{"id":"772bb04f-8701-4bc8-aa95-74e72e80280d","arxiv_id":"1908.05738","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Robin-boundary-condition HKLL smearing functions with spacelike support are constructed in AdS causal wedges and used to map the double-trace-deformed CFT Wightman function to the bulk Robin Wightman function.","lead":"This paper builds explicit smearing functions that reconstruct a quantum field inside anti-de Sitter space from boundary data when the boundary condition is a mix of Dirichlet and Neumann. The construction reaches the unitary bound and is tested on causal wedges along an RG flow driven by a double-trace deformation.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The Robin smearing map is not yet justified: the paper proves the boundary Wightman function satisfies µSC, but never computes the wavefront set of the smeared Robin kernel f_F, which is the other half of the Hörmander product criterion.","rationale":"The reader's stated weakest assumption is the analytic continuation in ν used to define the Neumann kernel in global AdS, with the regulator scheme of Section 3.1-3.2. That concern is real for the global construction, but it is not the most load-bearing premise for the central causal-wedge claim: the kernels (4.19)-(4.20) are defined directly in momentum space by replacing ν with -ν in the mode functions, which are analytic in the relevant variable, and no regulator subtraction is used there. The step that is genuinely necessary for the claimed RG-flow map is the assertion that the product KR·ω2 (after smearing K against compactly supported bulk test functions) is a well-defined distribution. This requires two ingredients: the boundary Wightman function satisfies the microlocal spectrum condition, which the paper proves for (5.7), and the smeared kernel fF has wavefront set disjoint from -WF(ω2). The paper never proves the second ingredient for the Robin/Neumann kernel; it only asserts spacelike support and cites the WKB analysis of Morrison. Because the Neumann symbol grows polynomially on the timelike cone, the required decay of \\hat fF in timelike directions is not automatic and must be demonstrated. This is a concrete, checkable gap: if it fails, (5.8) is formal; if it passes, the central claim is substantially supported. The verdict remains CONDITIONAL because the gap is addressable by a direct wavefront-set or Fourier-decay computation, but it is exactly the kind of omitted technical check that prevents an unconditional acceptance.","tokens_in":24774,"tokens_out":41592,"duration_ms":398544,"concrete_test":"Choose d=2, ν=0.75, and a compactly supported bulk test function F(T,X,Z)=exp[-(T²+X²+Z²)/(2σ²)] in the Poincaré patch. Compute \\hat fF(ω,kX)=∫dZ \\tilde KR(ω,kX;Z) \\hat F(ω,kX,Z) using (4.19), with \\hat F the Fourier transform of F in T,X. Numerically evaluate |\\hat fF| along the timelike rays (kX=0, ω>0) and (kX=ω/2, ω>0) for ω up to, say, 100/σ. If |\\hat fF| does not decay faster than any polynomial in ω (e.g., (1+ω)^{-5} or better), then WF(fF) contains timelike covectors and the Hörmander product with ω2 is not guaranteed. Separately, check the spacelike ray kX=2ω: one expects exponential growth e^{cω}, which is harmless only if the timelike rays decay rapidly. The same check should be repeated for the Rindler kernel (4.20) using the asymptotic (4.17) with Δ replaced by Δ-.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central step (5.8) multiplies the Robin kernel KR with the deformed boundary Wightman function ω2 as distributions. Morrison's sufficient condition is that the smeared kernel fF(x)=∫KR(x|X)F(X)dX has wavefront set containing only spacelike covectors, so that it cannot cancel the causal covectors of ω2 in (5.10). The paper proves (5.10) but never analyses WF(fF) for the Robin/Neumann kernel. For the Dirichlet kernel (4.6), the momentum-space symbol \\tilde KD = 2^νΓ(Δ+)(ω²-k_X²)^{-ν/2}ZJν(√(ω²-k_X²)Z) is smooth and polynomially bounded on the closed timelike cone and exponentially growing only for spacelike momenta, so smearing against compactly supported F makes \\hat fF rapidly decaying in a conic neighbourhood of the timelike/null cone. For the Neumann term in (4.19), \\tilde KN = 2^{-ν}Γ(Δ-)(ω²-k_X²)^{ν/2}ZJ_{-ν}(√(ω²-k_X²)Z) grows polynomially on the timelike cone (like |k|^{ν-1/2} for ν>1/2) and exponentially on the spacelike cone. The paper asserts in Section 4.2 that the WKB analysis of [7] remains unchanged, but it does not show that after smearing with compactly supported F the Fourier transform decays rapidly in the timelike cone, nor does it compute the stationary-phase set defining fF. If fF had timelike or null singular directions, the product with ω2 would fail the Hörmander criterion and the formal delta-function computation leading to (5.8) would be unjustified. This gap, not the global-AdS regulator of Section 3, is what directly protects the causal-wedge RG-flow map, since (4.19)-(4.20) are defined directly in momentum space without the regulator. The spacelike-support statement for the Rindler Robin kernel (4.20) is likewise asserted ('it is possible to check...') rather than proven.