(Anti-)De Sitter null strings and Carroll-Weyl symmetry

Bo Sundborg; Ulf Lindstr\"om

REVIEW

(Anti-)De Sitter null strings and Carroll-Weyl symmetry

Ulf Lindström, Bo Sundborg

High Energy Physics - Theory · Submitted 7 Sep 2026 · 13 references

Pith's verdict: Real construction, clean Dirac reduction, but the gauge fixing only reaches X^2>0 orbits, so the claim that the Carroll-Weyl action is the dS string is one patch stronger than proven.

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arXiv ID2609.07765 (version 1)

Pith IDHD34SRXR, Pith's permanent record of this paper and its review

CitationBo Sundborg, Ulf Lindstr\"om. "(Anti-)De Sitter null strings and Carroll-Weyl symmetry." arXiv:2609.07765 (2026). https://pith.science/paper/HD34SRXR

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Claim paper

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Editor's note

read the letter

Read this one. It does what it says, up to an important caveat. Starting from the flat tensionless string, lifting to R^{1,d}, gauging the scale symmetry (2.2), and fixing with psi_dS=0 produces a genuine dS_d tensionless string: the Dirac brackets, the metric (5.9), and the reduced action (5.15) all check out. The connection to the Carroll-Weyl action is a real conceptual step. Earlier work read (2.3) as a (d+1)-dimensional flat string; the paper gives good reasons, including broken translation invariance and gauge non-invariance of X^M, that the physical target is d-dimensional. The construction being purely algebraic is notable, and the AdS sibling is a nice bonus.

The stress-test has a real point. The local scaling is X -> lambda X with lambda > 0. That preserves the sign of X^2, so psi_dS=0 can only be reached from orbits with X^2 > 0. The constraints (3.2) do not kill the other sectors: for example a null X with P parallel to X satisfies all three constraints and cannot be scaled onto the quadric. So the paper has constructed the dS branch of the gauged system, not a complete reduction of the full constraint surface. The central derivation stands on that branch, but the advertised claim that the Carroll-Weyl symmetric action is the dS_d null string is stronger than what is proved. A short treatment of the X^2 <= 0 sectors, or an argument that they are unphysical, would close the gap.

The R issue is less serious than it looks. R enters as the gauge-fixing constant and ends up as the dS radius of the reduced metric; it can be absorbed into redefinitions, so calling it non-physical is acceptable, though the paper could be clearer.

Citations look fair; the disputed Carroll-Weyl papers are engaged with directly, and self-citations are to actual related constructions.

Recommendation: send it to a serious referee. The construction is novel and the core computation is solid; the global gauge-fixing gap is fixable but should be addressed before the interpretation is presented as universal. Worth discussing in a reading group and worth citing if you work on tensionless strings or dS holography.

Referee report

Significance

If the construction is accepted, the paper provides an explicit algebraic route to curved target-space string actions and gives a concrete reinterpretation of the Carroll-Weyl string model. The derivation is largely checkable: the Dirac brackets are spelled out, the dS isometry algebra is exhibited in Eqs. (4.1)-(4.4), and the metric (5.9) follows from an explicit inversion on the quadric. A particular strength is that the final action is not postulated but obtained from the gauged flat-space model. The reinterpretation also gives a potential explanation for the discrete spectrum recently reported for quantized null strings. The main weakness is that the gauge-fixing condition is only locally admissible, so the claim that the action universally describes dS_d null strings is not fully established.

Major objections (2)

