DSSYK–de Sitter Duality in Quantum Gravity

• DSSYK–de Sitter duality is a precise quantum holographic mapping between the double-scaled SYK model and static patch de Sitter gravity that captures thermodynamic properties and chaos bounds.
• It connects combinatorial chord algebra and ensemble-averaged operator dynamics in the DSSYK model with gravitational observables such as Gibbons–Hawking entropy, quasinormal modes, and spatial complexity.
• The duality leverages large-N, large-q limits to reproduce static-patch correlators and complexity measures, providing actionable insights into the microphysics of cosmological horizons.

The DSSYK–de Sitter duality, also called the DSSYK–dS correspondence, designates a precise, quantum-level holographic mapping between the double-scaled Sachdev-Ye-Kitaev (DSSYK) model of quantum mechanics and static patch de Sitter (dS) gravity—most concretely in two and three bulk spacetime dimensions. The duality is constructed in an ensemble-averaged, maximally chaotic, large-$N$, large-$q$ limit, and organizes the microscopic, combinatorial degrees of freedom of the DSSYK boundary theory so as to reproduce gravitational, thermodynamic, and dynamical features of the de Sitter static patch, including entropy, temperature, partition function, quasinormal modes, chaos bounds, and notions of spatial complexity.

1. Formulation of the DSSYK–de Sitter Duality

The double-scaled SYK model is an exactly solvable system of $N$ Majorana fermions with $q$-body random interactions, where the double-scaling limit takes $N \to \infty$, $q \to \infty$, at fixed $\lambda = q^2/N$ of $O(1)$. At infinite temperature ($\beta \to 0$), the system achieves maximal entropy, and the spectrum is described by a wide, bounded density of states.

The bulk dual is de Sitter gravity, primarily in the static patch, with two natural settings:

• $d = 2$: Jackiw–Teitelboim gravity with a positive cosmological constant (JT-dS), corresponding to the Lorentzian action

$$S_{dS} = -\frac12 \int d^2x\, \sqrt{g}\, [\phi R + 2\Lambda \phi] + \text{bdy}.$$

The near-horizon or static-patch limit corresponds to a bounded subregion (the "static patch" $0 \leq r < \ell$) with dilaton profile proportional to $r/\ell$ (Rahman, 2022, Okuyama, 12 May 2025).

• $d=3$: Schwarzschild–de Sitter (SdS$_3$) spacetime with or without a conical defect representing an observer, and/or circle reduction to obtain a two-dimensional effective theory (Narovlansky et al., 2023, Tietto et al., 6 Feb 2025, Verlinde, 1 Feb 2024).

The central operational step is the identification of the DSSYK time with the global static-patch cosmic time in de Sitter space; the boundary operators and spectrum in the maximally mixed ($T = \infty$) DSSYK state reproduce bulk static-patch correlators, entropy, and quantum chaos.

2. Holographic Dictionary and Model Correspondence

The dictionary connecting DSSYK and de Sitter gravity is as follows (Susskind, 2022, Narovlansky et al., 2023, Verlinde, 1 Feb 2024, Tietto et al., 6 Feb 2025):

DSSYK Parameter/Observable	dS Gravity Quantity	Notes
$N$	Gibbons–Hawking entropy $S_{dS}$	$S_{dS} \sim N$
$q$	Bulk string/interaction scale ($L_s \sim 1/q$)	Must have $L_s \ll L_{dS}$
$\lambda = q^2/N$	Sub-dS locality parameter, 't Hooft coupling	Must be fixed, $O(1)$
Partition function $Z(\beta)$	Static patch partition function	Integral over spectrum, matches bulk calculation
DSSYK time $\tau$	Static-patch cosmic time $t_{dS}$	Exact identification
Spectral density $\rho(E)$	Static-patch density of states	Matches square-root edge, total entropy
Single-fermion correlator	Bulk lowest QNM scalar correlator	Exponential decay, Lyapunov growth
Chord (multi-fermion) operators	Bulk massive states/string excitations (confinement)	All but $O(1)$ confined (see below)
Two-point, four-point OTOC	De Sitter Green's, chaos bound (MSS)	$\lambda_L = 2\pi T_{dS}$
DSSYK $(H_L=H_R)$ constraint	Gauge invariance/projection on gravitational sector	Physical operators are bilocal, gauge-invariant in the bulk
Entanglement/complexity	Bulk geodesic separation, complexity of formation	Quantified via Krylov and spread complexity

The spectrum of DSSYK matches exactly with that of 3D de Sitter gravity, with identical $q$-deformed algebraic structures governing both the Hilbert space and operator algebra (Verlinde, 1 Feb 2024, Rahman et al., 17 Jul 2024).

