Worldsheet CFT$_2$ and Celestial CFT$_2$ : An AdS$_3$-CFT$_2$ perspective

Nishant Gupta; Sagnik Misra; Shamik Banerjee

REVIEW 3 major objections 3 minor 34 references

Worldsheet CFT$_2$ and Celestial CFT$_2$ : An AdS$_3$-CFT$_2$ perspective

T0 review · 3 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Zooming in near the boundary of Euclidean AdS_{d+1} contracts its symmetry group to the Poincaré group, and the paper argues that this makes the near-boundary limit of gravity theories holographically dual to celestial CFTs — with only…

desk verdict A serious, clearly written proposal for getting celestial CFTs from a near-boundary limit of AdS/CFT; the central claim is an openly admitted assumption, and the paper's own ultralocal scalar correlator leaves the claimed Liouville dual unexplained. read the letter →

arxiv 2506.14891 v1 pith:LHVCFEGR submitted 2025-06-17 hep-th

Shamik Banerjee , Nishant Gupta , Sagnik Misra This is my paper

classification hep-th

keywords celestialCFTAdS/CFTcorrespondencenear-boundaryscalinglimitInönü-WignercontractionlongstringsLiouvilleCarrollianstringtheoryonS3

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 argues that a class of celestial CFTs — the conjectured holographic duals of quantum gravity in asymptotically flat spacetime — can be obtained from AdS/CFT by zooming in near the boundary of Euclidean AdS_{d+1}. The zoom, implemented by the scaling η = εη̃ with ε→0 at fixed η̃, contracts the isometry group SO(d+2,1) of EAdS_{d+1} to the Poincaré group ISO(d+1,1). The paper proposes that the near-boundary limit of a theory of conformal gravity on EAdS is therefore dual to a boundary celestial CFT_d with full Poincaré symmetry, while non-conformal theories such as Einstein gravity or string theory give celestial CFTs with only Lorentz SO(d+1,1) invariance. Applied to bosonic string theory on Euclidean AdS3 with NS-NS flux, the near-boundary worldsheet becomes free and describes 'long strings' wrapping the boundary sphere, whose dual spacetime CFT2 is identified as a celestial CFT2 with only Lorentz invariance and a Liouville sector for radial fluctuations. The proposal matters because it offers a concrete, computable route into celestial holography using the established machinery of AdS3/CFT2.

What carries the argument

The central object is the Inönü-Wigner contraction of the conformal group of EAdS_{d+1}: writing the boundary-zooming coordinate η = εη̃ and taking ε→0 at fixed η̃ turns the Killing vector fields (2.2) of SO(d+2,1) into the rescaled fields (2.7) that obey the Poincaré algebra ISO(d+1,1), with ε playing the role of the contraction parameter. In the string-theory application, the load-bearing mechanism is the near-boundary limit of the worldsheet action (7.3), in which the interaction term ββ̄ $e^{{−(2/α_+)φ}}$ is dropped, leaving the free action (7.5) with a βγ system and a linear-dilaton φ; the βγ system forces γ(γ̄) to be (anti)holomorphic maps to the boundary sphere, which are the long strings, and the linear-dilaton field supplies the Liouville sector with background charge Q = (k−3)√(2/(k−2)) in the dual spacetime CFT2. This free action is what allows the spacetime symmetries to be lifted from the worldsheet and identified with those of a celestial CFT2.

What would settle it

A concrete calculation that would settle the claim is to construct the explicit near-boundary limit of a known holographic boundary CFT (such as the two-point function limit worked out in Section 4) and check whether the resulting correlation functions obey the Carrollian or conformal Ward identities predicted for the celestial dual; a mismatch, or a demonstration that the boundary limit is not a local CFT at all, would falsify the proposal. On the string side, showing that the long-string spacetime CFT2 does not actually possess the SO(3,1) Lorentz structure claimed — for instance, finding a worldsheet computation in which the lifted Virasoro symmetries are anomalous in a way incompatible with a Lorentz-invariant celestial CFT2 — would break the identification.

