Non-Riemannian Holography: Long Strings and Soft Hair

Hyun-Cheol Kim; Jae-Hyuk Oh; Jeong-Hyuck Park; Shaun D. Hampton

REVIEW

Non-Riemannian Holography: Long Strings and Soft Hair

Shaun D. Hampton, Hyun-Cheol Kim, Jae-Hyuk Oh, Jeong-Hyuck Park

High Energy Physics - Theory · Submitted 19 Oct 2026 · 65 references

Pith's verdict: First O(D,D)-covariant holographic dictionary for non-Riemannian backgrounds; the soft-hair claim is physically interesting but conditional on a counterterm completion that the paper acknowledges is open.

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arXiv ID2610.25669 (version 1)

Pith ID3YDN6OHN, Pith's permanent record of this paper and its review

CitationHyun-Cheol Kim, Jae-Hyuk Oh, Jeong-Hyuck Park, Shaun D. Hampton. "Non-Riemannian Holography: Long Strings and Soft Hair." arXiv:2610.25669 (2026). https://pith.science/paper/3YDN6OHN

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Editor's note

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This paper does something genuinely new: it constructs an everywhere non-Riemannian bulk completion of the Gomis–Ooguri boundary data, proposes an O(D,D)-covariant holographic dictionary that unifies it with the Riemannian AdS3 case, and identifies a normalizable mode W1 as charge-neutral soft hair that couples to long strings. The derivations are detailed and the limitations are frankly stated. If the framework holds up, this gives a home for the long-string sector in a non-Riemannian geometry.

The main result—that the non-Riemannian branch carries independent soft hair while the Riemannian branch does not—is well-supported within the DFT-based construction. The Codazzi identity and the boundary conservation law are derived explicitly. The supersupersymmetry analysis is also a nice touch.

The soft spot is exactly what the stress-test note and the paper itself flag: the holographic dictionary's identification of the finite part of radial momenta with CFT one-point functions depends on a counterterm that has only been worked out to quadratic order. A general nonlinear O(D,D)-covariant counterterm does not yet exist. This means the extracted W1 response could shift under a different counterterm choice, so the physical interpretation of W1 as an unambiguous observable is conditional. The paper acknowledges this openly (SM 2.2), so this is not a hit—it is a real open question that limits the strength of the central claim until resolved.

Other open questions: the relative weight of the two saddles in the bulk path integral is undetermined, and the interpretation of W1 as a long-string condensate is speculative. These are not flaws but honest statements of what remains to be done. The reliance on the authors' own DFT framework is not a flaw—the derivations are self-consistent and the cited earlier work is published and reproducible.

Who is this for? Researchers working on non-Riemannian geometries, double field theory, and AdS3/CFT2. It is a significant advance within that subfield. The paper deserves a serious referee; the counterterm question and the saddle weight should be high on the revision agenda.

Referee report

Major objections (4)

[SM 2.2 and SM 1.2] The two-point functions (20) and the associated interpretation of W₁ as a normalizable mode rely on the assumption that the interior boundary variation vanishes (SM 2.2, after Eq. SM 26). The non-Riemannian saddle terminates at q=1 where e^{-2d}=0, but the paper does not verify that linearised fluctuations satisfy appropriate boundary conditions there. The renormalised on-shell action (SM 5) includes an interior endpoint contribution; whether this endpoint contributes to the second variation and therefore to the correlation functions is not addressed. The authors should either justify that the interior endpoint does not affect the separated-point kernels, or compute the interior contribution explicitly. Without this check, the correlation functions are conditional on an unverified assumption.
[Eqs. (13)–(14b)] The derivation of the boundary conservation law from the O(D,D)-covariant Codazzi identity is presented at the level of schematic. Equation (13) involves the unrescaled radial momenta A_y^{a\bar{b}} and B_y, whereas the one-point functions defined in (14a) involve rescaled and shifted combinations. The claim that (14b) satisfies ∇_A ⟨T^{CFT AB}⟩ = 0 on shell would be strengthened by an explicit demonstration of how the rescaling and the shift B_y → B_y + 4/l affect the identity, and how the counterterms necessary for finiteness are compatible with the identity.
[Eqs. (19)–(20) and SM 2.3–2.4] The paper reports only ‘particular kernels’ for the connected two-point functions and notes that the remaining source dependence of integration functions (e.g. F_R, λ_R±, λ_NR±) is relegated to the matrices R_s in (19). The full correlators are not given. While the claim that W₁ enters the off-diagonal one-point function and the mixed kernels M^NR_± is demonstrated, the reader cannot assess whether the complete correlators (including the missing contributions) are consistent with the symmetry algebra, unitarity, or the expected structure of a boundary CFT. The authors should either compute the full correlators or explain why the missing terms are state-dependent and do not affect the central conclusion about W₁. The authors acknowledge this: “Their correlators are not fixed by (SM 104); retaining them in quantum background responses requires an additional prescription.”
[Sec. 4, paragraph on asymptotic charges] The paper states that the restricted classical charge algebras give c_NR^charge = 0 and that this does not fix an OPE central charge. However, the worldsheet analysis in SM 3.2 suggests that the boundary CFT has a nonzero central charge (e.g. from the SL(2,R) WZW sector). The relationship between the classical charge algebra central charge and the quantum CFT central charge for the non-Riemannian branch should be discussed. Is the vanishing of the classical central charge an artifact of the charge definition, or does it indicate that the boundary CFT's central charge is not captured by the surface charges? This issue is relevant to the interpretation of the boundary theory.