3. Confinement, Species Problem, and Microscopic Degrees of Freedom

An essential mechanism in the DSSYK–de Sitter duality is the confinement of non-singlet (non-$SU(N)$ singlet) fermionic states on the boundary, which ensures that only $O(1)$ light bulk species propagate in the bulk gravitational theory (Susskind, 2023):

• Single-fermion two-point functions decay purely exponentially, with all but the global $SU(N)$ singlet channel vanishing at late times.
• Chord operators—representing multi-fermion composites—are destroyed exponentially, demonstrating that "de Sitter space has no chords:" the only unconfined, dynamical bulk fields are singlet combinations (graviton, photon, or mesons).
• This mechanism resolves the "species problem," i.e., the naive overcounting of bulk Hawking species, maintaining agreement with semiclassical Gibbons–Hawking expectations (Susskind, 2023).

The bulk interpretation is that the active degrees of freedom reside in a Planck-thick shell (the "stretched horizon") (Susskind, 14 Nov 2025). The entropy $S_{dS}\sim N$ is attributed to microstates confined to this shell, reflecting the genuinely quantum-gravitational (Planckian) nature of de Sitter entropy.

4. Chord Algebra, Complexity, and Bulk Holographic Structures

A remarkable combinatorial structure underpins the duality: DSSYK correlators can be rewritten in terms of sums over chord diagrams, with $q$-deformed rules encoding intersection statistics (Verlinde, 1 Feb 2024, Rahman et al., 17 Jul 2024). This structure has the following features:

• Chord algebra: The algebra of $p$-chords at the string scale encodes the full Hilbert space dynamics, governed by $q = e^{-2p^2/N}$-deformed oscillator relations.
• Wee-chords: On cosmic time scales, a distinct "wee-chord" algebra (with $q_{wee}=e^{-2n_H^2/N}$, $n_H=O(1)$) emerges that more closely matches the $SL(2,\mathbb{C})$ Chern–Simons algebra of Wilson loops in bulk dS$_3$ gravity. Identification of bulk observables with wee-chords restores the correct semiclassical separation of scales and entropy matching (Rahman et al., 17 Jul 2024).

In terms of complexity, DSSYK provides a microscopic realization of several complexity measures and their geometrization in the bulk (Aguilar-Gutierrez, 19 Mar 2024, Heller et al., 15 Oct 2025):

• Krylov (spread) complexity tracks the growth of operators under time evolution, identifying the average chord number with the bulk geodesic distance in dS$_2$ (Heller et al., 15 Oct 2025).
• Spread complexity encodes the entanglement between maximally mixed DSSYK copies, dual to the time separation (and hence the wormhole structure) between antipodal static-patch observers in dS$_3$ (Aguilar-Gutierrez, 19 Mar 2024).
• Query complexity and geometric (Nielsen) complexity map to, respectively, the network of chord (matter) insertions (dual to Wilson line junctions), and the geodesic length in the group manifold, both exhibiting bulk-to-boundary dual interpretations.

