Tensor gauge fields in arbitrary representations of GL(D,R) : duality & Poincare lemma

Nicolas Boulanger; Xavier Bekaert

REVIEW

Tensor gauge fields in arbitrary representations of GL(D,R) : duality & Poincare lemma

Xavier Bekaert, Nicolas Boulanger

High Energy Physics - Theory · Submitted 7 Aug 2002 · 22 references

Pith's verdict: A genuinely useful organizing theorem for mixed-symmetry gauge fields, with the general-S inductive step asserted rather than written out.

T0 · AI review

How this review was made: An AI referee (selfhosted/deepseek-v4.1-flash) read the paper on October 29, 2026 and answered a fixed list of questions about it. T0 is the starting level every paper gets. The levels describe how deeply Pith has checked the paper, never the authors.

T0 AI review An AI read the paper and answered a fixed list of questions.
T1 Audit Automatic checks that the references and files the paper relies on exist and match.
T2 Compute The paper's code or calculations re-run with the same results.
T3 Formal statement The claims written out in a formal math language a computer can read.
T4 Proof Those claims proved, and the proof checked by computer.

How Pith checks papers →

· 3 major objections · 5 minor · Ask about this paper · The compiled record · Read the full review →

Challenge this review Comment (0) Read on arXiv ↗

Cite ▾

arXiv IDhep-th/0208058 (version 2)

Pith IDA5UKOH7P, Pith's permanent record of this paper and its review

CitationNicolas Boulanger, Xavier Bekaert. "Tensor gauge fields in arbitrary representations of GL(D,R) : duality & Poincare lemma." arXiv:hep-th/0208058 (2002). https://pith.science/paper/A5UKOH7P

Cite this reviewBibTeX for Pith's review of this paper

Claim paper

Why Pith says this

Highlight any text to comment on it.

Editor's note

read the letter

The thing to know: the real new content here is the generalized Poincaré lemma for non-rectangular Young diagrams (Section 5.1), and the payoff is that duality, gauge symmetry, and the Labastida-Morris reducibility tower all fall out of one statement. The rectangular case is Dubois-Violette–Henneaux; the extension to arbitrary GL(D,R) irreps and the unified duality treatment in Section 6.3 are theirs. They say plainly which sections are review, which is more than most papers in this area bother to do, and the N-complex machinery is set up cleanly enough that you can actually follow the argument without reconstructing it yourself. Section 4's rewriting of linearized gravity as d_L R = 0 = d_R R with Tr R = 0 is a nice piece of bookkeeping and makes the electromagnetic analogy concrete rather than rhetorical. The two-column case of the lemma is proved properly in A.1–A.3, with the branching decompositions and the pairwise cancellation of higher-L-grading terms written out. That part I trust. The soft spot is A.4. The step from S to S+1 columns is stated as an induction hypothesis plus two claims, and then closed with 'we just have to follow the same lines.' For S=2 the cancellation pattern among the higher-L-grading terms is checked explicitly; for S≥3, and especially for the step that adds a new column of length one, nothing in the text shows the pattern closes. That is a real gap in the top-level proof, and since the theorem is the headline claim, it is the right place to push. I would not call it fatal: the S=2 proof is not mere bookkeeping, the corollaries are mutually consistent, and the follow-up literature has been using these results for twenty years without, as far as I know, anyone finding a counterexample. But 'plausible and widely used' is not 'proved,' and a referee should ask for the S→S+1 step to be written out. On the stress-test's second point, Corollary 1: I think that one is weaker than it looks. The '∝' in (5.4) is shorthand for the structural argument in the bullet points below it — the antisymmetrization either hits two derivatives or hits λ's column symmetry — and that argument does establish a nonzero constant. I would not list it as a gap. Who should read this: anyone working on mixed-symmetry or higher-spin gauge fields, and anyone who needs the duality map for a specific irrep. Send it to a referee who will actually check A.4.