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper aims to extend the HKLL boundary-to-bulk reconstruction to Robin (mixed) boundary conditions for a free scalar field in AdS, down to the unitary bound, and to connect the resulting bulk smearing kernels with the double-trace RG flow of the boundary CFT. In global AdS the authors define Dirichlet and Neumann kernels through analytic continuation in the parameter ν, combine them into a Robin kernel, and show (with regulators) that the smearing reproduces the bulk field. For the Poincaré patch and the Rindler wedge they write the kernel as a Fourier integral operator, argue for its spacelike support, and exhibit the Robin linear combination. In Section 5 they analytically continue the Euclidean double-trace two-point function to a Lorentzian Wightman function, prove that its wavefront set satisfies the microlocal spectrum condition, and then formally smear this boundary two-point function twice with the Robin kernel to obtain the bulk Robin two-point function of [24], identifying cot γ = -f Aν.","tokens_in":25098,"tokens_out":21497,"duration_ms":211078,"significance":"If the missing technical steps are filled, the paper would establish a nontrivial extension of the HKLL construction: spacelike-supported boundary-to-bulk maps for all Robin boundary conditions, valid down to the unitary bound, and an explicit bridge between the double-trace-deformed CFT and the bulk Robin two-point function. The strengths are the closed-form expressions for the kernels in (4.19) and (4.20), the contour arguments for spacelike support, the self-contained proof of the microlocal spectrum condition for the deformed Wightman function, and the explicit match with the known Robin two-point function. The paper is also well structured and largely self-contained. However, the central distributional product is not yet fully justified, because the wavefront set of the smeared Robin kernel is never computed.","major_comments":[{"comment":"The distributional product defining the correlator map is not justified. Eq. (5.8) is obtained by formally exchanging integrals and identifying two delta functions; for this to be valid, Hörmander's criterion (Theorem A.1) must be satisfied for the pair (f_F, ω2). The paper proves in (5.10) that WF(ω2) contains only causal covectors, but it never computes WF(f_F) for the Robin/Neumann kernel. This is not automatically inherited from the Dirichlet case: in Eq. (4.19) the Neumann symbol behaves as q^{ν-1/2} for large timelike momenta q when ν>1/2, whereas the Dirichlet symbol decays. The sentence in Section 4.2 that the WKB analysis 'remains the same' is not a substitute for a wavefront-set computation. I request an explicit proof, or a precise reference to a theorem covering this case, that for compactly supported bulk test functions F the boundary distribution f_F has only spacelike singular covectors.","section":"Section 5, Eq. (5.8)"},{"comment":"The construction of the Neumann kernel rests on an analytic continuation whose validity is asserted rather than proved. The paper shows that g(ν)=Φ_D(ν) for ν>0 and then claims that adding one more regulator term extends the identity to ν∈(-1,0); the equality with Φ_N(ν)=Φ_D(-ν) is stated without a detailed argument. Please specify the connected domain of analyticity of both sides of (3.15), show that the regulated integrals (3.16) and (3.17) are analytic there, including the cancellation of apparent poles by C(Δ±), and state the sense in which the regulator terms become derivatives of delta functions. Because the Robin kernel (3.19) inherits all distributional properties from KN, this gap directly affects the global-AdS construction.","section":"Section 3.2, Eqs. (3.15)-(3.17)"},{"comment":"The spacelike support property for the Rindler Robin kernel is asserted but not shown. The