Minor objections (5)

[General] The notation ⟨T^{CFT}_{(0)}⟩ is used for the scalar one-point function dual to the DFT dilaton, but the paper sometimes refers to it loosely as ‘trace’. Since it is not a priori the trace of the stress tensor, this could be clarified explicitly.
[Eq. (14b)] The definition ⟨T^{CFT}_{AB}⟩ is given as a combination of the one-point functions. It would be helpful to state explicitly that this tensor is symmetric and traceless with respect to the boundary generalized metric, or explain how the two Witt algebras arise from its conservation.
[SM 4.1] The Clifford algebra realisation uses a specific tensor product structure. The statement that γ^{(11)} ε = ε and B₁₀ ε = ε^* defines a Majorana–Weyl condition. It would be useful to note that this fixes the number of independent real components to 16, consistent with the type-II supersymmetry counting.
[SM 5] The free abelian boundary CFT is constructed on the fixed non-Riemannian DFT vielbeins. The paper notes that the dilaton decouples and that the propagating sector is a free N=(1,1) scalar multiplet. It would be instructive to mention whether this free CFT can be holographically identified with the dual CFT of the construction, or whether it is merely a toy model that captures some features.
[Throughout] Minor typographical issues: ‘Ba˜nados’ in the abstract and text should read ‘Bañados’. Some equation numbers in the supplemental material (e.g., (SM 6)) are referenced without the prefix ‘SM’ in the main text, which may cause confusion.

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The paper

The paper claims

Only the non-Riemannian bulk completion of the Gomis-Ooguri boundary carries independent normalizable soft hair that couples to long strings but contributes no surface charges or on-shell action.

The reading

This paper proposes an O(D,D)-covariant holographic dictionary that applies to two distinct bulk completions of the same Gomis-Ooguri boundary data: a standard Riemannian AdS3 with NS-NS flux and an everywhere non-Riemannian string Newton-Cartan geometry. The dictionary derives the boundary conservation law from an O(D,D)-covariant Codazzi identity. Both completions produce identical diagonal one-point functions for the generalized metric, but only the non-Riemannian branch contains an independent normalizable mode—the soft hair W1—that couples directly to the worldsheet of long strings yet carries no covariant surface charge and does not affect the renormalized on-shell action under the stated interior prescription. The non-Riemannian branch also admits arbitrary chiral Killing-spinor profiles, realizing supersupersymmetry. If the framework is correct, it provides a window into long-string degrees of freedom invisible to conventional supergravity and raises the prospect that non-Riemannian geometries contribute to the quantum-gravity path integral.

What carries the argument

The key machinery consists of (i) the O(D,D)-covariant completion of the contracted Codazzi identity (Eq. 13) in the DFT Fefferman-Graham gauge, which yields the conserved boundary energy-momentum tensor; (ii) the non-Riemannian generalized metric (Eq. 6) that realizes the string Newton-Cartan geometry and accommodates the soft hair W1 as an independent normalizable component; and (iii) the DFT vielbein formalism that projects fluctuations into flat-index components and distinguishes the Riemannian and non-Riemannian branches.

Extended reading notes

Core claim

The central discovery is that the Gomis-Ooguri boundary data—the near-boundary description of long strings in AdS3—admits two radial completions in double field theory that differ in their geometric nature and boundary degrees of freedom. The Riemannian completion is the familiar AdS3 with NS-NS flux, while the non-Riemannian completion is an everywhere non-Riemannian string Newton-Cartan geometry. Using an O(D,D)-covariant holographic dictionary, the paper shows that both branches share diagonal generalized-metric one-point functions, but only the non-Riemannian branch carries an independent normalizable mode W1 that behaves as soft hair: it couples to the long-string worldsheet via a vertex operator, yet contributes neither to the asymptotic surface charges nor to the renormalized on-shell action. Furthermore, the non-Riemannian branch can preserve infinite-dimensional supersymmetries (supersupersymmetry) when the boundary geometry is supersymmetric, offering a new class of holographic backgrounds.

