Boundary-to-bulk maps for AdS causal wedges and RG flow

Alan Garbarz; Gabriel Palau; Guillem P\'erez-Nadal; Nicol\'as Del Grosso

REVIEW 3 major objections 3 minor 33 references

Boundary-to-bulk maps for AdS causal wedges and RG flow

T0 review · 3 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read 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 $\Delta=(d-2)/2$.

desk verdict 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. read the letter →

arxiv 1908.05738 v1 pith:L3QB27JS submitted 2019-08-15 hep-th math-phmath.MP

Nicolás Del Grosso , Alan Garbarz , Gabriel Palau , Guillem Pérez-Nadal This is my paper · ORCID

classification hep-thmath-phmath.MP

keywords AdS/CFTHKLLbulkreconstructionsmearingfunctionsRobinboundaryconditionsunitarybounddouble-tracedeformationsmicrolocalspectrumconditionScausalwedges

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the 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.

The 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.

The 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.

No circularity: the boundary Wightman comes from Gubser-Klebanov and the bulk from Dappiaggi-Ferreira independently. The paper is honest about where it skips steps.

Bottom 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.

Referee Report

3 major / 3 minor

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.

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 (3)

[Section 5, Eq. (5.8)] 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 3.2, Eqs. (3.15)-(3.17)] 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 4.2, Eq. (4.20)] 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.

minor comments (3)

[Throughout] 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 5] 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.
[Appendix A] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is self-contained and anchored to independent external benchmarks.

full rationale

The paper's central construction is not circular. The boundary Wightman function (5.7) is obtained by analytic continuation of the Euclidean double-trace result of Gubser and Klebanov [18], which is an independent external result, and the bulk Robin two-point function is taken from Dappiaggi and Ferreira [24], also independent of the present authors. The smearing map (5.8) is an explicit calculation: applying the Robin kernel (4.19) twice to the boundary two-point function produces a spectral integral whose form matches [24] once the standard AdS/CFT dictionary cot gamma = -f A_nu (5.9) is used. That dictionary is a known external identification, not a parameter fitted to make the match happen, and no quantity used as an input is defined in terms of the claimed output. The analytic continuation used to define the Neumann kernel is a regulator construction based on the identity theorem and the pole cancellation in C(Delta_+), not on assuming the target result. The assertion that the WKB analysis of [7] is unchanged under nu -> -nu is an unproved step, and the wavefront set of the smeared Robin kernel f_F is not analyzed, but these are correctness gaps about distributional products, not cases where a prediction reduces to its inputs by construction. Since the paper contains no self-citations and its final matching relation is a consistency check against independent literature, no circular step is exhibited.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No numbers are fitted to data; the deformation coupling f and the Robin parameter gamma are physical inputs of the boundary and bulk theories, related by the standard dictionary cot gamma = -f A_nu (5.9). The epsilon regulators in the analytic continuation tend to zero and are not free parameters. The axioms listed are the background physical and mathematical assumptions the construction rests on.

assumptions (7)

domain assumption A scalar field in AdS in the Breitenlohner-Freedman window 0<nu<1 admits Dirichlet, Neumann, and Robin boundary conditions, with Robin parameter gamma defined by (2.4).
Section 2; from singular Sturm-Liouville theory [13,25]. This defines the physical setup.
domain assumption The double-trace deformed CFT Schwinger 2-point function (5.1) from Gubser-Klebanov [18] is the boundary dual to Robin boundary conditions, with cot gamma = -f A_nu.
Section 5; standard multitrace AdS/CFT dictionary [16,17,18,20].
standard math The regularized integral g(nu) (3.11) is an analytic extension of the Dirichlet kernel integral, and the identity theorem forces g(nu) = ~Phi(nu).
Section 3.1; uses Taylor expansions and cancellation of poles by the Gamma function in C(Delta_plus).
domain assumption The replacement nu to -nu is legitimate for the Neumann kernel, and the Robin kernel is the linear combination (3.19)/(4.19).
Sections 3.2 and 4.2; relies on the AdS-invariant Green function argument of [2], footnote 8.
domain assumption The Lorentzian Wightman function of the deformed CFT is obtained from (5.1) by the contour continuation in (5.3)-(5.7).
Section 5; the branch choices and i epsilon prescription are made here.
standard math A continuous superposition over m^2 of free massive Klein-Gordon 2-point functions has wavefront set contained in the union of the individual wavefront sets, which are m-independent.
Appendix A; used to prove the microlocal spectrum condition for omega_2.
standard math The Hoermander multiplication theorem and the wavefront-set bound for oscillatory integrals (Theorems A.1, A.2) apply.
Appendix A; justifies multiplying K and the boundary 2-point function.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Boundary-to-bulk maps for AdS causal wedges and RG flow." pith.science (2026). https://pith.science/paper/L3QB27JS