Referee report

Significance

If the theorem holds, it is a genuinely useful structural result: a single cohomological statement that yields, as corollaries, the existence of curvature potentials, the full Labastida–Morris reducibility spectrum, and the Hodge-dual gauge field of an arbitrary free tensor theory, all in one language. The paper has real strengths worth naming. The two-column proof in A.1–A.3 is careful and explicit, including the GL(D,R) branching decompositions (A.24), (A.33) and the pairwise cancellation of higher-L-grading terms. The degree-of-freedom count (2.39) is manifestly invariant under n ↔ D−n−2, which is a nontrivial consistency check on the duality picture. There are no free parameters, no fitted quantities, and no target result smuggled into the assumptions: the derivation starts from the standard Poincaré lemma and the N-complex framework of other authors. The paper is a synthesis and reformulation of known results ([6, 8, 17, 18]) together with new proofs, and it is honest about that division of labour.

Major objections (3)

[Appendix A.4, Eqs. (A.52)–(A.54)] The general-S inductive step, which is the proof of the paper's headline theorem, is asserted rather than written. Section A.4 states the induction hypothesis (A.52) and the two claims to be proved, (A.53) and (A.54), and then closes with 'We just have to follow the same lines as in sections A.1, A.2 and A.3.' The S=2 argument being imitated is not mere bookkeeping: it depends on the branching decompositions (A.24) and (A.33), on the pairwise cancellation of the higher-L-grading terms in (A.25) and (A.32), and on the induction hypothesis to rewrite the survivors. For S ≥ 3 the decomposition of d_I ν̂ into GL(D,R) irreducibles involves more columns, and the text does not show that the cancellation pattern closes. The step (A.54), which adds a new column of length one, is the most exposed: it must be shown that the terms generated by the new column cancel against terms from the existing columns. Since the Theorem is the single statement from which Corollaries 1 and 2 and the duality construction of §6.3 all follow, this gap is load-bearing. At minimum the S=3 case should be written out explicitly, or the general cancellation pattern stated as a lemma with proof.
[§3.4, footnote 9, and §4, Eqs. (4.6) and (4.8)] Footnote 9 acknowledges that Proposition 1 was proved in [11, 12] under a different convention (antisymmetrizing the columns first) and that the paper's convention (3.10) is the one needed for Hodge dualization; it then says 'as we will show later, the generalized Poincaré lemma for rectangular diagrams remains true with the definition (3.10).' The place where this is supposed to be shown is the Theorem of §5.1, whose general-S proof is the sketch discussed above. But §4 uses the rectangular case directly, in the equivalences (4.6) and (4.8) that produce the dual gauge field h̃. So the derivation of the dual potential in §4 rests on a re-derivation that is deferred to the unproved general step. This should be made explicit: either give a direct proof of the rectangular case under convention (3.10), or state clearly that (4.6) and (4.8) are conditional on the Theorem and identify exactly which part of the Theorem is needed.
[§5.2, Corollary 1 and Eq. (5.4)] The conclusion κ = (∏ d_i) λ is supported by the operator relation (5.4), which is written with a proportionality sign, and the constant is neither computed nor shown to be nonzero. The accompanying argument is verbal: the two diagram tensors 'are proportional since the initial symmetrization of the partial derivatives will be cancelled out by the antisymmetrization in the columns.' For the existence statement this is enough only if the constant is nonzero, and the text does not establish that. A short explicit computation of the constant, or an injectivity argument on the relevant Schur module, would close the point. This matters because Corollary 1 is what identifies the curvature as d1...dS κ in §6.1, Eq. (6.3), and hence underlies the whole gauge-structure analysis of §6.

Minor objections (5)

[§5.2, Proposition 5] Proposition 5, which bridges the d_i formulation used in the gauge theories to the d_I formulation of the theorem, is stated with 'it is possible to show' and no proof or reference. Since the corollaries are applied in the d_i language, a one-line proof or a pointer to where it is proved would help the reader.
[§3.5, proof of Proposition 3, Eq. (3.23)] The preliminary lemma is justified by 'it can be checked explicitly that (Tr_ij)^{D−l_i−p}{∗_i∗_j α} is equal to a sum of terms proportional to (Tr_ij)^{l_j−k}{α}.' Displaying that contraction once, with the epsilon identity used, would make the induction self-contained.
[§2.4, Eq. (2.39)] The physical degree-of-freedom count is quoted without derivation. A sentence indicating how it is obtained (e.g. from the component count minus gauge and reducibility parameters) would let the reader check the n ↔ D−n−2 invariance independently.
[§3.2 and §4] The symbol d is used for three different objects: the projected derivative (3.10), the multiform operators d_i, and the combined d_I of (3.27). The ambiguity is acknowledged in §4, but a short notational table at the start of §3.2 would prevent confusion in §§5–6, where all three appear in the same equations.
[References] Reference [12] (Dubois-Violette and Henneaux), on which Proposition 1 and much of the framework rest, is listed only by its arXiv identifier. Please supply the published version if one exists.

