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Holomorphic extremal surfaces for pseudoentropy in dS/CFT

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Giorgos Anastasiou, Ignacio J. Araya, Avijit Das, Javier Moreno

High Energy Physics - Theory · Submitted 17 Oct 2026 · 114 references

Pith's verdict: A careful, well-motivated paper that gives a consistent holomorphic extremal-surface prescription for pseudoentropy in dS/CFT, solving a known limitation of the piecewise construction for the cases it treats.

T0 · AI review

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arXiv ID2610.24698 (version 1)
Pith ID46CZOMWS, Pith's permanent record of this paper and its review
CitationAvijit Das, Giorgos Anastasiou, Ignacio J. Araya, Javier Moreno. "Holomorphic extremal surfaces for pseudoentropy in dS/CFT." arXiv:2610.24698 (2026). https://pith.science/paper/46CZOMWS
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The paper

The paper claims

Holomorphic extremal surfaces compute pseudoentropy for any spherical region in dS/CFT.

The reading

This paper formulates the extremization problem for codimension-two surfaces in de Sitter space directly in holomorphic coordinates, treating the bulk metric as the analytic continuation of the Lorentzian dS metric. It shows that the resulting equation, $K^I=0$, defines a unique complex extremal surface for a given homology class of integration contours, and that the area (hence the holographic pseudoentropy) is independent of the contour within each class. For spherical entangling regions and their small harmonic deformations in any boundary dimension, the universal terms of the pseudoentropy are computed and shown to match and extend the earlier piecewise Euclidean–Lorentzian construction. The construction also reproduces the first law of pseudoentropy for conical defects in dS$_3$. The work provides a unified framework that overcomes the limitations of the piecewise prescription and opens the way to study more general backgrounds and subregions.

What carries the argument

The central machinery is the holomorphic extremality equation $K^I=0$, the vanishing of the complex mean curvature vector of a codimension-two surface in the complexified de Sitter manifold $M_\mathbb{C}$ (obtained by promoting all global coordinates to complex numbers). The surface is anchored on a real entangling region at future infinity, and the integration contour is chosen from a homology class labelled by an integer $n$ that counts winding around the $\tau$-cylinder. The invariance under continuous contour deformations (Cauchy's theorem) and the discrete isometry $\tau\to\tau+i\pi$ organize the pseudoentropy into classes.

Extended reading notes

Core claim

The holomorphic Ryu–Takayanagi prescription (the condition $K^I=0$) yields a unique pseudoentropy for a given homology class of integration contours on the complexified extremal surface. For spherical regions and their harmonic deformations, the result matches and extends the piecewise Hartle–Hawking construction to arbitrary region size and arbitrary boundary dimension. Inequivalent homology classes, labelled by an integer $n$ associated with the shift $\tau \to \tau + i\pi$, differ by a sign of the universal term in odd boundary dimensions and by real finite shifts in even dimensions. For conical defects in dS$_3$, the first law of pseudoentropy holds and the contour degeneracy within each homology class persists.

Load-bearing premise

The complexified de Sitter metric (Eq. 2.5) is taken as the holomorphic extension of the Lorentzian metric, and the existence of a smooth turning point (Eq. 2.27) for the extremal surface in the complex domain is assumed; in even boundary dimensions, the branch points of the area integrand are resolved by a local uniformization that may not hold for more general backgrounds or higher-order deformations.

Editorial extensions

If this is right

  • The universal expansion of pseudoentropy through second order in shape deformations is confirmed, establishing a Mezei-like formula for dS/CFT that constrains the CFT central charges.
  • The contour independence within each homology class supports the interpretation of the bulk geometry as emergent, since different geometric realizations (Lorentzian, Euclidean, or genuinely complex) yield the same pseudoentropy for the vacuum.
  • The integer label $n$ provides a new quantum number for complex saddles; its effect on the pseudoentropy (sign flip in odd dimensions, shift in even dimensions) is a testable signature in holographic computations.
  • The first law of pseudoentropy is verified for conical defects, indicating that the holomorphic framework can be extended to excited states and may capture the linearized bulk dynamics.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the holomorphic prescription is correct, it should apply to non-spherical subregions (e.g., strips) and reproduce finite pseudoentropies where the piecewise construction fails; this prediction can be tested numerically.
  • The integer $n$ labeling homology classes may correspond to different sheets in the analytic continuation of the boundary replica twist correlator; a direct CFT derivation of the $n$-dependence would strengthen the duality.
  • For even boundary dimensions, the pseudoentropy contains a logarithmic term whose coefficient is universal; this coefficient should match the A-type anomaly of the nonunitary CFT, providing a cross-check.
  • In excited states or black-hole backgrounds, the contour degeneracy is expected to be lifted; the present work sets a baseline for future studies of complex saddles in Schwarzschild–dS.
Original abstract

