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A Weyl-invariant renormalization route makes holographic pseudoentropy in de Sitter space finite and regulator-independent.

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2026-08-02 22:04 UTC pith:KZEZLGEB

load-bearing objection Solid conformal-renormalization machinery applied to dS pseudoentropy; the main caveat is the unproven AdS-to-dS analytic continuation of the CG/Einstein equivalence. the 3 major comments →

arxiv 2602.17989 v2 pith:KZEZLGEB submitted 2026-02-20 hep-th

Renormalized pseudoentropy in dS/CFT

Giorgos Anastasiou , Ignacio J. Araya , Avijit Das , Javier Moreno This is my paper
classification hep-th
keywords pseudoentropydS/CFT correspondenceconformal renormalizationconformal gravityextremal surfacesshape dependencecentral chargeholographic entanglement entropy
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Holographic pseudoentropy in de Sitter space, computed from extremal surfaces anchored at future infinity, diverges because of the infinite bulk volume near the boundary. This paper claims that a finite, regulator-independent universal part follows from conformal renormalization: evaluating the conformal-gravity action on the replica orbifold produces codimension-two functionals — the Graham–Witten action in dS4 and the Graham–Reichert action in dS6 — which reduce to renormalized areas on extremal surfaces. For spherical entangling regions the renormalized pseudoentropy is proportional to the complex central charge a* of the non-unitary dual CFT. For small shape deformations, the quadratic correction reproduces the analytic continuation of the standard AdS shape-dependence formula, which the authors read as a non-trivial check of the dS/CFT dictionary.

Core claim

The central claim is that the finite universal part of holographic pseudoentropy in dS4/CFT3 and dS6/CFT5 is obtained from conformally renormalized areas of extremal surfaces, derived from conformal-gravity functionals: in dS4, Su(B^2) = −πL*^2/(2GN), and in dS6, Su(B^4) = π^2L*^4/(3GN). The renormalized area is built from the Graham–Witten functional in four dimensions and the Graham–Reichert functional in six dimensions, after restricting to Einstein–dS backgrounds and extremal surfaces with vanishing trace of the extrinsic curvature. The quadratic shape corrections match the analytically continued AdS formula for holographic entanglement entropy, with the stress-tensor two-point coefficie

What carries the argument

The load-bearing mechanism is conformal renormalization. Start with the Weyl-invariant conformal-gravity action, which is finite on asymptotically (A)dS backgrounds without counterterms: Weyl-squared in four dimensions, and the distinguished cubic conformal combination (4I1 + I2 − I3/3) in six dimensions. Evaluate it on a replicated bulk geometry with a conical singularity of opening angle 2πϑ along the extremal surface; the O(1−ϑ) coefficient of the defect defines a codimension-two conformal functional, the Graham–Witten action in 4D and the Graham–Reichert action in 6D. On Einstein–dS backgrounds and extremal surfaces, these functionals reduce to a renormalized area (bare area plus a bound

Load-bearing premise

The entire scheme assumes that the equivalence between the conformal-gravity action and the renormalized Einstein–(A)dS action, proven in AdS, carries over to de Sitter by analytic continuation, a step the paper imports rather than proves.