How Pith checks papers → · Full audit of this paper →

The paper

The paper claims

The paper proves a generalized Poincaré lemma for tensor fields in arbitrary irreducible representations of $GL(D, Mathbb{R})$ and uses it to unify the gauge structure and Hodge duality of free tensor gauge theories.

The reading

The paper sets out to show that the gauge structure and electric-magnetic duality of free tensor gauge fields can be read off from one cohomological statement: a generalized Poincaré lemma for tensors shaped by Young diagrams with at most $S$ columns. The authors build a language of multiforms and $N$-complexes in which the exterior derivative is replaced by column-wise derivatives $d_i$, and the ordinary lemma is replaced by the vanishing of a generalized cohomology $H^{(m)}$ for every diagram whose columns are shorter than the spacetime dimension $D$. If that theorem holds, a closed curvature of the right Young symmetry automatically has a potential, the tower of gauge-for-gauge reducibilities follows from the same exactness, and Hodge-dualizing a curvature produces a dual gauge field in a generally different representation. The payoff is a unified derivation of known results for mixed-symmetry fields and linearized gravity, plus a systematic route to dualities among arbitrary free tensor gauge theories. The detailed proof is given for one and two columns; the general-$S$ step is asserted by following the same inductive pattern rather than written out in full.

What carries the argument

The load-bearing machinery is the passage from the de Rham complex to a multigraded complex of multiforms: tensors whose indices are grouped into $S$ antisymmetric columns, with commuting column derivatives $d_i$ and combined operators $d_I$. Young symmetrization projects onto irreducible $mathrm{GL}(D,mathbb{R})$ representations, while Hodge star operators $*_i$ and traces $mathrm{Tr}_{ij}$ translate algebraic Bianchi identities and field equations into statements about Young diagrams. The generalized Poincaré lemma is the exactness statement for this complex; Corollary 1 converts closed curvatures into potentials, and Corollary 2 converts the kernel of the gauge variation into the next generation of reducibility parameters. This is what lets one theorem do the work of separate case-by-case derivations for gravity, mixed-symmetry fields, and their duals.

Extended reading notes

Core claim

The central claim is the generalized Poincaré lemma: for a Young diagram $Y_{(S)}(ell_1,ldots,ell_S)$ with $0<ell_Slecdotsleell_1<D$, the generalized cohomology $H^{(ell_1,ldots,ell_S)}_{(m)}(Omega_{(S)}(mathbb{R}^D))$ vanishes for every $1le mle S$. In the paper's language this says that any tensor of that symmetry killed by all relevant $m$-fold column derivatives is a sum of derivatives of lower-symmetry tensors, with the coboundary built from $S-m+1$ column derivatives. From this single exactness statement the authors derive Corollary 1, that a curvature obeying all Bianchi identities has a potential $K=d_1cdots d_Skappa$, and Corollary 2, the reducibility identities for mixed-symmetry gauge parameters. They then apply these to arbitrary free tensor gauge fields, obtaining gauge transformations, the Labastida-Morris tower of reducibilities, and dual gauge fields for each Hodge-dualized curvature.

Load-bearing premise

The general proof rests on the induction from $S$ to $S+1$ columns in Appendix A.4, where the authors say the step follows the same lines as the detailed one- and two-column cases rather than displaying it; if that analogy fails for some diagram shape, the theorem and its corollaries lose their general-$S$ support.