We formulate the extremization problem for codimension-two hypersurfaces in de Sitter space in holomorphic coordinates, with real anchoring conditions at future infinity. The resulting equation ($K^I=0$) describes complex bulk realizations extending the usual piecewise Hartle--Hawking construction. We compute holographic pseudoentropy for harmonic deformations of spherical regions of arbitrary size and dimension, confirming its universal expansion through second order in the deformation parameter. The result is independent of the integration contour within each homology class on the complexified extremal surface. Inequivalent classes, labeled by an integer associated with a discrete global-coordinate isometry, differ by a sign of the universal term in odd boundary dimensions and by real finite shifts in even dimensions. For conical defects in dS$_3$ parametrized by a Schwarzschild-dS mass, we also verify the first law of pseudoentropy and the persistence of contour degeneracy within each homology class.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-20: 1 request, 5,037 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.23 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.35 · compares with prior results 0.53 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We formulate the extremization problem for codimension-two hypersurfaces in de Sitter space in holomorphic coordinates, with real anchoring conditions at future infinity. The resulting equation ($K^I=0$) describes complex bulk realizations extending the usual piecewise Hartle--Hawking construction. We compute holographic pseudoentropy for harmonic deformations of spherical regions of arbitrary size and dimension, confirming its universal expansion through second order in the deformation parameter. The result is independent of the integration contour within each homology class on the complexified extremal surface. Inequivalent classes, labeled by an integer associated with a discrete global-coordinate isometry, differ by a sign of the universal term in odd boundary dimensions and by real finite shifts in even dimensions. For conical defects in dS$_3$ parametrized by a Schwarzschild-dS mass, we also verify the first law of pseudoentropy and the persistence of contour degeneracy within each homology class.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We formulate the extremization problem for codimension-two hypersurfaces in de Sitter space in holomorphic coordinates, with real anchoring conditions at future infinity. contribution 0.60 0.92 0.08 asserted 0.99
s1 The resulting equation ($K^I=0$) describes complex bulk realizations extending the usual piecewise Hartle--Hawking construction. result 0.42 0.85 0.15 asserted 1.00
s2 We compute holographic pseudoentropy for harmonic deformations of spherical regions of arbitrary size and dimension, confirming its universal expansion through second order in the deformation parameter. result 0.83 0.95 0.34 asserted 1.00
s3 The result is independent of the integration contour within each homology class on the complexified extremal surface. result 0.99 0.84 0.12 asserted 1.00
s4 Inequivalent classes, labeled by an integer associated with a discrete global-coordinate isometry, differ by a sign of the universal term in odd boundary dimensions and by real finite shifts in even dimensions. result 1.00 0.89 0.20 asserted 1.00
s5 For conical defects in dS$_3$ parametrized by a Schwarzschild-dS mass, we also verify the first law of pseudoentropy and the persistence of contour degeneracy within each homology class. result 0.83 0.94 0.10 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.99 · evidence from proof or derivation 0.98.

What was examined, and what was not

sources read
  • arXiv listing abstract v1
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. Representation of a complex RT surface corresponding to an arbitrary real en￾tangling region on I + R . This boundary is the chosen Euclidean real section of the complex conformal boundary of MC. The data of the dual nonunitary CFT are specified on this real section, while the corresponding extremal surface extends into the complexified bulk. applied to the HH wavefunction computes S(A) from the area of the codimension-two surface ΣA fixed by the replica symmetry [73], which becomes extremal and anchored on ∂A as the number of replicas tends to one. In the original formulation [37,…
Figure 2. (Left) We show pictorially the disk entangling region A = B 2 of radius tE = T0 defined in S 2 , which is in the equator—that is, at θ1 = 0 following (2.23)—of S 3 . (Right) We show the same disk-like entangling region of unit radius perturbed harmonically, B 2 ϵ , following (2.35). complexified. Now consider a ball-shaped entangling region, A = B 2 with entangling surface ∂A = S 1 , defined on the θ1 = 0 slice of the future conformal boundary (I + R ≃ S 3 )—see
Figure 3. (Left) Representative contours Cϕ ≃ C′ ϕ in the complex ϕ-plane, connecting ϕ = 0 to ϕ = 2π; since ϕ ∼ ϕ + 2π, both wind once around the angular cylinder. (Right) Representative contours for n = 0, from the UV endpoint τUV to τ (0) ⋆ : the canonical C (0) τ (red), C ′ τ through the Lorentzian and Euclidean sections (blue), and a generic complex contour C ′′ τ (green). Since cosh τ is entire, all give the same area. 2.4.2 Small deformation around sphere Having established that the holomorphic RT prescription reproduces the pseudoentropy of the spherical entangling region, we now turn…
Figure 4. Contours from τUV to τ (0) ⋆ for even d. (Left) Direct approach to the turning point. (Right) One winding around the turning point followed by an approach on the second sheet, shown in a lighter shade. The wavy lines denote branch cuts. Both contours belong to the n = 0 class and give the same area. Evaluating Eq. (3.5) with the cutoff τUV = − log δ, we obtain the expansion in Eq. (2.21), with H = sin T0 and cd−2k = − iL d−1π d/2−1Γ(d/2 − k) 4GN Γ(d − 2k + 1)Γ(k) 2F1  1 − k, d 2 − k; d − 2k + 1; sin2 T0  , (3.7) c0 = − iL d−1π d/2−1 4GN Γ(d/2) H (1) d/2−1  cos2

Figures reproduced from arXiv:2610.24698 by the authors.

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