What would settle it

Compute the on-shell conformal-gravity action on an asymptotically de Sitter spacetime with Neumann boundary conditions and compare it, term by term, with the renormalized Einstein–dS action; if boundary terms or the ghost-free sector behave differently on dS, the renormalized areas (2.23) and (3.22) lose their justification. Alternatively, evaluate the finite pseudoentropy of a spherical region using standard holographic counterterm renormalization directly in dS and check whether it equals −πL*^2/(2GN) in dS4 and π^2L*^4/(3GN) in dS6.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The divergent, scheme-dependent pieces of dS pseudoentropy are removed by a geometric counterterm inherited from bulk Weyl symmetry, leaving a finite universal value.
  • For spherical entangling surfaces, the universal pseudoentropy is fixed by the central charge a* alone: Su(B^2) = −πL*^2/(2GN) in dS4 and Su(B^4) = π^2L*^4/(3GN) in dS6.
  • For small deformations of the sphere, the quadratic shape correction is controlled by CT, analytically continued from AdS to dS, matching the AdS form of the shape-dependence formula.
  • The construction provides a new entry in the dS/CFT dictionary: bulk conformal invariance organizes the renormalization of codimension-two observables on the dS side just as it does in AdS.
  • The result extends conformal renormalization from AdS entanglement entropy to dS pseudoentropy, covering even bulk dimensions 4 and 6.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same conformal-renormalization route should extend to pseudo-Rényi entropies and to higher even bulk dimensions, where the relevant codimension-two conformal functionals are less explicitly known; a direct check would be to reproduce the dS6 sphere result from standard counterterm holographic renormalization.
  • Because the renormalized pseudoentropy is generically complex and lacks the global bounds of Willmore-type energies, it may not admit an ordinary entropy interpretation; the complex phase could instead carry information about the no-boundary wavefunction's phase structure.
  • The analytic continuation L|AdS → iL|dS used for CT suggests that universal CFT data in dS can be obtained from AdS data without a separate calculation; a sharper test would be to compare with a free non-unitary CFT computation of pseudoentropy for a deformed sphere.
  • For non-holographic non-unitary CFTs, such as those with CT = 0, the paper leaves open whether shape-dependence universality survives; an exact or lattice CFT check there would delineate the holographic from the generic.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper develops a conformal renormalization prescription for holographic pseudoentropy in dS4/CFT3 and dS6/CFT5. It uses four- and six-dimensional conformal gravity on replicated manifolds to derive codimension-two functionals (Graham–Witten in d=4, Graham–Reichert in d=6) that reduce, on Einstein–dS backgrounds and extremal surfaces, to renormalized area functionals. The finite part of pseudoentropy is then identified with these renormalized areas, Eqs. (2.26) and (3.24). Explicit results are given for the sphere, Su(B^2)=-πL_*^2/(2G_N) and Su(B^4)=π^2L_*^4/(3G_N), and for small shape deformations, where the quadratic correction is written in a Mezei-like form using an analytically continued C_T^{dS}, Eqs. (2.49)/(2.50) and (3.44).

Significance. If the underlying assumptions hold, the paper provides a systematic, regulator-independent method for extracting the universal part of pseudoentropy in dS/CFT, a quantity that has so far lacked a renormalization framework. The explicit counterterm cancellations in (2.44)–(2.47) and (3.38)–(3.41), the finiteness argument in Appendix B, and the detailed six-dimensional decomposition in Appendix D are worked out carefully and appear internally consistent. The paper also makes a welcome attempt to connect the dS results to the Mezei formula for shape dependence, which would be a nontrivial piece of the dS/CFT dictionary if the connection is made independently. However, the central construction rests on an unproven analytic continuation of the conformal-gravity/Einstein-action equivalence to dS (footnote 4), and the Mezei matching is partly fixed by the definition of C_T^{dS}. These issues are load-bearing for the paper's main claims.

major comments (3)
  1. [§1, footnote 4; §2.2 Eq. (2.23); §3.2 Eq. (3.22)] The renormalized pseudoentropy formulas rest on the equivalence I_CG|_E = I_ren^E for Einstein-dS backgrounds. For AdS this is established (Refs. [98,101]), but for dS the paper only states in footnote 4 that it is 'expected to hold through analytic continuation.' The derivation of A_ren in (2.23) and (3.22) uses the same coupling and topological term with σ=-1. Since the near-boundary expansion in dS is oscillatory and the Neumann-sector elimination of the ghost mode is proven only for AdS, the boundary Chern forms in (2.25) and (3.23) and the counterterms in (2.45), (3.39) could acquire additional i factors or different coefficients on the dS side. A concrete check is needed: evaluate the on-shell CG action on the HH dS background with Neumann boundary conditions and compare to the renormalized Einstein-dS action for the sphere. Until then the central results (2.31), (2.48), (3.31), (3
  2. [§2.3.2, Eqs. (2.48)–(2.50); §3.3.2, Eq. (3.44)] The claim that the quadratic shape dependence 'recovers' the analytically continued Mezei formula is weakened by the definition of C_T^{dS}. In Eq. (2.50), C_T^{dS} is fixed to the value that converts the computed coefficient -L_*^2/(8G_N) into π^3 C_T^{dS}/24; similarly, C_T = 30L^4/(π^4G_N) in (3.44) is chosen to match the coefficient L^4/(72πG_N). The ℓ(ℓ^2-1) dependence in (2.49) and (ℓ-1)_5 in (3.44) are genuine outputs of the gravitational calculation, so the functional form is tested, but the overall normalization is not an independent prediction unless C_T^{dS} is computed from the stress-tensor two-point function of the dual non-unitary CFT or derived directly on the dS side. The wording 'non-trivial check of the dS/CFT dictionary' overstates the result as presented.
  3. [§3.1 and Appendix D] The six-dimensional functional F(Σ) relies on the splitting-problem resolution of Ref. [159] and on a boundary term fixed in Eq. (D.6) through an asymptotic equivalence that is argued on the AdS side (σ=+1). The conically singular evaluation of this boundary term in (D.7) introduces σ^{3/2}, which is another analytic continuation; if the dS boundary term differs, the renormalized area (3.22) is not justified. This is closely related to Major Comment 1 but is a distinct technical step in the six-dimensional construction. The paper should either prove the continuation for these boundary terms or explicitly list them as assumptions that the dS extension requires.
minor comments (4)
  1. [Throughout] There are numerous typos and grammatical slips, e.g. 'aim on developing' in §2, 'Associate Legendre polynomials' (p. 15), 'Rimemann tensor' in Appendix B, and 'r(0)µνρσ' in Eq. (B.7). A careful proofreading pass is needed.
  2. [Footnote 4] The analytic continuation is not specified precisely. State explicitly the map (e.g. L_* → -iL_*, σ: +1 → -1) and, if possible, cite a dS-side analysis of the Neumann condition for conformal gravity.
  3. [Appendix C, Eq. (C.14)] The formula displayed for A_ren(Σ) contains factors 8G and 2G that do not match the definition in Eq. (2.23), which has no G. This is likely a typo, but it is confusing in an appendix whose purpose is to prove finiteness.
  4. [§3.3.2, Eq. (3.44)] The value C_T = 30L^4/(π^4G_N) is stated without derivation. If it is obtained by analytic continuation from the AdS value, say so explicitly and give the AdS reference; otherwise provide the derivation.