contour-deformation argument in Section 4.1 is given explicitly only for the Dirichlet mode; for the Neumann mode the large-ω asymptotic (4.17) changes because Δ- < Δ+, so the exponential bound should be re-examined, and for the Robin combination one must also check that no pole or branch cut appears in the relevant half-plane. Please provide the analogous contour-deformation proof for the Neumann/Robin Rindler kernel.","section":"Section 4.2, Eq. (4.20)"}],"minor_comments":[{"comment":"There are several typographical errors: 'accurs' in Section 4.1, 'Neumman' in Section 3.2, 'obiquitous' in Appendix A, 'adress' in the Conclusions, 'spcetrum' in the Introduction, and 'Letc.' in reference [33]. These should be corrected.","section":"Throughout"},{"comment":"The sentence 'These integrals contribute with two delta functions' is informal. After the wavefront-set issue is resolved, it would be helpful to state explicitly how the delta functions are obtained in a rigorous distributional sense.","section":"Section 5"},{"comment":"The proof that a(x,k,m) is an asymptotic symbol is sketched by reducing to one dimension and states that the reduction is 'roughly justified'. A fully rigorous treatment should handle vector-valued k and provide uniform estimates on compact sets in (t,x), including the t-derivatives.","section":"Appendix A"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the scope of JHEP and the central idea is attractive. The missing wavefront-set analysis for the Robin kernel is the main obstacle; it appears fixable with a direct computation, not a fundamental flaw. I would encourage the editors to request a revision that supplies this computation and tightens the analytic-continuation arguments in Section 3."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Let me tell you what to expect. The paper does something genuinely useful: it constructs HKLL-type smearing kernels for Robin boundary conditions in AdS causal wedges, continues them down to the unitary bound, and shows explicitly that smearing the double-trace-deformed CFT Wightman function reproduces the known Robin bulk two-point function. That is new and worth having. The µSC proof for the deformed boundary two-point function is a nice check of Morrison's assumption.\n\nThe soft spot is where the stress-test note lands. The correlator map (5.8) multiplies the Robin kernel KR with ω2 as distributions. For that product to be defined, you need both halves of the Hörmander criterion: ω2 has causal WF, so fF = ∫KR F must have only spacelike singular directions. The paper proves the first half in the Appendix, but never analyzes WF(fF) for the Neumann term in (4.19). The Dirichlet side is fine—Morrison did it, and the symbol is smooth on the timelike cone. The Neumann symbol, however, grows polynomially on the timelike cone, and it is simply asserted that the WKB analysis of [7] goes through unchanged. You can't just assert that: the smearing with compactly supported F has to produce rapid decay in a conic neighbourhood of the timelike/null cone, and the paper doesn't show it. Without that, the formal delta-function computation leading to (5.8) is incomplete. The spacelike support claim for the Rindler Robin kernel is likewise asserted rather than proven.\n\nThe global-AdS regulator business—subtracting boundary terms that become delta-function derivatives—is terse, but I think it's probably fixable, and the momentum-space definition of the Neumann kernel in (4.19) may sidestep it. The match with Dappiaggi-Ferreira is plausible; they identify cot γ = -f Aν, and the normalization is left to the reader, but that's a minor detail.\n\nNo circularity: the boundary Wightman comes from Gubser-Klebanov and the bulk from Dappiaggi-Ferreira independently. The paper is honest about where it skips steps.\n\nBottom line: this deserves peer review, but with a request to close the wavefront-set gap for the Neumann/Robin kernel and to prove the Rindler support. If that check works, the paper is a solid contribution. I'd cite it if I were working on HKLL.","headline":"Genuinely extends HKLL to Robin boundary conditions and gives an explicit correlator map along the double-trace RG flow, but the product in (5.8) lacks the wavefront-set check for the smeared Neumann/Robin kernel.","tokens_in":25784,"tokens_out":3851,"would_cite":true,"duration_ms":35502,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims that the HKLL boundary-to-bulk reconstruction, which builds bulk AdS fields