[Sec. 3, Eq. (3.3)] The gauge-fixing condition ψ_dS = X^2 - R^2 = 0 is not globally admissible on the constraint surface. The scale transformations (2.2a) multiply X^M by a positive function, and the SO(1,d) symmetry preserves the quadratic form X^2, so the sign of X^2 is invariant along every gauge orbit. Initial data with X^2 = 0 or X^2 < 0 can never be brought to the quadric X^2 = R^2 > 0. Nothing in the first-class constraints (3.2) excludes such initial data: for any null momentum P, the condition X·P = 0 admits solutions with X^2 of either sign or zero. The Dirac analysis in Sec. 3 eliminates only the local pair (φ_X·P, ψ_dS) on the patch where X² = R² ≠ 0, so it does not prove that the reduced phase space is globally the cotangent bundle of dS_d. The paper should either restrict the phase space to the spacelike sector X^2 > 0 with a physical justification, show that the X^2 ≤ 0 sectors decouple or are unphysical, or explicitly frame the result as a statement about one sector. The same caveat applies to the AdS analogue ψ_AdS = 0 proposed in Sec. 6. This point is load-bearing because the central claim is that the Carroll-Weyl action describes dS_d null strings, not merely a local patch of them.
[Sec. 3 and Sec. 5, Eqs. (3.3) and (5.9)] The paper states below Eq. (3.3) that the gauge-fixing constant R is an arbitrary, unphysical quantity, but Eq. (5.9) identifies the reduced target-space metric as the dS metric with curvature radius R and the reduced action (5.15) contains this metric as a background. The status of R therefore needs a precise statement. If R is purely a gauge artifact, the authors should explain why the target-space curvature scale is not observable in the tensionless-string theory, for example by showing that a constant rescaling of the metric can be absorbed in a field redefinition of V^a. If R is physical, the claim that it is an arbitrary gauge-fixing constant needs qualification. The paper does not currently reconcile these two roles of R.

Minor objections (4)

[Sec. 3, after Eq. (3.5)] 'Weak inequality' should read 'weak equality', since the symbol ≈ denotes equality after imposing the constraints and gauge conditions.
[Sec. 2, footnote 2] The phrase 'higher dimensional Dirac space' appears to be a typo; it should presumably be 'target space' or 'Minkowski space'.
[Sec. 5, Eq. (5.9)] The denominator R^2 - X^k η_kl X^l should be defined explicitly as R^2 - X_m X^m, since the contraction with η is easy to misread in the displayed form.
[Sec. 5, Eq. (5.15)] The Legendre transformation from the reduced Hamiltonian to the standard tensionless action is asserted rather than demonstrated; a few intermediate steps would make the final claim easier to verify.

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The paper

The paper claims

Gauging scale symmetry shows that Carroll-Weyl symmetric strings describe De Sitter null strings, not flat-spacetime strings.

The reading

This paper constructs a tensionless string in De Sitter space by lifting the standard tensionless string to a higher-dimensional Minkowski space and gauging worldsheet scale symmetry. The gauged action turns out to be the Carroll-Weyl symmetric string action recently studied, which the authors reinterpret as describing a string in dS_d rather than in (d+1)-dimensional flat spacetime. Through Dirac analysis and a De Sitter invariant gauge fixing, they show that the target space reduces to the dS_d quadric, with the metric emerging algebraically from the constraints. The construction also extends to anti-De Sitter space and offers a new algebraic route to curved target spaces for null strings.

What carries the argument

The central object is the gauged Carroll-Weyl symmetric action (2.3), which couples the tensionless string fields X^M to a gauge field W_a for local scale symmetry. The De Sitter gauge fixing condition ψ_dS = η_{MN} X^M X^N - $R^{2}$ = 0 (3.3) picks out the dS_d quadric, and the Dirac bracket analysis of the resulting second-class constraints produces the dS_d metric (5.9) and the reduced canonical phase space.

Extended reading notes

Core claim

The paper claims that the Carroll-Weyl symmetric action (2.3), after gauge fixing to the De Sitter quadric (3.3), describes a tensionless string in dS_d, not a (d+1)-dimensional flat space string as previously thought. The resulting reduced theory has all the properties of a dS_d tensionless string, with the metric derived algebraically.

Load-bearing premise

The construction assumes that the local scale symmetry (2.2) can be consistently gauged with a single gauge field W_a, that the gauge fixing condition ψ_dS = 0 picks out a single orbit of that symmetry, and that the algebraically derived metric is the correct physical metric for the target space.