@misc{pith2026190805738,
  author       = {Pith},
  title        = {Pith review of: Boundary-to-bulk maps for AdS causal wedges and RG flow},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/L3QB27JS}},
  note         = {Machine review of arXiv:1908.05738}
}

abstract

We consider the problem of defining spacelike-supported boundary-to-bulk propagators in AdS$_{d+1}$ down to the unitary bound $\Delta=(d-2)/2$. That is to say, we construct the `smearing functions' $K$ of HKLL but with different boundary conditions where both dimensions $\Delta_+$ and $\Delta_-$ are taken into account. More precisely, we impose Robin boundary conditions, which interpolate between Dirichlet and Neumann boundary conditions and we give explicit expressions for the distributional kernel $K$ with spacelike support. This flow between boundary conditions is known to be captured in the boundary by adding a double-trace deformation to the CFT. Indeed, we explicitly show that using $K$ there is a consistent and explicit map from a Wightman function of the boundary QFT to a Wightman function of the bulk theory. In order to accomplish this we have to study first the microlocal properties of the boundary two-point function of the perturbed CFT and prove its wavefront set satisfies the microlocal spectrum condition. This permits to assert that $K$ and the boundary two-point function can be multiplied as distributions.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

33 extracted references · 12 canonical work pages

[7]

I. A. Morrison, Boundary-to-bulk maps for AdS causal wedges and the Reeh-Sc hlieder property in holography, JHEP 1405, 053 (2014) doi:10.1007/JHEP05(2014)053 [arXiv:1403.34 26 [hep-th]]

work page doi:10.1007/jhep05(2014)053 2014
[24]

Hadamard states for a scalar field in anti-de Sitter spacetime with arbitrary boundary conditions

C. Dappiaggi and H. R. C. Ferreira, Hadamard states for a scalar ﬁeld in anti–de Sitter spacetim e with arbitrary boundary conditions , Phys. Rev. D 94, no. 12, 125016 (2016) doi:10.1103/PhysRevD.94.125016 [arXiv:1610.01049 [gr-qc]]

work page Pith review arXiv 2016
[1]

Hamilton, D

A. Hamilton, D. N. Kabat, G. Lifschytz and D. A. Lowe, Local bulk operators in AdS/CFT: A Boundary view of horizons and locality , Phys. Rev. D 73, 086003 (2006) doi:10.1103/PhysRevD.73.086003 [hep-th/0506118]

arXiv 2006
[2]

Hamilton, D

A. Hamilton, D. N. Kabat, G. Lifschytz and D. A. Lowe, Holographic representation of local bulk operators, Phys. Rev. D 74, 066009 (2006) doi:10.1103/PhysRevD.74.066009 [hep-th/ 0606141]

doi:10.1103/physrevd.74.066009 2006
[3]

Kabat, G

D. Kabat, G. Lifschytz and D. A. Lowe, Constructing local bulk observables in interacting AdS/CFT, Phys. Rev. D 83, 106009 (2011) doi:10.1103/PhysRevD.83.106009 [arXiv:1 102.2910 [hep-th]]

work page doi:10.1103/physrevd.83.106009 2011
[4]

Kabat and G

D. Kabat and G. Lifschytz, CFT representation of interacting bulk gauge ﬁelds in AdS , Phys. Rev. D 87, no. 8, 086004 (2013) doi:10.1103/PhysRevD.87.086004 [ar Xiv:1212.3788 [hep-th]]

arXiv 2013
[5]

Leichenauer and V

S. Leichenauer and V . Rosenhaus, AdS black holes, the bulk-boundary dictionary, and smearin g functions, Phys. Rev. D 88, no. 2, 026003 (2013) doi:10.1103/PhysRevD.88.026003 [arXiv:1304.6821 [hep-th]]

arXiv 2013
[6]

Bousso, B

R. Bousso, B. Freivogel, S. Leichenauer, V . Rosenhaus an d C. Zukowski, Null Geodesics, Local CFT Operators and AdS/CFT for Subregions , Phys. Rev. D 88, 064057 (2013) doi:10.1103/PhysRevD.88.064057 [arXiv:1209.4641 [hep-th]]

arXiv 2013

Show all 33 references

[8]

Hormander, The Analysis of Linear Partial Differential Operators I , 2nd Edition, Springer-V erlag (1990)

L. Hormander, The Analysis of Linear Partial Differential Operators I , 2nd Edition, Springer-V erlag (1990)

1990
[9]