Watch

Extended reading notes

Core claim

The central claim is that the near-boundary scaling limit (2.4), η = εη̃ with ε→0, of a theory of gravity on EAdS_{d+1} is holographically dual to a boundary celestial CFT_d. For conformal gravity the contracted symmetry group ISO(d+1,1) matches the full symmetry of a celestial CFT, so the dual is a celestial CFT with Poincaré invariance; for non-conformal theories the bulk has only SO(d+1,1) as its isometry group, so the dual is a celestial CFT with only Lorentz invariance. In the concrete string-theory example, the near-boundary limit of the worldsheet action for bosonic strings on Euclidean AdS3 with NS-NS flux is free, the worldsheet fields γ and γ̄ become holomorphic and anti-holomorphic maps to the boundary sphere, and the wrapped 'long strings' have a spacetime CFT2 with a Liouville sector describing their radial fluctuations; the paper's explicit statement is that this spacetime CFT2 is an example of a celestial CFT2 with only Lorentz invariance. The identification rests on the symmetry contraction together with the assumption that the AdS/CFT duality survives the scaling limit.

Load-bearing premise

The load-bearing premise is that holographic duality continues to hold after the near-boundary scaling limit is taken, so that the ε→0 limit of the bulk theory really is dual to the corresponding limit of the boundary CFT; the paper assumes this explicitly and notes that the analogous limit on the field-theory side is not yet understood.

Editorial extensions

If this is right

If the proposal is correct, the near-boundary limit of conformal gravity on EAdS_{d+1} yields a new holographic construction of celestial CFTs with full ISO(d+1,1) invariance.
The near-boundary limit of Einstein gravity on EAdS3 has a dual celestial CFT2 with only SO(3,1) invariance, and the tracelessness of the leading boundary-data tensor h^{(1)}_{ij} implies that this celestial CFT2 has zero central charge.
For bosonic string theory on AdS3 with NS-NS flux, the dual long-string CFT2 is a celestial CFT2 with only Lorentz invariance, and its worldsheet affine symmetries lift to spacetime symmetries with level multiplied by the wrapping number p.
Because conventional string theory lacks target-space conformal invariance, its near-boundary dual is only Lorentz-invariant; obtaining a celestial CFT2 with full ISO(3,1) invariance would require the near-boundary limit of twistor string theory on AdS3.

Reading between the lines

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

If the scaling limit commutes with the duality, then AdS/CFT techniques become a practical source of celestial CFT data such as spectra and OPEs, since the near-boundary worldsheet is free and exactly solvable.
The zero central charge found for the near-boundary Einstein gravity dual suggests that holographic celestial CFT2s may be non-unitary as conventional CFTs, and that the Virasoro generators act projectively in a way that would be visible in boundary correlation functions.
The long-string identification gives a concrete two-dimensional setting in which the celestial sphere and the string worldsheet coincide, which could be used to test the claim that high-energy flat-space string amplitudes are organized by a celestial description.
One could test the proposed universality of the Liouville sector in celestial CFT2 by examining the near-boundary limit of other AdS3 compactifications, and check whether the same linear-dilaton factor always appears.

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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

Here's the bottom line. This is a serious, clearly written proposal, but the headline claim — that the near-boundary scaling limit of AdS/CFT manufactures celestial CFTs — rests on an assumption the authors state openly and do not prove. Conditional is the right verdict.

What's genuinely new: the near-boundary group contraction SO(d+2,1) → ISO(d+1,1) in Section 2 is a clean observation, and using it to propose the near-boundary limit as an engine for celestial CFTs is a synthesis I haven't seen before. The specific identification of the long-string sector of AdS3 string theory as a Lorentz-only celestial CFT2 is new and gives the celestial program a concrete target in well-studied territory. Section 4's analysis of the near-boundary limit of CFT two-point functions — the electric and magnetic Carrollian sectors — is explicit and looks correct, and the unitarity-bound remarks are thoughtful. The appendix's zero-central-charge observation for the Einstein-gravity case is a nice touch.