Editorial extensions

If this is right

The Carroll-Weyl symmetric string action is now correctly interpreted as describing tensionless strings in De Sitter space, resolving an earlier apparent contradiction with flat-spacetime interpretation.
The purely algebraic derivation of the curved De Sitter metric from gauging suggests a method to obtain other curved target spaces for null strings by analogous gauge fixing.
The construction immediately generalises to anti-De Sitter space, yielding AdS_d tensionless strings and potentially new insights into AdS/CFT with tensionless strings.
Quantisation of the dS_d tensionless string may yield a discrete spectrum consistent with compact spatial sections of De Sitter space, as hinted by recent results on Carroll-Weyl strings.
The reduced number of physical degrees of freedom (d-2) in the dS_d string matches the expected counting for a null string in d dimensions, not (d+1) dimensions.

Reading between the lines

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

If the dS_d tensionless string can be consistently quantised, it may provide a tractable toy model for quantum gravity in De Sitter space, possibly shedding light on dS holography.
The gauge fixing constant R, while physically arbitrary, could become physical upon quantisation through the spectrum, a point worth checking.
The algebraic method might extend to other homogeneous spaces (e.g., symmetric spaces) by starting from a higher-dimensional flat space and gauging an appropriate symmetry.
The obstacles noted for tensile strings (due to Weyl invariance) suggest that the construction is specific to null strings, highlighting the special role of worldsheet scale symmetry of the Carrollian type.

Original abstract

We construct a $dS_d$ null string in a novel way by lifting the standard tensionless string to $R^{1,d}$. Gauging worldsheet scale symmetry introduces terms in the action breaking translation invariance, but preserving the $dS_d$ symmetry $SO(1,d)$. Choosing a De Sitter symmetric gauge fixing of the scale symmetry yields a $d$ dimensional theory which we show has all the expected properties of $dS_d$ tensionless strings. The action happens to be the intriguing Carrol-Weyl symmetric string action in $R^{1,d}$, now with a clear target space interpretation in $d$ dimensions. That it does not, as recently claimed, represent a ($d+1$)-dimensional string in flat ($d+1$)-dimensional Minkowski space settles an apparent issue with the Carrol-Weyl string model. Furthermore, while there are clear obstacles for similar formulations of tensile strings in (A)dS backgrounds, our new insights are easily modified to the formulation of Anti-De Sitter tensionless strings. That our construction generates a curved geometry purely algebraically may have even wider applications.

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, 2026-09-23: 1 request, 5,693 input tokens.

What the abstract itself states, as probabilities: claims first 0.17 · states limitation 0.68 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.86 · 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 construct a $dS_d$ null string in a novel way by lifting the standard tensionless string to $R^{1,d}$. Gauging worldsheet scale symmetry introduces terms in the action breaking translation invariance, but preserving the $dS_d$ symmetry $SO(1,d)$. Choosing a De Sitter symmetric gauge fixing of the scale symmetry yields a $d$ dimensional theory which we show has all the expected properties of $dS_d$ tensionless strings. The action happens to be the intriguing Carrol-Weyl symmetric string action in $R^{1,d}$, now with a clear target space interpretation in $d$ dimensions. That it does not, as recently claimed, represent a ($d+1$)-dimensional string in flat ($d+1$)-dimensional Minkowski space settles an apparent issue with the Carrol-Weyl string model. Furthermore, while there are clear obstacles for similar formulations of tensile strings in (A)dS backgrounds, our new insights are easily modified to the formulation of Anti-De Sitter tensionless strings. That our construction generates a curved geometry purely algebraically may have even wider applications.

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 construct a $dS_d$ null string in a novel way by lifting the standard tensionless string to $R^{1,d}$.	contribution 0.85	0.95	0.12	asserted 1.00
s1	Gauging worldsheet scale symmetry introduces terms in the action breaking translation invariance, but preserving the $dS_d$ symmetry $SO(1,d)$.	method 0.68	0.80	0.08	asserted 1.00
s2	Choosing a De Sitter symmetric gauge fixing of the scale symmetry yields a $d$ dimensional theory which we show has all the expected properties of $dS_d$ tensionless strings.	result 0.52	0.94	0.23	asserted 1.00
s3	The action happens to be the intriguing Carrol-Weyl symmetric string action in $R^{1,d}$, now with a clear target space interpretation in $d$ dimensions.	interpretation 0.67	0.82	0.20	asserted 1.00
s4	That it does not, as recently claimed, represent a ($d+1$)-dimensional string in flat ($d+1$)-dimensional Minkowski space settles an apparent issue with the Carrol-Weyl string model.	result 0.58	0.71	0.29	asserted 1.00
s5	Furthermore, while there are clear obstacles for similar formulations of tensile strings in (A)dS backgrounds, our new insights are easily modified to the formulation of Anti-De Sitter tensionless strings.	contribution 0.53	0.81	0.10	asserted 0.71
s6	That our construction generates a curved geometry purely algebraically may have even wider applications.	interpretation 0.95	0.75	0.09	hedged possibility 1.00