from boundary data by smearing kernels, extends from Dirichlet to Robin boundary conditions and down to the unitary bound $\\Delta=(d-2)/2$.","keywords":["AdS/CFT","HKLL bulk reconstruction","smearing functions","Robin boundary conditions","unitary bound","double-trace deformations","microlocal spectrum condition","AdS causal wedges"],"falsifier":"Evaluate the regulated Neumann expression (3.13) or (3.16) for a compactly supported boundary test function at $\\nu\\in(0,1)$, take the $\\varepsilon\\to0$ limit, and compare with the mode-sum definition of the Neumann bulk field built directly from the $\\Phi_4$ modes; any mismatch would show the analytic continuation misses the physical branch. A simpler check is the massless $d=1$ Neumann result (3.14), which could be tested against an independent construction.","tokens_in":24516,"feed_emoji":"🔭","tokens_out":5632,"duration_ms":55718,"temperature":0.7,"pith_summary":"This paper claims that the HKLL boundary-to-bulk construction, which reconstructs local bulk fields in AdS from boundary data through smearing kernels, continues to work when the bulk scalar obeys Robin boundary conditions instead of the usual Dirichlet condition. That extension reaches the lowest allowed conformal dimension, the unitary bound $\\Delta=(d-2)/2$, where the two AdS falloffs $\\Delta_\\pm$ become interchangeable parts of a one-parameter family. On the boundary side the same one-parameter family is a relevant double-trace deformation, so the paper's map connects CFT correlators along the RG flow to Robin bulk correlators. If correct, bulk reconstruction does not require conformal invariance or Dirichlet boundary conditions, and the smearing kernels remain spacelike supported and real.","feed_headline":"HKLL bulk reconstruction works for Robin boundary conditions","feed_subtitle":"Boundary double-trace deformations map to bulk Robin correlators down to Delta=(d-2)/2.","key_machinery":"The load-bearing object is the distributional kernel $K_R=\\cos\\gamma\\,K_D+\\sin\\gamma\\,K_N$, built by analytic continuation in $\\nu$ of the Dirichlet kernel. On causal wedges the kernel is a Fourier integral whose modes $V_k(z)$ are analytic in $\\omega$; replacing $\\nu$ by $-\\nu$ in that momentum-space expression defines the Neumann kernel without the position-space divergences, and gives spacelike support. The boundary side is carried by the wavefront set, the set of positions and momentum directions where a distribution fails to be smooth: the paper shows the deformed Wightman function's wavefront set is confined to null-related points with timelike momenta, and that is exactly the condition that allows products of $K$ and the correlator as distributions.","core_discovery":"The central claim is that the smearing kernels $K_R$ in (4.19) and (4.20) are real, spacelike-supported distributions, and that smearing the double-trace-deformed CFT Wightman function (5.7) twice with $K_R$ reproduces the bulk Robin-boundary Wightman function, matching the known result of [24] under the identification $\\cot\\gamma=-fA_\\nu$. The paper also proves that the boundary two-point function of the perturbed CFT satisfies the microlocal spectrum condition, so the distribution products that define the smearing map are well defined.","pith_inferences":["One testable extension is the three-point function: if smearing with $K_R$ also matches the bulk Robin three-point function under the double-trace deformation, reconstruction along the RG flow is not merely a two-point artifact.","The same Fourier-space $\\nu\\to-\\nu$ replacement could define smearing kernels for other fields with alternative falloffs, since the analyticity argument only needs the modes to remain analytic in $\\omega$.","Because the kernels are spacelike supported in causal wedges, the paper leaves open that subregion duality survives broken conformal invariance; testing entanglement wedge reconstruction would be a natural next step."],"forward_implications":["HKLL reconstruction applies to all Robin boundary conditions, including the Neumann endpoint, down to the unitary bound.","The original Dirichlet kernel, which only made sense for $\\Delta_+>d-1$, is extended to all $\\Delta_+>d/2$ by the same regulator.","Boundary correlators of the double-trace-deformed CFT map to bulk Robin Wightman functions with the identification $\\cot\\gamma=-fA_\\nu$.","The microlocal spectrum condition holds at every point of the RG flow, so the smearing map between distributions is well