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

Abstract: We present a dual gravity interpretation of the complex reparametrization mode $ψ(u)$ 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.

• The paper demonstrates a precise holographic mapping between the DSSYK soft mode and 3D near-de Sitter gravity through Israel junction conditions.
• The methodology translates the one-dimensional DSSYK effective action into a dual 2D de Sitter slice embedded in global dS3, matching boundary correlators.
• It reveals implications for two-time holography and non-AdS dualities, offering insights into finite temperature effects and shockwave geometries.

Dual Gravity Description of the DSSYK Soft Mode in 3D Near-de Sitter Gravity

Introduction and Motivation

This paper establishes a holographic correspondence between the complex soft mode $\psi(u)$ governing the double-scaled Sachdev-Ye-Kitaev (DSSYK) model under time-dependent Maldacena-Qi (MQ) coupling and a gravitational system described by 2+1-dimensional Einstein-de Sitter (dS$_3$) gravity. The construction translates the effective one-dimensional DSSYK action, which generalizes Schwarzian quantum mechanics, into a dual description involving a 2D de Sitter slice $\Sigma$ embedded in global dS$_3$, with all dynamics localized on this surface. The mapping is precise: the soft mode equations of motion match Israel junction conditions on $\Sigma$, and the boundary correlators in DSSYK correspond to geodesic/boundary-to-boundary correlators in dS$_3$, up to squaring.

Figure 1: Schematic depiction of the AdS$_{1+2}$ holographic perspective with two time directions. The $\psi(t)$ mode traces a 1D path on the 2D boundary of the emergent 3D geometry, with the gluing surface $\Sigma$ spanning between.

Soft Mode Effective Theory in DSSYK

At large $N$ and $_3$0, with $_3$1 and energies rescaled as $_3$2, the non-linear dynamics are captured by a complex reparametrization mode $_3$3. The effective action

$_3$4

with Hamiltonian (for time-dependent $_3$5 and MQ coupling $_3$6):

$_3$7

$_3$8 and $_3$9 are canonically conjugate via $\Sigma$0. Semi-classical trajectories of $\Sigma$1 are given by Hamilton's equations, with $\Sigma$2 determined from $\Sigma$3 as

$\Sigma$4

The observable two-point function is

$\Sigma$5

Holographic Duality: dS$\Sigma$6 Gravity and the Soft Mode

The central proposal is that the dynamical variable $\Sigma$7 corresponds to the embedding of a codimension-one dS$\Sigma$8 hypersurface $\Sigma$9 in global dS$_3$0. All non-trivial gravity dynamics are thus supported on this surface, which acts as the non-linear dynamical screen for the 3D bulk. Specifically, the equations of motion for the soft mode are shown to be equivalent to Israel junction conditions across $_3$1 in dS$_3$2:

$_3$3

Here, $_3$4 is directly sourced by the boundary MQ coupling $_3$5.

The reduced 1D effective action for $_3$6 is derived via explicit evaluation of all gravitational and boundary (Gibbons-Hawking-York, Hayward corner) terms in the dS$_3$7 Einstein-Hilbert action, imposing $_3$8 conformal (hyperbolic slicing) boundary conditions at $_3$9. The resulting action exactly matches the SYK effective action, identifying $\Sigma$0.

Figure 2: Schematic depiction of the standard Euclidean gravity path integral for Schwarzschild-de Sitter; overlaps of Hartle-Hawking wave functions produce the relevant glued geometry.

Geometry: Thermal and Shockwave Solutions

For special symmetric solutions, the dual geometries admit transparent interpretation:

• Thermal Solutions: The soft mode solution at inverse temperature $\Sigma$1 maps onto a Schwarzschild-dS$\Sigma$2 (SdS$\Sigma$3) with conical deficit angle $\Sigma$4 at the poles, controlled by the DSSYK parameter $\Sigma$5. Deficit angle–energy–entropy relations are explicitly matched.

Figure 3: The DSSYK spectral density and energy parametrized by $\Sigma$6. Positive and negative temperature regimes correspond to different monotonicity of density with energy.

• Delta Function Source: A shockwave in the coupling $\Sigma$7 creates a local conical defect on $\Sigma$8. The orientation of the conical wedge determines which region of the dS$\Sigma$9 spacetime is identified as physical/removable.

  Figure 4: The gluing surfaces $_3$0 and $_3$1 for a shockwave with $_3$2, exhibiting the creation of a localized conical defect.