What kind of paper the classifier reports this is: theoretical result 0.96 · evidence from proof or derivation 1.00.

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

References

Works cited (13)

[1]

A. Schild,Classical Null Strings,Phys. Rev. D16(1977) 1722

View on Pith 1977
[2]

A. Karlhede and U. Lindstrom,The Classical Bosonic String in the Zero Tension Limit, Class. Quant. Grav.3(1986) L73

View on Pith 1986
[3]

B. Sundborg,The Hagedorn transition, deconfinement and N=4 SYM theory,Nucl. Phys. B 573(2000) 349 [hep-th/9908001]. – 7 –

View on Pith Pith review arXiv 2000
[4]

B. Sundborg,Stringy gravity, interacting tensionless strings and massless higher spins,Nucl. Phys. B Proc. Suppl.102(2001) 113 [hep-th/0103247]

View on Pith Pith review arXiv 2001
[5]

J. Isberg, U. Lindstrom, B. Sundborg and G. Theodoridis,Classical and quantized tensionless strings,Nucl. Phys. B411(1994) 122 [hep-th/9307108]

View on Pith Pith review arXiv 1994
[6]

H. Gustafsson, U. Lindstr¨ om, P. Saltsidis, B. Sundborg and R. von Unge,Hamiltonian BRST quantization of the conformal string,Nucl. Phys. B440(1995) 495 [hep-th/9410143]

View on Pith Pith review arXiv 1995
[7]

U. Lindstr¨ om,Symmetries of tensionless strings,2605.26185

View on Pith Pith review arXiv
[8]

U. Lindstr¨ om,The conformal null string ind+ 2andddimensions,2606.22498

View on Pith Pith review arXiv
[9]

M.M. Sheikh-Jabbari and H. Yavartanoo,Null Strings Gauged and Reloaded, I: Null Strings Have Carroll-Weyl Gauge Symmetry,2605.25817

View on Pith Pith review arXiv
[10]

M.M. Sheikh-Jabbari and H. Yavartanoo,Null Strings Gauged and Reloaded, II: Consistent Classical Treatment of the Null Strings,2605.26822

View on Pith Pith review arXiv
[11]

I.M. Rasulian, M.M. Sheikh-Jabbari and H. Yavartanoo,Null-strings Gauged, Reloaded and Quantized, I: Canonical Quantization in the Light-Cone Gauge,2607.02970

View on Pith Pith review arXiv
[12]

P. Haggi-Mani and B. Sundborg,Free large N supersymmetric Yang-Mills theory as a string theory,JHEP04(2000) 031 [hep-th/0002189]

View on Pith Pith review arXiv 2000
[13]

J.A. Minahan and K. Zarembo,The Bethe ansatz for N=4 superYang-Mills,JHEP03 (2003) 013 [hep-th/0212208]. – 8 –

View on Pith Pith review arXiv 2003

Forward citations

Cited by 6 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

eAdS- embedded AdS, tensionless strings and contracted Carroll-Weyl symmetry
hep-th 2026-11 accept novelty 6.0 of 10

The eAdS null string combines local scaling symmetry commuting with AdS isometries and is the tensionless limit of a tensile string.
Null Strings Gauged, Reloaded and Quantized, II: Null Strings on Flat Backgorund Have Discrete Spectrum
hep-th 2026-11 conditional novelty 6.0 of 10

