Asymptotic Algebras and Holography of Information in CGHS Model

Abstract: Holography of Information (HoI) and the island formula are two recent approaches to the black hole information loss paradox that yield different Page curves. The difference between the Page curves has been argued to be rooted in the algebra that each approach focuses on. In this paper, we study the candidate algebras for the CGHS model. Making the usual assumptions that are made in HoI literature, we establish HoI for the CGHS model coupled to a massless scalar by constructing the radiative phase space and asymptotic algebras at future null infinity. We prove that the quantum state of the right-moving modes can be recovered from an arbitrarily small neighbourhood of the right future null infinity. We then argue that if the Hilbert space factorization between left- and right-moving modes holds in the quantum theory of CGHS, then the island entropy cannot be identified with the von Neumann entropy of any one-sided asymptotic algebra, and that this mismatch constitutes a version of the factorization problem for CGHS/RST models. We discuss the possible distinction between the HoI and island approaches for the CGHS model.

• The paper demonstrates that in the CGHS model, asymptotic algebras enable full reconstruction of the quantum state from a tiny neighborhood at null infinity, yielding a flat Page curve for radiation entropy.
• It employs a covariant phase space analysis to show that the right-moving sector solely determines the asymptotic symmetry algebra, highlighting clear factorization between left and right modes.
• The paper contrasts the operator-algebraic approach of holography of information with the island formula, underscoring fundamental differences in handling black hole information and entropy reconstruction.

Asymptotic Algebras and Holography of Information in the CGHS Model

Introduction and Context

This work systematically analyzes the interface between Holography of Information (HoI) and the quantum extremal surface (island) formula in resolving the black hole information paradox, specifically within the context of the two-dimensional Callan-Giddings-Harvey-Strominger (CGHS) model. These two frameworks offer sharply contrasting predictions for the radiation entropy (Page curve) during black hole evaporation: HoI predicts a constant (flat) Page curve, while the island formula yields the expected rising-and-falling Page curve. The central question explored is whether this dichotomy can be traced purely to the choice of asymptotic algebra, or whether the distinction is more fundamental in low-dimensional dilaton gravity.

The CGHS model provides an ideal setup for this comparison, owing to its tractability: the dynamics separate into left- and right-moving matter sectors ($f^+, f^-$), and the asymptotic structure at null infinity is particularly transparent.

Radiative Phase Space and Asymptotic Symmetry Algebra

The authors provide a detailed covariant phase space analysis, clarifying that—unlike the situation in higher dimensional gravity—the only dynamical degrees of freedom at the null boundary in the CGHS model are those of the massless scalar field. All gravitational variables (dilaton, metric) are non-dynamical at null infinity. The explicit construction of the boundary symplectic form at $\mathcal{I}^+_R$ reveals that it is determined entirely by the right-moving sector, with cross-terms vanishing due to rapid decay (Schwartz falloff) at infinity. The resulting algebra consists of local algebras $A_{\rm rad}(I)$—generated by smeared $f^-(x^-)$ and $\partial_- f^-(x^-)$—which exhibit the expected properties of additivity and irreducibility.

A key technical result is that the asymptotic symmetry group corresponds to diffeomorphisms of the null coordinate $x^-$. The boundary canonical charge $Q_\epsilon$ for vector fields $\epsilon(x^-) \partial_-$ is quadratic in the field, and no degenerate soft vacua arise, in sharp contrast to the BMS structure in $3+1$ gravity. The generator for rigid translations along $x^-$ is the right-moving (Bondi) Hamiltonian $\mathcal{I}^+_R$0, which encodes energy flux through null infinity.

Holography of Information in CGHS

Under assumptions analogous to those employed in $\mathcal{I}^+_R$1 asymptotically flat gravity—positivity of the spectrum and the uniqueness of the ground state for $\mathcal{I}^+_R$2—the analysis establishes nonperturbatively that the global algebra of bounded operators on the radiative Hilbert space $\mathcal{I}^+_R$3 can be generated from the algebra associated with an arbitrarily small neighborhood of the past boundary of $\mathcal{I}^+_R$4, once the Bondi mass operators $\mathcal{I}^+_R$5 (for all cuts $\mathcal{I}^+_R$6) are included. This gravitationally completed algebra $\mathcal{I}^+_R$7, when $\mathcal{I}^+_R$8 contains any open set, satisfies $\mathcal{I}^+_R$9—full reconstruction from asymptotic data is possible.