Circularity Check

2 steps flagged

Mezei-form recovery in dS4/dS6 is normalization-by-definition of C_T; core conformal-renormalization derivation is self-contained.

specific steps
  1. self definitional [§2.3.2, Eqs. (2.48)–(2.50)]
    "Su(B2ε) = −πL2⋆/2GN − L2⋆/8GN ε²∑ℓ ℓ(ℓ²−1)(aℓ²+bℓ²) ... By direct analogy with the AdS case, Eq. (2.48) can be rewritten as Su(B2ε) = −πL2⋆/2GN + π³CdST/24 ε²∑ℓ ℓ(ℓ²−1)(aℓ²+bℓ²), where CdST = −3L2⋆/(π³GN)."

    The coefficient of the ε² term in (2.48) is produced by the explicit area computation, not from a CFT two-point function. Eq. (2.50) defines C_T^dS so that substituting it into (2.49) reproduces (2.48) identically. Hence the statement that the result 'can be rewritten' as the Mezei formula has its normalization guaranteed by construction. The only independent content is the ℓ(ℓ²−1) mode sum and the sign; the claimed universal coefficient carries no information beyond the computed number, so the advertised 'non-trivial check' of the dS/CFT dictionary for the coefficient is circular.

  2. self definitional [§3.3.2, Eqs. (3.43)–(3.44)]
    "Su(B4ε) = π²L4⋆/3GN + ε² L4⋆/(72πGN)∑ℓ aℓ²(ℓ−1)5 ... Su(B4ε) = π²L4⋆/3GN + ε² π³/(2160)CT ∑ℓ aℓ²(ℓ−1)5, where ... CT = 30L4⋆/(π4GN)."

    Same structure as in dS4: (3.43) is the computed renormalized-area result, and (3.44) introduces C_T = 30L4⋆/(π4GN) with 'one has that' — this value is chosen so that the π³/(2160) C_T prefactor reproduces the L4⋆/(72πGN) coefficient of (3.43). Thus the Mezei-form matching of the normalization is by definition rather than by an independent CFT computation. The mode sum (ℓ−1)_5 is non-tautological, but the C_T-dependent overall coefficient is not an independent prediction.

full rationale

The principal derivation — CG action (2.1)/(3.12) evaluated on replicated orbifolds, identification of the conical-defect functional L(Σ)/F(Σ) with renormalized areas, and explicit cancellation of divergences in (2.44)–(2.46) and (3.38)–(3.41) — is presented with explicit formulas and does not reduce to its inputs. The dS continuation of the CG/Einstein equivalence is acknowledged in footnote 4 as 'expected to hold' through analytic continuation; this is a missing proof and a correctness risk, but it is not a circularity, because the paper does not present it as a derived result. The sphere answers (2.31)/(3.31) are topological constants; the identification with a* is asserted in words without an equation making it circular. The genuine circular element is restricted to the Mezei-form 'recovery': in both dS4 and dS6 the coefficient C_T is defined (Eqs. (2.50) and (3.44)) as the normalization that converts the computed shape-dependent coefficient into the Mezei expression, so the normalization match is by construction. The mode sums themselves are computed and give the claim non-trivial content. Overall, the core conformal-renormalization result is self-contained; the circularity is partial and confined to the dictionary-check framing of the Mezei coefficient.