5. Duality in Two, Three Dimensions and Liouville-dS Gravity

Multiple bulk effective gravitational models participate in the DSSYK–de Sitter correspondence:

• JT-de Sitter Gravity ($d=2$ dS-JT): The large-$N$ double-scaled SYK model, zoomed in at the upper edge of its spectrum, reduces precisely to JT gravity with $\Lambda>0$ (Okuyama, 12 May 2025). All-loop bosonic correlators and partition functions including quantum fluctuations reproduce the dS-JT perturbative expansion (Okuyama, 12 May 2025).
• 3D de Sitter Gravity and Chern–Simons Formulation: The DSSYK spectrum and partition function match precisely those derived from the quantization of the 3D Schwarzschild-de Sitter phase space, with the chord algebra mirrored by the algebra of gravitational Wilson loops (Verlinde, 1 Feb 2024).
• 2D Liouville–de Sitter Gravity: Upon quantization of 3D dS in the Chern–Simons picture, the resulting theory is a sum of two Liouville CFTs with complex central charges adding to $c=26$, and partition functions and correlators exactly matching those of DSSYK (Verlinde et al., 4 Feb 2024).
• The triangle

$$\text{DSSYK} \;\longleftrightarrow\; \text{3D dS gravity} \;\longleftrightarrow\; \text{2D Liouville–dS gravity}$$

is realized by explicit matching of partition functions, correlation functions, operator content, and entropy. The sector selection—especially the $H_L=H_R$ constraint in doubled DSSYK—selects the correct physical Hilbert space for bulk dS holography (Narovlansky et al., 2023, Aguilar-Gutierrez, 26 Jun 2025).

6. Physical Interpretation, Open Problems, and Observational Sectors

The DSSYK–de Sitter duality stands as a concrete quantum model of static patch de Sitter holography that can be solved exactly at large-$N$, giving unprecedented access to the microstructure of horizons, quantum chaos, and operator growth in cosmological spacetimes. Key physical points and open directions include:

• Entropy localization: The system's entropy is stored in Planck-thick degrees of freedom near the cosmological horizon, and not at the much larger string scale—a correction arising from large-$N$ analysis of the 't Hooft model (Susskind, 14 Nov 2025).
• Emergence and scrambling: The time to "scramble" or fully access the static patch is $t_*\sim \beta_{dS} \log S_{dS}$, analogous to the black hole fast-scrambling time (Susskind, 2022, Rahman, 2022). The static patch is accessible after this timescale via boundary correlators.
• Relational observables: The doubled-DSSYK Hilbert space with $H_L=H_R$ admits a relational algebra of gauge-invariant (BRST) observables matching those accessible to a static observer in dS, including robust construction of complexity and (in principle) "baby universe" sectors (Aguilar-Gutierrez, 26 Jun 2025).
• Thermodynamic subtleties: There exist two distinct inverse temperatures—a "physical" temperature seen by an observer and the Gibbons–Hawking temperature at the horizon—mirrored by the two distinct periodicities in DSSYK correlators (Tietto et al., 6 Feb 2025).
• Limitations: Embedding the disorder-averaged model into a unique, unitary theory remains unresolved; mapping the behind-the-horizon region (FRW patch) or constructing fully dynamical observers in the bulk remains challenging (Susskind, 2023, Tietto et al., 6 Feb 2025).
• Extension and ambiguity resolution: The identification of the proper chord algebra (string scale vs cosmic scale "wee-chords") is refined to restore semiclassical consistency and correct entropy, resolving earlier disagreements in the literature (Rahman et al., 17 Jul 2024).

7. Significance and Broader Context

The DSSYK–de Sitter duality achieves an explicit, analytic, and non-perturbative realization of low-dimensional static-patch de Sitter holography, unifying quantum gravity, statistical, and operator-algebraic approaches. It provides:

• A laboratory for the microphysics of cosmological horizons and entropy counting.
• A combinatorial and algebraic underpinning for quantum chaos and complexity in dS spacetimes.
• Explicit bridges between random matrix theory, Chern–Simons gravity, Liouville quantum gravity, and the quantum theory of operator growth.
• Paradigmatic examples to paper observer complementarity, gauge-invariant relational observables, and quantum information in spacetimes with horizons.

Ongoing research seeks to clarify the precise role of ensemble averaging, extend the correspondence to higher dimensional de Sitter spacetimes, refine the microscopic theory of observers, and further resolve the structure of the (possibly) non-perturbative sectors, including relational and baby-universe algebras (Narovlansky et al., 2023, Aguilar-Gutierrez, 26 Jun 2025, Susskind, 14 Nov 2025).