This algebraic result translates directly to the statement that the entropy $A_{\rm rad}(I)$0 of the reduced density matrix associated to the algebra $A_{\rm rad}(I)$1 is independent of $A_{\rm rad}(I)$2. That is, the fine-grained von Neumann entropy of the right-moving Hawking radiation remains flat along $A_{\rm rad}(I)$3, for all states. Extensions of this argument to situations with residual horizons, as well as to the full double-boundary system, are discussed: the former yields a flat (nonzero) entropy due to residual entanglement with left-moving modes, matching expectations from partial trace over inaccessible interior degrees of freedom.

Numerical claims are qualitative in this context, but the construction provides rigorous operator-algebraic control over where information is localized.

Comparison with the Island Formula and the Factorization Problem

An incisive point of the analysis is the explicit demonstration that radiative algebras $A_{\rm rad}(I)$4 are intrinsically unable to access the left-moving sector $A_{\rm rad}(I)$5, due to sharp kinematical factorization. In contrast, the island prescription, exemplified in RST/CGHS computations (e.g., [Hartman et al., (Hartman et al., 2020)]), yields an entropy that involves both $A_{\rm rad}(I)$6 and $A_{\rm rad}(I)$7, since the extremal surface (the "island") can probe behind the horizon and access both sectors. This is not merely an artifact of boundary conditions or special features of model black holes—the discrepancy persists fundamentally because the gravitational path integral (with replica wormholes) does not respect the canonical factorization $A_{\rm rad}(I)$8.

This is cast as structurally analogous to the factorization problem in AdS/CFT, wherein the Euclidean path integral links otherwise causally disconnected boundary Hilbert spaces through wormhole geometries, violating naive factorization. In the present context, the two sectors correspond to spatially separated, but causally connected, asymptotic boundaries of the same 2D spacetime. The algebraic construction based on Fock quantization preserves factorization, but the Euclidean gravitational path integral is insensitive to this, thereby generating entropy curves at odds with operator-algebraic expectations.

Practical and Theoretical Implications

Practical Consequences

• In the operator-algebraic (HoI) approach, information about the right-moving Hawking radiation can always be recovered from an arbitrarily small region at null infinity, suggesting no information loss, but the resulting entropy curve is strictly flat.
• The island formula, valid in gravitational effective field theories that admit a holographic UV completion, generates a nontrivial Page curve that matches the naive expectations from unitarity of black hole evaporation, but at the cost of violating the boundary factorization structure in the CGHS/Fock quantization.

Conceptual Implications and Future Directions

• Non-holographic Theories: The CGHS model quantized as a Fock space for matter fields does not support microstates with finite Bekenstein-Hawking entropy. Thus, the applicability of the island formula may be limited to theories with strong holographic properties.
• Holography in HoI vs. the Island Formula: The sense in which "holography" is realized in HoI is weaker—it is a property of redundancy of asymptotic data in reconstructing bulk physics, not necessarily tied to finite entropy or absence of global symmetries, unlike in the strong AdS/CFT-inspired holography relevant for islands.
• UV Sensitivity and Ensemble Averaging: The path integral derivations of the island formula are expected to be reliable only when an ensemble average over geometries (e.g., baby universes, wormholes) is justified. The absence of such effects in a strictly factorized Fock quantization questions the universality of the Page curve derived via the island prescription.

Conclusion

The paper presents a comprehensive algebraic treatment of the information paradox in the CGHS model, exposing a sharp tension between the predictions of HoI and the island formula for the entropy of Hawking radiation. The results indicate that in the operator-algebraic formalism natural to Fock quantization in the CGHS model, the entire quantum state can be reconstructed from arbitrarily small neighborhoods at null infinity (flat Page curve), with no sensitivity to left-moving sectors in single-boundary setups. By contrast, the island proposal yields a Page curve that incorporates both left and right sectors, in contradiction with the operator-algebraic structure unless the factorization property of the Hilbert space is abandoned—mirroring the factorization problem seen in more holographic settings.

The broader implication is that the applicability of replica wormhole techniques and the island formula is intimately tied to the holographic properties (or lack thereof) of the UV completion of the theory. The CGHS model provides a clarifying case where these issues can be analyzed with exceptional precision, illuminating challenges for the generalization of the modern gravitational entropy program to non-holographic quantum gravity.

References

• "Asymptotic Algebras and Holography of Information in CGHS Model" (2606.09346)
• "Islands in Asymptotically Flat 2D Gravity" (Hartman et al., 2020)
• "The Holographic Nature of Null Infinity" (Laddha et al., 2020)
• "An apologia for islands" (Antonini et al., 4 Jun 2025)
• "Seeing Page Curves and Islands with Blinders On" (Geng et al., 6 Feb 2026)
• "Global symmetry, Euclidean gravity, and the black hole information problem" (Harlow et al., 2020)
• "Euclidean gravity and holography" [Harlow, Shaghoulian: 2021]