  

  Figure 5: The gluing surfaces $_3$3 and $_3$4 with $_3$5, leading to distinct spacetime identifications.

• Constant $_3$6: Cutting the spatial sphere at fixed $_3$7 corresponds in the boundary model to $_3$8 with constant imaginary part and leads to geometry with disconnected trajectories, further confirming the dictionary.

Spectral Density and Entropy

The path-integral evaluation of microcanonical entropy in the 1D theory, via the phase-space integral $_3$9, reproduces the exact spectral density and entropy formula

$_{1+2}$0

as opposed to the standard Gibbons-Hawking entropy expression for dS$_{1+2}$1. The difference is traced to the $_{1+2}$2 conformal boundary conditions, and is manifested as extra contour contributions (Figure 6).

Figure 6: The Euclidean path integral for near-dS$_{1+2}$3 gravity, with gluing of Hartle-Hawking wavefunctions via $_{1+2}$4 conformal boundary conditions; semiclassical entropy follows from contributions to the Hayward and bulk action.

Holographic Two-Point Functions

A central result is the identification of the boundary-to-boundary Green function in dS$_{1+2}$5 with the square modulus of the DSSYK two-point function (the gravity result corresponds to combining both the right and left soft mode sectors):

$_{1+2}$6

with geodesic distance

$_{1+2}$7

This identification of the dS propagator with the modulus squared of the chiral DSSYK correlator reflects the necessity to combine both “left” and “right” copies (chiral and anti-chiral), analogous to how full CFT correlators assemble from products of chiral blocks.

Signature Ambiguity and Two-Time Perspective

There exists a structural ambiguity: the emergent 3D geometry may be viewed either as a dS$_{1+2}$8 (with one time, two space) or as AdS$_{1+2}$9 (two time, one space), as the signature can be flipped by redefining the roles of $\psi(t)$0 and the holographic direction. In both cases, the soft mode action and its equation of motion are unchanged, but the physical interpretation of boundary locality and operator ordering changes. This points towards "two-time physics" perspectives for holography, and could influence bulk causality constructs.

Theoretical and Practical Implications

The proposal demonstrates that:

• A 1D quantum mechanical model (DSSYK) with specific couplings encodes the gravitational dynamics of 3D (near-)de Sitter gravity, expanding the scope of holography and non-AdS dualities.
• The mapping works for all energies, not just near the Schwarzian/AdS$\psi(t)$1 limit, by a well-controlled non-linear generalization.
• The soft mode “gravitational” dynamics is made explicit via the matching of effective actions and equations of motion.
• The full boundary-to-boundary propagators in the bulk receive contributions from both chiral sectors, with important implications for the structure of quantum state spaces, operator algebra (including non-trivial braiding/commutation), and the emergence of time in holography.

Potential extensions involve the incorporation of finite $\psi(t)$2 (i.e., $\psi(t)$3) corrections, generalization to finite temperature/energy configurations, and investigation of the precise operator and bulk locality structure for higher-point functions or out-of-time-order correlators. There are possible ties with quantum group/6j-symbol algebraic structures and connections to dynamical gravity in both de Sitter and AdS signatures.

Speculations and Future Directions

• The construction enables the study of quantum gravity in de Sitter backgrounds using solvable boundary models, with applications to cosmology and quantum information in expanding universes.
• The observed necessity to square modulus the chiral boundary correlator, as imposed by the bulk gravitational path integral, may indicate a universal feature of de Sitter-like holography: the emergence of the so-called "modular square" structure at the boundary.
• The mathematical equivalence between the double scaling limit of SYK and quantum group symmetries observed in both boundary and bulk may unlock the pathway towards constructing exact quantum gravity duals in more general non-AdS spacetimes.

Conclusion

This work presents a precise non-linear holographic correspondence between the dynamics of the complex soft mode in the DSSYK model (with time-dependent MQ coupling) and 2+1-dimensional Einstein-de Sitter gravity, mediated by a single complex embedding mode $\psi(t)$4. The mapping is exact at the classical level and current evidence supports its quantum completion. It extends the Schwarzian/JT/AdS$\psi(t)$5 holography paradigm to its double-scaled and de Sitter gravity counterparts, clarifies the spectral properties, and motivates exploration into higher-dimensional, non-AdS, and two-time holography. The results deepen the connection between exactly solvable quantum mechanical models and semiclassical/quantum gravity in the de Sitter context.