Load-bearing premise

The paper assumes that the O(D,D)-covariant Codazzi identity and the proposed holographic dictionary correctly capture the boundary CFT dynamics, and that the two saddles are valid competing contributions to the same bulk path integral, but the relative weight of the non-Riemannian branch is not determined.

Editorial extensions

If this is right

The two saddles represent competing radial flows from the same supersymmetric fixed point; their relative weight in the bulk path integral must be determined to understand the quantum gravity ensemble in AdS3.
The normalizable hair W1, being charge-neutral and absent from the on-shell action, provides a holographic realization of soft hair that may be relevant for black-hole microstate structures.
The Riemannian one-point functions exhibit a semiclassical T\bar{T}-like factorization, suggesting a connection between non-relativistic strings and irrelevant deformations of the boundary CFT.
The framework extends to other AdS backgrounds via formulations of supergravity with manifest U-duality, potentially uncovering new non-Riemannian holographic duals.
The supersupersymmetry of the non-Riemannian branch shows that holographic backgrounds can support infinite-dimensional superalgebras, which may constrain the boundary CFT.

Reading between the lines

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

If the non-Riemannian saddle contributes significantly to the path integral, it could explain the origin of long-string states that are missing from the usual supergravity spectrum, thereby resolving a long-standing puzzle in AdS3/CFT2.
The soft hair W1 might be detected through its effect on long-string correlation functions without altering the stress-tensor two-point function, providing a clean observational signature of non-Riemannian geometry in a holographic setting.
The O(D,D)-covariant dictionary can be applied to other boundary geometries with non-Riemannian data, such as those with more than one pair of degenerate directions, opening a broader class of holographic duals beyond the Gomis-Ooguri sector.

Original abstract

We propose an $\mathbf{O}(D,D)$-covariant holographic dictionary for two bulk completions of the Gomis--Ooguri boundary data: Riemannian $AdS_3$ supported by pure NS--NS flux and an everywhere non-Riemannian string Newton--Cartan geometry. An $\mathbf{O}(D,D)$-covariant Codazzi identity yields the boundary conservation law. Both branches share diagonal generalized-metric one-point functions, but only the non-Riemannian branch carries independent normalizable soft hair that couples to long strings without contributing to surface charges or the renormalized on-shell action. Supersymmetric non-Riemannian saddles admit arbitrary chiral Killing-spinor profiles, realizing "supersupersymmetry."

The compiled record

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

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-20: 1 request, 3,744 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.25 · human participants 0.01 · controlled comparison 0.20 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.38 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

We propose an $\mathbf{O}(D,D)$-covariant holographic dictionary for two bulk completions of the Gomis--Ooguri boundary data: Riemannian $AdS_3$ supported by pure NS--NS flux and an everywhere non-Riemannian string Newton--Cartan geometry. An $\mathbf{O}(D,D)$-covariant Codazzi identity yields the boundary conservation law. Both branches share diagonal generalized-metric one-point functions, but only the non-Riemannian branch carries independent normalizable soft hair that couples to long strings without contributing to surface charges or the renormalized on-shell action. Supersymmetric non-Riemannian saddles admit arbitrary chiral Killing-spinor profiles, realizing "supersupersymmetry."

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We propose an $\mathbf{O}(D,D)$-covariant holographic dictionary for two bulk completions of the Gomis--Ooguri boundary data: Riemannian $AdS_3$ supported by pure NS--NS flux and an everywhere non-Riemannian string Newton--Cartan geometry.	contribution 1.00	0.88	0.26	asserted 0.74
s1	An $\mathbf{O}(D,D)$-covariant Codazzi identity yields the boundary conservation law.	result 0.50	0.81	0.09	asserted 1.00
s2	Both branches share diagonal generalized-metric one-point functions, but only the non-Riemannian branch carries independent normalizable soft hair that couples to long strings without contributing to surface charges or the renormalized on-shell action.	result 1.00	0.83	0.59	asserted 1.00
s3	Supersymmetric non-Riemannian saddles admit arbitrary chiral Killing-spinor profiles, realizing "supersupersymmetry."	result 0.98	0.73	0.10	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.97 · evidence from proof or derivation 0.99.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

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

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Discussion

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