicit form of the localized energy distribution on the dS₂ slice and its precise matching to the time-dependent MQ coupling should be shown without implicit assumptions on the stress-energy tensor; a derivation gap here would affect the central claim that the EOM take the form of junction conditions.

    Authors: We appreciate the referee highlighting the importance of this identification. In the revised manuscript we have added a dedicated subsection that derives the localized energy distribution explicitly. Beginning from the time-dependent Maldacena-Qi coupling in the DSSYK Hamiltonian, we integrate the interaction term over the transverse directions to obtain the induced stress-energy tensor on the dS₂ slice without presupposing its form. We then compute the resulting jump in the extrinsic curvature tensor and verify that it reproduces the SYK equations of motion exactly. This step-by-step matching confirms that the effective equations take the form of Israel junction conditions, thereby closing the derivation gap and reinforcing the central claim. revision: yes

Circularity Check

0 steps flagged

No significant circularity; duality construction is self-contained

full rationale

The paper derives a holographic dictionary by mapping the DSSYK soft-mode dynamics (complex reparametrization ψ(u) and time-dependent MQ coupling) onto 3D Einstein-de Sitter gravity with an embedded dS2 slice. It shows that SYK equations of motion become Israel junction conditions, that the 1D effective action coincides with the gravitational action under conformal boundary conditions on I± (split into k=-1 slices), that an adapted Gibbons-Hawking entropy calculation reproduces the semiclassical DSSYK entropy, and that boundary-to-boundary Green functions equal the square of DSSYK two-point functions. These equalities are obtained by explicit calculation rather than by fitting parameters to SYK data or by re-labeling prior results. No self-definitional steps, fitted-input predictions, or load-bearing self-citation chains appear in the provided abstract and derivation outline; the central claims retain independent content on both the SYK and gravity sides.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the assumption that the dual is precisely 3D Einstein-de Sitter gravity with energy localized on an embedded dS2 slice and on the choice of conformal boundary conditions that split I± into k=-1 hyperbolic slices. No free parameters or new invented entities are explicitly introduced in the abstract; the construction adapts standard gravitational tools (Israel junctions, Gibbons-Hawking) to this setting.

axioms (2)
  • domain assumption The dual gravity system is 2+1-dimensional Einstein-de Sitter gravity with an energy distribution localized on a dS2 slice within dS3.
    Directly stated as the form taken by the dual.
  • domain assumption Conformal boundary conditions on I± split the boundaries into two hyperbolic k=-1 slices with the holographic screen at their intersection.
    Required for the effective action derivation and entropy calculation.

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