defined.","The same construction works in Poincaré and Rindler causal wedges, with real spacelike-supported kernels, so subregion reconstruction does not need conformal invariance."],"supporting_citations":[{"why":"Introduced the HKLL smearing construction for local bulk operators in AdS/CFT, which this paper generalizes to Robin boundary conditions.","marker":"[1]"},{"why":"Provided the holographic representation of local bulk operators and the coordinate-space kernel formulas that the paper revisits and extends.","marker":"[2]"},{"why":"Supplies the distributional treatment of boundary-to-bulk maps in AdS causal wedges that this paper extends to Robin boundary conditions.","marker":"[7]"},{"why":"Defines the microlocal spectrum condition that the paper proves for the deformed boundary two-point function.","marker":"[10]"},{"why":"Provides the singular Sturm-Liouville treatment of Klein-Gordon fields with Robin boundary conditions in global AdS, which the paper uses for the bulk modes.","marker":"[13]"},{"why":"Supplies the analytic continuation of distributions $x_+^\\lambda$ that underlies the regulator scheme defining the Neumann kernel.","marker":"[15]"},{"why":"Gives the Euclidean Schwinger two-point function for a double-trace deformation, from which the paper derives the Wightman function.","marker":"[18]"},{"why":"Provides the bulk Robin-boundary Wightman two-point function that the paper's smearing map reproduces under $\\cot\\gamma=-fA_\\nu$.","marker":"[24]"}],"fun_headline_variants":["Robin BCs extend HKLL reconstruction to unitary bound","HKLL smearing works for Robin BCs down to unitary bound","Double-trace deformation yields bulk Robin Wightman via smearing","Robin BCs enable spacelike HKLL smearing to unitarity","HKLL smearing at unitarity via Robin boundary conditions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything rests on the analytic continuation in $\\nu$: the divergent boundary integrals are regulated by subtracting terms that become derivatives of delta functions, and the claim is that this regulated expression is the unique correct distribution for all $\\nu>-1$; if that regulator picks the wrong extension, the Neumann kernel and hence $K_R$ are not well defined.","fun_headline_variants_meta":{"raw":{"variants":["Robin BCs extend HKLL reconstruction to unitary bound","HKLL smearing works for Robin BCs down to unitary bound","Double-trace deformation yields bulk Robin Wightman via smearing","Robin BCs enable spacelike HKLL smearing to unitarity","HKLL smearing at unitarity via Robin boundary conditions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001204,"raw_usage":{"total_tokens":4928,"prompt_tokens":883,"completion_tokens":4045,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":499,"completion_tokens_details":{"reasoning_tokens":3959}},"tokens_in":499,"tokens_out":4045,"duration_ms":27976,"temperature":1.0,"reasoning_tokens":3959,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:06:46.258144+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate the regulated Neumann expression (3.13) or (3.16) for a compactly supported boundary test function at $\\nu\\in(0,1)$, take the $\\varepsilon\\to0$ limit, and compare with the mode-sum definition of the Neumann bulk field built directly from the $\\Phi_4$ modes; any mismatch would show the analytic continuation misses the physical branch. A simpler check is the massless $d=1$ Neumann result (3.14), which could be tested against an independent construction.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the distributional treatment of boundary-to-bulk maps in AdS causal wedges that this paper extends to Robin boundary conditions."},{"cited_title":"Ground states of a Klein-Gordon field with Robin boundary conditions in global anti-de Sitter spacetime","cited_arxiv_id":"1805.03135","evidence_quote":"Provides the singular Sturm-Liouville treatment of Klein-Gordon fields with Robin boundary conditions in global AdS, which the paper uses for the bulk modes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the analytic continuation of distributions $x_+^\\lambda$ that underlies the regulator scheme defining the Neumann kernel."},{"cited_title":"Hadamard states for a scalar field in anti-de Sitter spacetime with arbitrary boundary conditions","cited_arxiv_id":"1610.01049","evidence_quote":"Provides the bulk Robin-boundary Wightman two-point function that the paper's smearing map reproduces under $\\cot\\gamma=-fA_\\nu$."}],"review_version":1}