The soft spots are real. The load-bearing assumption, 'the duality survives the near boundary scaling limit,' is exactly that: an assumption. The authors know it; the FAQ concedes the field-theory side of the limit is not understood. That alone justifies a conditional verdict. But there's a sharper internal tension. The paper's own computation in Section 5.2, for a massive scalar in the near-boundary limit, gives a boundary two-point function that is a pure contact term, because the limiting bulk action loses all spatial derivatives and becomes ultralocal. The dual invoked in Section 7.2 for the string theory is Liouville, with ordinary power-law correlators. The paper never explains how nonlocal Liouville correlators emerge from an ultralocal bulk. The string case might be special — the long-string CFT is derived from the worldsheet, not from applying the extrapolate dictionary to the limiting action — but the paper does not show that, and symmetry matching alone does not fix correlation functions. So the celestial identification of the long-string CFT is asserted, not derived.

Who this is for: people working on celestial holography and on AdS3/CFT2. It is a proposal with a genuine gap, but the gap is explicit and the target is concrete. It deserves a serious referee; the referee should push on the ultralocal-versus-Liouville tension and on what 'the duality survives the limit' could mean on the field-theory side. I would engage with it, but I would not treat the central claim as established.

Referee Report

3 major / 3 minor

Summary. The paper argues that a class of Celestial CFT_d theories can be engineered by taking a near-boundary scaling limit in Euclidean AdS_{d+1} and assuming that the holographic duality commutes with this limit. Section 2 shows that the conformal group SO(d+2,1) of EAdS_{d+1} contracts to the Poincaré group ISO(d+1,1) under the coordinate rescaling η = εη̃. The proposal is then applied to conformal gravity (full ISO(d+1,1) celestial CFT), Einstein gravity and other non-conformal theories (only Lorentz SO(d+1,1) celestial CFT), and finally to bosonic string theory on AdS_3 with NS-NS flux, where the near-boundary limit of the worldsheet action is a free theory describing long strings and the dual spacetime CFT is argued to be a celestial CFT_2 with only Lorentz invariance. The appendix derives the near-boundary limit of Einstein gravity in Fefferman–Graham gauge and its constraints.

Significance. The symmetry contraction in §2 is a clean mathematical fact, and the identification in §4 of the near-boundary two-point function with a Carrollian structure is a useful observation that may connect celestial holography with Carrollian holography and the long-string literature. If the central assumption—that AdS/CFT duality commutes with the scaling limit—could be justified in even one nontrivial example, the proposal would open a new constructive route to celestial CFTs. The paper is honest about this assumption in §3.1 and §3.3, and the explicit computations of the near-boundary actions and Green's functions are internally consistent. However, the paper does not bridge the gap between the ultralocal near-boundary bulk description and the proposed Liouville-type celestial dual, which is essential for the central claim.

major comments (3)

[§3.1, §3.3] The central claim rests on the unproved assumption that the holographic duality commutes with the near-boundary scaling limit, as stated in §3.1 ('assuming that the duality survives the near boundary scaling limit') and conceded in §3.3 ('Currently we do not understand this limit in the field theory side'). This is not a minor caveat but the load-bearing premise of the paper, since without it the symmetry contraction of §2 yields only a kinematic statement about the bulk isometry group and says nothing about the dual field theory. The paper should either provide a concrete example where the dual correlators and spectrum can be computed both before and after the limit and shown to match, or be reframed explicitly as a conjecture with this assumption as a named hypothesis.
[§5.2 vs §7.2] There is an internal tension between the two-point function computed in §5.2 and the celestial CFT invoked in §7.2. In §5.2 the near-boundary limit of a massive scalar is ultralocal (action (5.7) has no spatial derivatives) and the boundary correlator obtained from the extrapolate dictionary is a pure contact term, ⟨O_{Δ+}(γ,γ̄)O_{Δ−}(γ′,γ̄′)⟩ = δ²(γ−γ′) (Eq. (5.17)). In §7.2, however, the dual of the near-boundary string theory is asserted to be a celestial CFT_2 containing a Liouville sector with central charge c=6kp, whose correlators are non-ultralocal power laws such as ⟨e^{αφ(z)}e^{−αφ(0)}⟩ ∝ |z|^{−2α²}. The paper does not explain how a non-local Liouville correlator emerges from an ultralocal near-boundary bulk description, nor how the contact-term correlators of §5.2 are related to the long-string CFT. The identification of the long-string CFT as a celestial CFT is therefore asserted rather than derived.
[§7.2] The statement that the spacetime CFT_2 dual to S_NB is an example of a celestial CFT_2 with only Lorentz invariance is not supported by any explicit check of the celestial CFT axioms. The properties listed in §7.1 are standard results about the long-string CFT (central charge, Liouville sector, affine symmetry lifting), but no computation shows that the spectrum and correlators of this CFT transform under the Poincaré group (or only its Lorentz subgroup) in the way required of a celestial CFT, nor that the translation generators are realized as internal symmetries. Without such a check, the identification rests on the same symmetry-matching logic that the paper itself qualifies in §3.1 as 'based on symmetries alone'.