Editorial extensions

If this is right

Every free tensor gauge field whose curvature has columns shorter than $D$ acquires a potential and a gauge transformation whose reducibility tower follows the Labastida-Morris rules, rather than requiring a separate construction.
Hodge-dualizing a curvature produces a dual gauge field in the dual Young diagram, so electric-magnetic duality becomes a general operation relating different free tensor gauge theories.
The dual field equations are generally higher-trace equations; only in special cases such as $D=4$ symmetric tensors do they have the same form as the original equations.
In $D=4$, linearized gravity's dual gauge field is again a symmetric tensor, so the free spin-two theory has a genuine electric-magnetic duality; in $D>4$ the dual field has mixed symmetry.
For the two-column field $M_{mu_1ldotsmu_n|mu_{n+1}}$, the physical degree-of-freedom count is invariant under $nleftrightarrow D-n-2$, confirming the duality with a symmetric tensor at $D=n+3$.

Reading between the lines

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

The theorem's real economy is compression: once the generalized cohomology vanishes, many 'there exists a potential' and 'the gauge symmetry is reducible' results become the same calculation in different Young diagrams.
A concrete stress test would be to write out the missing $S=3$ induction for a diagram with three unequal columns, such as $(2,2,1)$ in $D=5$; if the decomposition closes there, the asserted general pattern gains direct support.
The duality operation suggests a classification of free tensor gauge theories into orbits under Hodge dualization, with the trace-modified equations as the invariant data.
The paper's discussion of magnetic sources and modified Bianchi identities hints that the same complex could organize consistent couplings of mixed-symmetry fields, although the paper leaves that to later work.

Original abstract

Using a mathematical framework which provides a generalization of the de Rham complex (well-designed for p-form gauge fields), we study the gauge structure and duality properties of theories for free gauge fields transforming in arbitrary irreducible representations of GL(D,R). We prove a generalization of the Poincare lemma which enables us to solve the above-mentioned problems in a systematic and unified way.

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-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,341 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.13 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.13 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

Using a mathematical framework which provides a generalization of the de Rham complex (well-designed for p-form gauge fields), we study the gauge structure and duality properties of theories for free gauge fields transforming in arbitrary irreducible representations of GL(D,R). We prove a generalization of the Poincare lemma which enables us to solve the above-mentioned problems in a systematic and unified way.

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	Using a mathematical framework which provides a generalization of the de Rham complex (well-designed for p-form gauge fields), we study the gauge structure and duality properties of theories for free gauge fields transforming in arbitrary irreducible representations of GL(D,R).	contribution 0.92	0.89	0.11	asserted 0.94
s1	We prove a generalization of the Poincare lemma which enables us to solve the above-mentioned problems in a systematic and unified way.	contribution 0.45	0.95	0.45	asserted 1.00

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

What was examined, and what was not

sources read

arXiv listing abstract v2

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 (22)

[1]

C. Fronsdal, Phys. Rev. D18 (1978) 3624; J. Fang and C. Fronsdal Phys. Rev. D18 (1978) 3630; T. Curtright, Phys. Lett. B85 (1979) 219

View on Pith 1978
[2]

J. M. F. Labastida, Nucl. Phys. B322 (1989) 185

View on Pith 1989
[3]

C. Burdik, A. Pashnev and M. Tsulaia, Mod. Phys. Lett. A 16 (2001) 731 hep-th/0101201; Nucl. Phys. Proc. Suppl. 102 (2001) 285 hep-th/0103143

View on Pith Pith review arXiv 2001
[4]

D. Francia and A. Sagnotti, hep-th/0207002. 43

View on Pith Pith review arXiv
[5]

J. A. Nieto, Phys. Lett. A 262 (1999) 274 hep-th/9910049; U. Ellwanger, hep-th/0201163; H. Casini, R. Montemayor and L. F. Urrutia, hep-th/0206129

View on Pith Pith review arXiv 1999
[6]

C. M. Hull, JHEP 09 (2001) 027, hep-th/0107149

View on Pith Pith review arXiv 2001
[7]

C. M. Hull, Nucl. Phys. B583 (2000) 237–259, hep-th/0004195; Class. Quant. Grav. 18 (2001) 3233–3240, hep-th/0011171; JHEP 12 (2000) 007, hep-th/0011215

View on Pith Pith review arXiv 2000
[8]