Within a regular separated Schrödinger sector, the paper derives a discrete, nonnegative mass-squared spectrum mu^2=m(m-1) for bosonic null strings on flat space, with no critical dimension and no tachyons.
Reflected constraint algebras of open Carroll--Weyl null strings
hep-th 2026-10 conditional novelty 6.0 of 10

Open Carroll-Weyl null strings have a reflected classical phase space whose boundary algebra is the boundary Carrollian conformal algebra extended by one weight-one current, with D-3 local embedding pairs.
Carrollian Worldsheets on Carrollian Targets, When does ILST describe a null string theory?
hep-th 2026-10 conditional novelty 6.0 of 10

On Carrollian target spaces with a homothetic kernel generator, the CW-gauged null string action equals the ILST action off shell, making the two theories classically equivalent.
Null String Holography, Null Strings Probe Projective Boundary of Their Target Spaces
hep-th 2026-09 conditional novelty 6.0 of 10

Minimal null string theory on a background manifold is classically equivalent to the tensionless ILST string theory on the projective boundary of that manifold, giving concrete AdS, dS, and flat-space duals.

Show 1 more citing papers

Null string holography in AdS from local BRST descent
hep-th 2026-10 conditional novelty 5.0 of 10

The radial sector of the Carroll-Weyl gauged null string in Poincaré AdS_D is BRST-contractible, so its local cohomology reduces exactly to the (D-1)-dimensional ILST complex, with D=27 mapping to d=26.

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

A. Schild,Classical Null Strings,Phys. Rev. D16(1977) 1722

View on Pith 1977

[2] [2]

A. Karlhede and U. Lindstrom,The Classical Bosonic String in the Zero Tension Limit, Class. Quant. Grav.3(1986) L73

View on Pith 1986

[3] [3]

B. Sundborg,The Hagedorn transition, deconfinement and N=4 SYM theory,Nucl. Phys. B 573(2000) 349 [hep-th/9908001]. – 7 –

View on Pith Pith review arXiv 2000

[4] [4]

B. Sundborg,Stringy gravity, interacting tensionless strings and massless higher spins,Nucl. Phys. B Proc. Suppl.102(2001) 113 [hep-th/0103247]

View on Pith Pith review arXiv 2001

[5] [5]

J. Isberg, U. Lindstrom, B. Sundborg and G. Theodoridis,Classical and quantized tensionless strings,Nucl. Phys. B411(1994) 122 [hep-th/9307108]

View on Pith Pith review arXiv 1994

[6] [6]

H. Gustafsson, U. Lindstr¨ om, P. Saltsidis, B. Sundborg and R. von Unge,Hamiltonian BRST quantization of the conformal string,Nucl. Phys. B440(1995) 495 [hep-th/9410143]

View on Pith Pith review arXiv 1995

[7] [7]

U. Lindstr¨ om,Symmetries of tensionless strings,2605.26185

View on Pith Pith review arXiv

[8] [8]

U. Lindstr¨ om,The conformal null string ind+ 2andddimensions,2606.22498

View on Pith Pith review arXiv

[9] [9]

M.M. Sheikh-Jabbari and H. Yavartanoo,Null Strings Gauged and Reloaded, I: Null Strings Have Carroll-Weyl Gauge Symmetry,2605.25817

View on Pith Pith review arXiv

[10] [10]

M.M. Sheikh-Jabbari and H. Yavartanoo,Null Strings Gauged and Reloaded, II: Consistent Classical Treatment of the Null Strings,2605.26822

View on Pith Pith review arXiv

[11] [11]

I.M. Rasulian, M.M. Sheikh-Jabbari and H. Yavartanoo,Null-strings Gauged, Reloaded and Quantized, I: Canonical Quantization in the Light-Cone Gauge,2607.02970

View on Pith Pith review arXiv

[12] [12]

P. Haggi-Mani and B. Sundborg,Free large N supersymmetric Yang-Mills theory as a string theory,JHEP04(2000) 031 [hep-th/0002189]

View on Pith Pith review arXiv 2000

[13] [13]

J.A. Minahan and K. Zarembo,The Bethe ansatz for N=4 superYang-Mills,JHEP03 (2003) 013 [hep-th/0212208]. – 8 –

View on Pith Pith review arXiv 2003