minor comments (3)

[§2] There is a typo, 'coorrdinate', in the first paragraph of §2; please fix it.
[Figure 2] In Figure 2, the label '3-D (Carrollian) EFTs with gravity / Long strings' is unclear: for an AdS_3 bulk the boundary is two-dimensional, so the label should be '2-D' or the dimension should be specified.
[§5.2, Eq. (5.17)] The statement that the correlation function (5.17) is 'conformally invariant' is too terse, since a delta function is distributional; please spell out the sense in which the contact term is invariant under the SO(3,1) transformations described in §5.1.

Circularity Check

2 steps flagged · score 5.0 of 10

The 'celestial' label is fixed by symmetry matching, and the long-string example renames the known Liouville CFT; the symmetry contraction itself is not circular.

self definitional [Section 3.1, 'Our Proposal: Conformal gravity on AdS_{d+1}', following Eqs. (2.4)-(2.9)]
"This dual CFT_d, since the symmetries match, is an example of a Celestial CFT_d. ... This identification is based on symmetries alone. In other words, the correlation functions of the Celestial CFT_d which arises in this way need not necessarily compute the semiclassical scattering amplitudes in asymptotically flat (d+2) dimensional space-time."

Celestial CFT_d is introduced in Sec. 1 as a CFT whose global symmetry group is the Poincare group ISO(d+1,1). The paper's constructive evidence is the Inonu-Wigner contraction of SO(d+2,1) to ISO(d+1,1) in Eqs. (2.4)-(2.9). The conclusion that the near-boundary dual 'is an example of a Celestial CFT_d' therefore follows from the definition of a Celestial CFT plus the symmetry match, and not from any independent dynamical computation. The paper concedes this: the identification is 'based on symmetries alone' and Sec. 3.3 says the field-theory side of the limit is not understood. Thus the claimed prediction restates its input (the contracted symmetry group) as the output (the Celestial CFT label).
renaming known result [Section 7.2, 'Dual Celestial CFT_2']
"The space-time CFT2 dual to the string theory described by the worldsheet action S_NB is an example of a Celestial CFT2 (with only Lorentz invariance). The Celestial CFT 2 contains a Liouville sector which describes the radial fluctuation of the long string world-sheet."

The Liouville sector with central charge c=6kp and background charge Q=(k-3)sqrt(2/(k-2)) is taken from the earlier independent long-string literature (refs. [27,29]), and the near-boundary worldsheet action S_NB is the same free long-string action studied there. The new step is solely the assignment of the word 'Celestial' to this known CFT, based on the Sec. 3 symmetry-matching proposal. No celestial correlation function, celestial OPE, or flat-space S-matrix element is computed from S_NB; the section renames the familiar long-string CFT as a Celestial CFT_2 rather than deriving a new celestial dual.

full rationale

The bulk of the paper is not circular: the group contraction in Sec. 2 is an explicit mathematical calculation, and the electric/magnetic Carrollian two-point functions in Sec. 4 are genuine limits of the relativistic CFT correlator. No parameters are fitted and no prediction is obtained by fitting. The circularity is concentrated in the identification step: 'since the symmetries match' the near-boundary dual is declared to be a Celestial CFT, which is exactly the defining symmetry of a Celestial CFT as stated in Sec. 1; the paper itself limits the claim to symmetries alone. The string-theory section then labels the independently known long-string Liouville CFT as a Celestial CFT_2. There is also an internal tension, noted by the paper's own computation: the near-boundary scalar action (5.7) is ultralocal and gives a contact two-point function (5.17), whereas the long-string Liouville CFT invoked in Sec. 7.2 has nonlocal power-law correlators. This is a consistency problem rather than a circular step, but it reinforces that the celestial label is attached by symmetry alone rather than derived from the dynamics. No load-bearing self-citation chain is present; the cited long-string results are independent prior work.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted to data. The proposal rests on two explicit assumptions: survival of the duality under the scaling limit, and the classification of the long-string CFT as a Lorentz-only celestial CFT. No new particles, forces, or dimensions are introduced.