J. M. F. Labastida and T. R. Morris, Phys. Lett. B180 (1986) 101

View on Pith 1986
[9]

P.J. Olver, “Diﬀerential hyperforms I,” Univ. of Minnesota repor t 82-101

View on Pith
[10]

P.J. Olver in S. Koh, Invariant theory , (Springer-Verlag, 1987) 62-80

View on Pith 1987
[11]

M. Dubois-Violette and M. Henneaux, Lett. Math. Phys. 49 (1999) 245–252, math.qa/9907135

View on Pith arXiv 1999
[12]

M. Dubois-Violette and M. Henneaux, math.qa/0110088

View on Pith arXiv
[13]

M. Fierz and W. Pauli, Proc. Roy. Soc. Lond. A173 (1939) 211–232

View on Pith 1939
[14]

N. Boulanger, T. Damour, L. Gualtieri, and M. Henneaux, Nucl. Phys. B597 (2001) 127–171, hep-th/0007220

View on Pith Pith review arXiv 2001
[15]

R. Feynman, F. Morinigo, and W. Wagner, Feynman Lectures on Gravitation . (Addison-Wesley, 1995)

View on Pith 1995
[16]

S. Deser, Gen. Rel. Grav. 1 (1970) 9–18; D. G. Boulware and S. Deser, Ann. Phys. 89 (1975) 193

View on Pith 1970
[17]

T. Curtright, Phys. Lett. B165 (1985) 304

View on Pith 1985
[18]

C. S. Aulakh, I. G. Koh, and S. Ouvry, Phys. Lett. B173 (1986) 284

View on Pith 1986
[19]

R. Troncoso and J. Zanelli, Phys. Rev. D58 (1998) 101703, hep-th/9710180; Int. J. Theor. Phys. 38 (1999) 1181–1206, hep-th/9807029; M. Banados, Phys. Rev. Lett. 88 (2002) 031301, hep-th/0107214

View on Pith Pith review arXiv 1998
[20]

X. Bekaert, N. Boulanger and M. Henneaux, Phys. Rev. D67 (2003) 044010, hep-th/0210278

View on Pith Pith review arXiv 2003
[21]

M. Dubois-Violette, math.qa/0005256

View on Pith arXiv
[22]

X. Bekaert, “Spins ´ elev´ es et complexes g´ en´ eralis´ es,” to appear in the proceedings of the “Rencontres Math´ ematiques de Glanon (6` eme ´ edition)”, Glanon (France) 1-5 july 2002. 44

View on Pith 2002

Forward citations

Cited by 17 Pith papers

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

Bosonic conformal higher--spin fields of any symmetry
hep-th 2009-09 conditional novelty 7.0 of 10

A complete classification and explicit free Lagrangians for bosonic conformal higher-spin fields of arbitrary symmetry type in any dimension, derived via a correspondence with supersymmetric matrix mechanics.
Tensor global symmetries and the Stueckelberg mechanism for tensor fields
hep-th 2024-11 reject novelty 6.0 of 10

The authors construct Stueckelberg completions for massive mixed-symmetry tensor fields and derive mixed-symmetry currents and a 't Hooft anomaly for linearized gravity, but a sign error invalidates the (2,1) current ...
Fractons, dipole symmetries and curved spacetime
hep-th 2021-11 conditional novelty 6.0 of 10

Complex scalar theories with dipole symmetry couple to arbitrary Aristotelian (boost-free) spacetimes, and the dynamical symmetric-tensor gauge theory couples only to geometries obeying a specific curvature constraint.
A unified approach to standard and exotic dualizations through graded geometry
hep-th 2019-08 accept novelty 6.0 of 10

A single two-parameter first-order action in graded geometry generates the known standard, exotic and double dualizations for p-forms and (p,1) mixed-symmetry tensors such as the graviton and Curtright field.
Mixed-symmetry fields in de Sitter space: a group theoretical glance
hep-th 2016-12 conditional novelty 6.0 of 10

A character-based dictionary links de Sitter representations to mixed-symmetry fields, and the flat limit yields a clean massless spectrum in even spacetime dimensions but not in odd ones.