Brouder, N

C. Brouder, N. V . Dang and F. Hélein, A smooth introduction to the wavefront set , J. Phys. A 47, no. 44, 443001 (2014) doi:10.1088/1751-8113/47/44/443001 [a rXiv:1404.1778 [math-ph]]

2014 arXiv
[10]

Brunetti, K

R. Brunetti, K. Fredenhagen and M. Kohler, Commun. Math . Phys. 180, 633 (1996) doi:10.1007/BF02099626 [gr-qc/9510056]

1996 arXiv
[11]

S. J. Avis, C. J. Isham and D. Storey, Quantum Field Theory in anti-De Sitter Space-Time , Phys. Rev. D 18, 3565 (1978). doi:10.1103/PhysRevD.18.3565

1978 doi
[12]

Ishibashi and R

A. Ishibashi and R. M. Wald, Dynamics in nonglobally hyperbolic static space-times. 3. Anti-de Sitter space-time, Class. Quant. Grav. 21, 2981 (2004) doi:10.1088/0264-9381/21/12/012 [hep-th/0402184]

2004 arXiv
[13]

Dappiaggi, H

C. Dappiaggi, H. Ferreira and A. Marta, Ground states of a Klein-Gordon ﬁeld with Robin boundary conditions in global anti–de Sitter spacetime , Phys. Rev. D 98, no. 2, 025005 (2018) doi:10.1103/PhysRevD.98.025005 [arXiv:1805.03135 [hep-th]]

2018 arXiv
[14]

Breitenlohner and D

P . Breitenlohner and D. Z. Freedman, Phys. Lett. 115B, 197 (1982). doi:10.1016/0370-2693(82)90643-8 – 25 –

1982 doi
[15]

I. M. Gel’fand and G. E. Shilov, Generalized Functions, V olume 1, Academic Press (1964)

1964
[16]

Witten, Multitrace operators, boundary conditions, and AdS / CFT co rrespondence, hep-th/0112258

E. Witten, Multitrace operators, boundary conditions, and AdS / CFT co rrespondence, hep-th/0112258

arXiv
[17]

I. R. Klebanov and E. Witten, AdS / CFT correspondence and symmetry breaking , Nucl. Phys. B 556, 89 (1999) doi:10.1016/S0550-3213(99)00387-9 [hep-th/9 905104]

1999 doi
[18]

S. S. Gubser and I. R. Klebanov, A Universal result on central charges in the presence of doub le trace deformations, Nucl. Phys. B 656, 23 (2003) doi:10.1016/S0550-3213(03)00056-7 [hep-th/0212138]

2003 arXiv
[19]

S. S. Gubser and I. Mitra, Double trace operators and one loop vacuum energy in AdS / CFT , Phys. Rev. D 67, 064018 (2003) doi:10.1103/PhysRevD.67.064018 [hep-th/ 0210093]

2003 doi
[20]

Hartman and L

T. Hartman and L. Rastelli, Double-trace deformations, mixed boundary conditions and functional determinants in AdS/CFT , JHEP 0801, 019 (2008) doi:10.1088/1126-6708/2008/01/019 [hep-th/0602106]

2008 arXiv
[21]

J. M. Maldacena, The Large N limit of superconformal ﬁeld theories and superg ravity, Int. J. Theor. Phys. 38, 1113 (1999) [Adv. Theor. Math. Phys. 2, 231 (1998)] doi:10.1023/A:1026654312961, 10.4310/A TMP .1998.v2.n2.a1 [hep-th/9711200]

1999 arXiv
[22]

S. S. Gubser, I. R. Klebanov and A. M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428, 105 (1998) doi:10.1016/S0370-2693(98)00377-3 [hep-th/ 9802109]

1998 doi
[23]

Witten, Anti-de Sitter space and holography , Adv

E. Witten, Anti-de Sitter space and holography , Adv. Theor. Math. Phys. 2, 253 (1998) doi:10.4310/A TMP .1998.v2.n2.a2 [hep-th/9802150]

1998 arXiv
[25]

Zettl, Sturm-Liouville Theory, AMS (2005)

A. Zettl, Sturm-Liouville Theory, AMS (2005)

2005
[26]

Balasubramanian, P

V . Balasubramanian, P . Kraus and A. E. Lawrence, Bulk versus boundary dynamics in anti-de Sitter space-time, Phys. Rev. D 59, 046003 (1999) doi:10.1103/PhysRevD.59.046003 [hep-th/ 9805171]

1999 doi
[27]

Hamilton, D

A. Hamilton, D. N. Kabat, G. Lifschytz and D. A. Lowe, Local bulk operators in AdS/CFT: A Holographic description of the black hole interior , Phys. Rev. D 75, 106001 (2007) Erratum: [Phys. Rev. D 75, 129902 (2007)] doi:10.1103/PhysRevD.75.106001, 10.110 3/PhysRevD.75.129902 [...