Axiom & Free-Parameter Ledger

4 free parameters · 6 axioms · 0 invented entities

The central claim rests on the conjectural dS/CFT dictionary, the replica/LM prescription for pseudoentropy, the equivalence of CG to Einstein-(A)dS on the Einstein sector (continued to dS without proof), and an analytic-continuation definition of C_T. No new particles, forces, or conserved quantities are introduced; the main free choices are the CG couplings (fixed by matching Einstein gravity) and the dS C_T values (fixed by analytic continuation).

free parameters (4)
  • alpha_CG (4D) = sigma L_*^2/(64 pi G_N)
    Coupling of the Weyl-squared action chosen so that conformal gravity reproduces the renormalized Einstein action (Eq. 2.5); adopted from prior work, not fitted to data here.
  • alpha_CG (6D LPP) = -L_*^4/(384 pi G_N)
    Coupling chosen so that the LPP conformal-gravity combination reproduces the renormalized Einstein action in six dimensions (after Eq. 3.10); adopted from Ref. [127].
  • C_T^{dS} (d=4) = -3 L_*^2/(pi^3 G_N)
    Defined by analytic continuation from AdS in Eq. (2.50) and used to rewrite the computed deformation coefficient as the Mezei-like formula; no independent dS/CFT computation is provided.
  • C_T^{dS} (d=6) = 30 L_*^4/(pi^4 G_N)
    Introduced below Eq. (3.44) to match the Mezei form; assumed by analytic continuation, not independently derived from the dS/CFT stress-tensor two-point function.
axioms (6)
  • domain assumption dS/CFT correspondence: Psi_dS = Z_CFT and the Hartle-Hawking saddle-point approximation (Eqs. 1.1-1.2).
    The whole pseudoentropy calculation presumes this conjectured duality and the saddle-point evaluation of the wavefunction.
  • domain assumption Replica/Lewkowycz-Maldacena prescription for pseudoentropy: S(A) = -lim_{theta->1} d_theta I[M^(theta)] (Eq. 2.10).
    This extends the LM argument to dS/Hartle-Hawking wavefunctions, following Ref. [58]; if the replica trick does not apply to transition matrices, the derivation fails.
  • ad hoc to paper The conformal-gravity action evaluated on Einstein-(A)dS backgrounds equals the renormalized Einstein action, and this equivalence continues to dS by analytic continuation (footnote 4).
    Explicitly flagged as 'expected to hold' for dS; not proven in this paper.
  • domain assumption Neumann boundary conditions select the Einstein sector and remove the conformal-gravity ghost (Sec. 2, citing Refs. [98,101]).
    Required to justify restricting to Einstein-dS backgrounds in the conformal renormalization construction.
  • domain assumption In six dimensions, the LPP combination L_CG = 4 I1 + I2 - 1/3 I3 admits Einstein spacetimes and yields a splitting-independent codimension-two defect functional (Refs. [110,159], plus the new boundary term in Appendix D).
    The splitting problem is resolved by relying on prior results; the additional boundary term is asserted rather than fully proven.
  • ad hoc to paper Analytic continuation L_AdS -> -i L_dS gives the dS stress-tensor coefficient C_T^{dS} (Eq. 2.50).
    No independent CFT computation is given; this is the load-bearing dictionary assumption for the Mezei-like match.

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We study holographic pseudoentropy for subregions in non-unitary Euclidean conformal field theories (CFTs) within the framework of the de Sitter/conformal field theory (dS/CFT) correspondence. Pseudoentropy, defined as the von Neumann entropy of a transition matrix, is computed holographically from codimension-two extremal surfaces in dS space and is divergent due to the asymptotic bulk volume at future infinity. We show that a finite and regulator-independent definition follows from the on-shell action of conformal gravity in four and six dimensions, implemented through the replica construction. We illustrate the formalism for spherical entangling surfaces and small shape deformations thereof. The renormalized pseudoentropy isolates the universal contribution, which for a spherical entangling surface is proportional to the complex-valued central charge $a^\star$ of the non-unitary CFT. On an equal footing, for infinitesimal deformations away from the sphere, we recover, at quadratic order in the deformation parameter, an analytic continuation of the Mezei-like formula in its anti-de Sitter counterpart.

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