assumptions (4)

domain assumption AdS/CFT duality holds for the bulk theories considered.
Invoked throughout, e.g., Section 3.1: "AdS-CFT duality tells us that any theory of quantum gravity on asymptotically EAdS_{d+1} space is holographically dual to a boundary CFT_d."
ad hoc to paper The holographic duality survives the near-boundary scaling limit (2.4).
Explicitly assumed in Section 3.1: "assuming that the duality survives the near boundary scaling limit". This is the central conjecture of the proposal.
domain assumption The near-boundary limit of the worldsheet theory is the free action (7.5), and the dual space-time theory is the long-string CFT.
Taken from prior literature, refs. [27-30], as used in Section 7. The paper relies on this identification for the long-string sector.
ad hoc to paper The dual of the near-boundary string theory is a Celestial CFT2 with only Lorentz invariance.
Section 7.2 asserts this on the basis of symmetry matching. The long-string CFT has Virasoro symmetry, so the "only Lorentz invariance" statement requires a specific interpretation that the paper does not fully spell out.

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Cite this review

Pith. "Pith review of Worldsheet CFT$_2$ and Celestial CFT$_2$ : An AdS$_3$-CFT$_2$ perspective." pith.science (2026). https://pith.science/paper/LHVCFEGR

@misc{pith2026250614891,
  author       = {Pith},
  title        = {Pith review of: Worldsheet CFT$_2$ and Celestial CFT$_2$ : An AdS$_3$-CFT$_2$ perspective},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LHVCFEGR}},
  note         = {Machine review of arXiv:2506.14891}
}

abstract

Celestial CFT$_d$ is the putative dual of quantum gravity in asymptotically flat $(d+2)$ dimensional space time. We argue that a class of Celestial CFT$_d$ can be engineered via AdS$_{d+1}$-CFT$_d$ correspondence. Our argument is based on the observation that if we zoom in near the boundary of (Euclidean) AdS$_{d+1}$ then the conformal isometry group of EAdS$_{d+1}$, which is SO$(d+2,1)$, contracts to the Poincare group ISO$(d+1,1)$. This suggests that the near boundary scaling limit of a theory of \textit{conformal} gravity on EAdS$_{d+1}$ should be dual to a boundary CFT$_d$ with ISO$(d+1,1)$ symmetry. This dual CFT$_d$, since the symmetries match, is an example of a Celestial CFT$_d$. Similarly, if we have a \textit{non-conformal} theory of gravity on EAdS$_{d+1}$ then the near boundary scaling limit of such a theory is dual to a (boundary) Celestial CFT$_d$ with \textit{only} (SO$(d+1,1)$) Lorentz invariance. Celestial CFTs with only Lorentz invariance have been recently studied in the literature. Now following this logic we discuss, among other things, the near boundary scaling limit of the bosonic string theory on Euclidean AdS$_3$ in the presence of the NS-NS B field. The AdS$_3$ part of the worldsheet theory is free in this limit and has been studied in the literature in different contexts. This limit describes a ``long string'' which wraps the (Euclidean) AdS$_3$ boundary and it has been argued that the space-time CFT$_2$ which describes the radial fluctuations of a long string is a Liouville CFT. According to our proposal, the dual CFT$_2$ which describes the \textit{long string sector} is an example of a \textit{Celestial} CFT$_2$ with \textit{only} (SO$(3,1)$)Lorentz invariance. We do not get a full ISO$(3,1)$ invariant Celestial CFT$_2$ in this way because the string theory does not have target space conformal invariance.

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

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