2007 arXiv
[28]

http: //dlmf.nist.gov/, Release 1.0.23 of 2019-06-15

NIST Digital Library of Mathematical Functions. http: //dlmf.nist.gov/, Release 1.0.23 of 2019-06-15. F. W . J. Olver, A. B. Olde Daalhuis, D. W . Lozier,B. I. Schneider, R. F. Boisvert, C. W . Clark, B. R. Miller, and B. V . Saunders, eds

2019
[29]

Porrati and C

M. Porrati and C. C. Y . Y u, Notes on Relevant, Irrelevant, Marginal and Extremal Doubl e Trace Perturbations, JHEP 1611, 040 (2016) doi:10.1007/JHEP11(2016)040 [arXiv:1609.00 353 [hep-th]]

2016 doi
[30]

M. Reed, B. Simon, F ourier Analysis, Self-Adjointness (Methods of Modern Mathematical Physics), V ol. 2, Academic Press (1975)

1975
[31]

Rudin Walter: Functional Analysis McGraw-Hill (1991)

1991
[32]

URL: https://helda.helsinki.ﬁ/handle/10138/15 4461

V estberg Matias: The wavefront set and oscillatory integrals Master thesis, University of Helsinki (2015). URL: https://helda.helsinki.ﬁ/handle/10138/15 4461

2015
[33]

Strohmaier, Microlocal analysis, Letc

A. Strohmaier, Microlocal analysis, Letc. Notes Phys. 786, 85 (2009). – 26 –

2009

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [7]

I. A. Morrison, Boundary-to-bulk maps for AdS causal wedges and the Reeh-Sc hlieder property in holography, JHEP 1405, 053 (2014) doi:10.1007/JHEP05(2014)053 [arXiv:1403.34 26 [hep-th]]

work page doi:10.1007/jhep05(2014)053 2014

[2] [24]

Hadamard states for a scalar field in anti-de Sitter spacetime with arbitrary boundary conditions

C. Dappiaggi and H. R. C. Ferreira, Hadamard states for a scalar ﬁeld in anti–de Sitter spacetim e with arbitrary boundary conditions , Phys. Rev. D 94, no. 12, 125016 (2016) doi:10.1103/PhysRevD.94.125016 [arXiv:1610.01049 [gr-qc]]

work page Pith review arXiv 2016

[3] [1]

Hamilton, D

A. Hamilton, D. N. Kabat, G. Lifschytz and D. A. Lowe, Local bulk operators in AdS/CFT: A Boundary view of horizons and locality , Phys. Rev. D 73, 086003 (2006) doi:10.1103/PhysRevD.73.086003 [hep-th/0506118]

arXiv 2006

[4] [2]

Hamilton, D

A. Hamilton, D. N. Kabat, G. Lifschytz and D. A. Lowe, Holographic representation of local bulk operators, Phys. Rev. D 74, 066009 (2006) doi:10.1103/PhysRevD.74.066009 [hep-th/ 0606141]

doi:10.1103/physrevd.74.066009 2006

[5] [3]

Kabat, G

D. Kabat, G. Lifschytz and D. A. Lowe, Constructing local bulk observables in interacting AdS/CFT, Phys. Rev. D 83, 106009 (2011) doi:10.1103/PhysRevD.83.106009 [arXiv:1 102.2910 [hep-th]]

work page doi:10.1103/physrevd.83.106009 2011

[6] [4]

Kabat and G

D. Kabat and G. Lifschytz, CFT representation of interacting bulk gauge ﬁelds in AdS , Phys. Rev. D 87, no. 8, 086004 (2013) doi:10.1103/PhysRevD.87.086004 [ar Xiv:1212.3788 [hep-th]]

arXiv 2013

[7] [5]

Leichenauer and V

S. Leichenauer and V . Rosenhaus, AdS black holes, the bulk-boundary dictionary, and smearin g functions, Phys. Rev. D 88, no. 2, 026003 (2013) doi:10.1103/PhysRevD.88.026003 [arXiv:1304.6821 [hep-th]]

arXiv 2013

[8] [6]

Bousso, B

R. Bousso, B. Freivogel, S. Leichenauer, V . Rosenhaus an d C. Zukowski, Null Geodesics, Local CFT Operators and AdS/CFT for Subregions , Phys. Rev. D 88, 064057 (2013) doi:10.1103/PhysRevD.88.064057 [arXiv:1209.4641 [hep-th]]

arXiv 2013