Where is the Entropy in DSSYK-de Sitter? Correction to a wrong claim (2511.10907v1)

Abstract: A question arises in the holographic description of the static patch of de Sitter space: Where does the entropy reside? The answer of course is in the stretched horizon, but how far from the mathematical horizon is the stretched horizon? In papers and lectures I argued that the entropy in DSSYK/JT-de Sitter resides at a string distance from the horizon. That conclusion was based on misconception about the confinement-deconfinement transition in the 't Hooft model. When corrected the right answer is of order the Planck distance (which differs from the string distance by a factor of order $\sqrt{N}).$

• The paper revises the entropy localization in DSSYK–de Sitter duality by establishing that the stretched horizon exists at the Planck scale rather than the string scale.
• It employs detailed analysis of energy scales and the 't Hooft model to correct the previously conflated string/QCD scale with the true deconfinement transition temperature.
• The findings highlight key differences between two-dimensional open-string models and higher-dimensional closed-string theories, emphasizing the impact on de Sitter holography.

Correction of the Entropy Scale in DSSYK–de Sitter Duality

Introduction

This essay provides a technical summary of "Where is the Entropy in DSSYK-de Sitter? Correction to a wrong claim" (2511.10907), which revises a longstanding assertion about the scale at which entropy is stored in the context of the double-scaled SYK (DSSYK)–de Sitter correspondence, and its relationship to the 't Hooft model. The author critiques previous identifications of thermodynamic and string-theoretic scales, eschewing the previous conflation of the string/QCD scale with the true confinement-deconfinement transition temperature. The discussion highlights implications for entropy localization near de Sitter horizons and clarifies the disparity between string and Planck scales in two-dimensional models versus higher-dimensional string theories.

Notation and Scale Relations

The work reiterates and corrects notational conventions regarding coupling constants in the context of the 't Hooft model and affiliated string-theoretic constructions. Specifically, for the DSSYK model, the central relation is:

$\lambda = g_{\text{open}}^2$

with $\lambda$ as the double-scaling parameter and $g_{\text{open}}$ as the open-string coupling. The closed-string coupling in comparable string-theory models is accordingly $g_{\text{closed}} = g_{\text{open}}^2$, but the 't Hooft model is strictly an open-string theory with no physical closed-string states.

The energy scales in the model are established as follows:

• $M_{\text{min}}$: Gibbons-Hawking temperature
• $M_{\text{Planck}}$: Planck mass
• $M_{\text{string}}$: string mass, $$M_{\text{string}} = \sqrt{\lambda} M_{\text{micro}} = \sqrt{\frac{\lambda}{N}} M_{\text{Planck}}$$
• $M_{\text{micro}}$: geometric mean scale, $$M_{\text{micro}} = \sqrt{M_{\text{Planck}} M_{\text{min}}} = M_{\text{Planck}}/\sqrt{N}$$

The connections between these scales are central to the arguments regarding the phase structure near the de Sitter horizon.

Reevaluation of the Phase Boundary

The core correction in this work is the disambiguation between the string/QCD scale ($\Lambda$) and the actual confinement-deconfinement temperature ($T_c$) in two-dimensional large-$N$ QCD as realized in the 't Hooft model. Previous treatments erroneously located the phase boundary between confined and deconfined degrees of freedom at the string scale $\Lambda$, which would map the entropy-carrying "stretched horizon" to a distance $\ell_{\text{string}}$ from the event horizon.

The rectification, based on the statistical mechanics of the 't Hooft model as computed by McLerran and Sen, stipulates that the true deconfinement transition occurs at

$$T_c = \Lambda \sqrt{N} = M_{\text{string}} \sqrt{N}$$

Employing the string and Planck scale relations, the transition temperature is given by

$T_c = g_{\text{open}} M_{\text{Planck}}$

which is parametrically larger than $M_{\text{string}}$ for large $N$. The proper distance of the stretched horizon from the true horizon is then $$\rho_{\text{sh}} = T_c^{-1} = g_{\text{open}}^{-1} \ell_{\text{Planck}}$$, indicating that the entropy is concentrated at Planckian distances, not at the string scale.

Comparison to Higher-Dimensional String Theory

In higher-dimensional closed-string theories, the Hagedorn transition marks a phase of exponential density of states at

$T_c = M_{\text{string}}$

However, here the string and Planck masses are related by

$$M_{\text{string}} = g_{\text{closed}} M_{\text{Planck}}$$

so that the transition temperature also takes the form

$T_c = g_{\text{closed}} M_{\text{Planck}}$

Thus, both the open-string (DSSYK) and closed-string (conventional string theory) cases manifest structurally analogous formulae for the entropy-localizing transition temperature, with the distinction of the pertinent coupling constant (open versus closed string).

A key implication is that in higher dimensions, $g_{\text{closed}}$ may be of order unity so $M_{\text{string}}$ and $M_{\text{Planck}}$ are comparable. In contrast, in two dimensions for large $N$, the hierarchy can be significant, so that the scale at which horizon entropy is located is Planckian and much smaller than the string scale.

Entropy Localization and Theoretical Consequences

The corrected identification of the phase transition and entropy scale implies:

• The "stretched horizon" is only Planck-scale separated from the true horizon in the dual field theory.
• The horizon entropy is a consequence of Planck scale physics, not string-scale physics, in the DSSYK–de Sitter correspondence.
• The parametric separation of scales in two-dimensional models introduces quantitative and qualitative differences from higher-dimensional de Sitter–string theory correspondences.
• The result emphasizes the care required in translating between thermal phenomena in QCD-like models and their putative gravitational duals, especially regarding the physical meaning of "stringy" versus "quantum gravity" regimes.

Numerical and Conceptual Implications

The central quantitative assertion is that the transition temperature, and thus the localization of entropy, scales as $T_c \sim g M_{\text{Planck}}$ with $g$ the appropriate string coupling. In two-dimensional open-string models, the parametric separation $T_c \gg M_{\text{string}}$ provides a sharp counterpoint to prior assumptions. This also implies that the entropy associated with the horizon is reflective of high-energy, quantum gravitational microstates, not merely stringy excitations.

The result underscores the importance of re-examining conventional wisdom regarding where entropy resides in holographic correspondences involving de Sitter or Rindler spaces, especially for low-dimensional or double scaled models.

Future Directions

This clarification invites several avenues for further research:

• Analysis of horizon microstates in other exactly solvable models, particularly those with large scale separations.
• Extension of these results to more intricate gravitating systems and examination of their applicability beyond the double-scaled/SYK paradigm.
• Exploration of the impact on quantum information-theoretic interpretations of de Sitter entropy, specifically regarding the dominance of Planck-scale contributions.

Conclusion

The correction established in this work resolves a misidentification of the scale at which entropy is stored in the DSSYK–de Sitter system, establishing the Planck scale, not the string scale, as decisive for the stretched horizon in the dual QFT. This influences theoretical interpretations of holography in de Sitter spacetimes, especially for lower-dimensional open-string models, and highlights critical differences in the scaling of transition temperatures between open and closed string cases.