Show 12 more citing papers

Maxwell-like Lagrangians for higher spins
hep-th 2012-06 conditional novelty 6.0 of 10

Imposing divergence-free gauge parameters yields Maxwell-like Lagrangians for massless bosons of arbitrary spin and symmetry in flat and (A)dS backgrounds, with extra hand-imposed constraints needed in the mixed-symme...
String Lessons for Higher-Spin Interactions
hep-th 2010-06 conditional novelty 6.0 of 10

A single generating function reproduces every three-point amplitude of the open bosonic string's leading Regge trajectory, and its massless limit organizes all cubic higher-spin couplings under one differential operator.
Manifestly Conformal Descriptions and Higher Symmetries of Bosonic Singletons
hep-th 2009-07 conditional novelty 6.0 of 10

A fiber-based ambient-space reformulation makes conformal symmetry and locality simultaneously manifest for integer-spin singletons, and yields a classification of their higher symmetries.
Gauge fields in (anti)-de Sitter space and Connections of its symmetry algebra
hep-th 2009-04 conditional novelty 6.0 of 10

Every (partially)-massless gauge field in (anti)-de Sitter space corresponds to a single generalized connection of the (anti)-de Sitter algebra, with the field's Young diagram determining the representation.
Unfolding Mixed-Symmetry Fields in AdS and the BMV Conjecture: I. General Formalism
hep-th 2008-12 conditional novelty 6.0 of 10

The paper classifies the Lorentz-covariant Harish-Chandra modules generated by primary Weyl tensors of arbitrary Young shape and mass in constant-curvature backgrounds, identifying the critical masses at which massles...
Duality in linearized gravity
gr-qc 2004-08 conditional novelty 6.0 of 10

Duality rotations of the linearized Riemann tensor are shown to be symmetries of the Pauli-Fierz action, not only of the vacuum equations, via two superpotentials that rotate into each other.
Carrollian conformal fields and flat holography
hep-th 2023-05 accept novelty 5.0 of 10

Massless bulk fields in Minkowski space, pulled back to null infinity, realize Carrollian conformal fields of dimension s - (d-1)/2, providing a group-theoretic dictionary for flat holography.
Higher-Spin Interactions: four-point functions and beyond
hep-th 2011-07 conditional novelty 5.0 of 10

An infinite family of gauge-consistent four-point functions for massless higher-spin fields is built from powers of Yang-Mills four-point amplitudes, at the price of non-local quartic couplings.
Notes on Strings and Higher Spins
hep-th 2011-12 conditional novelty 3.0 of 10

Open-string disk amplitudes, expanded in the tensionless limit, generate the complete set of gauge-invariant cubic higher-spin vertices, which the author reads as evidence that string theory is a broken phase of a hig...
On the geometry of higher-spin gauge fields
hep-th 2002-12 conditional novelty 3.0 of 10

Free higher-spin gauge equations can be written with unconstrained fields and gauge parameters using non-local operators built from de Wit-Freedman curvatures, reducing to Fronsdal's local form after partial gauge fixing.
Introduction to the Classical Theory of Higher Spins
hep-th 2004-05 accept novelty 2.0 of 10

The notes assemble the standard formulations of free massless higher spin fields into one consistent picture and lay out, equation by equation, why coupling them to gravity or to each other remains obstructed in flat ...
De Sitter Representations
hep-th 2026-06 unverdicted

Review of so(1,D) representations for de Sitter space across all D, covering mixed symmetry and fermions, connected to propagating fields.

How later papers used it

87 later papers cite it

Evidence: substantial. 75 of the citing papers share no author with it, and 18 of those carry a confident label from Pith's citation classifier.

Pith read 15 of the 189 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

2 later papers by other authors are listed below: 2 used its method or tool. 1 more paper by the same authors is listed: 1 used its method or tool.

Uses its method or tool 3 of 15 papers with this label · 1 by the same authors

Used this paper's Poincare lemma to imply that a difference of fields is pure gauge.

Prepotentials for linearized supergravity (2018) · 0.84 · paraphrased: the licence does not allow quoting
Used this paper's generalized Poincare lemma to derive an equation involving an antisymmetric tensor.

The Action of the (Free) $\mathcal{N} = (3,1)$ Theory in Six Spacetime Dimensions (2018) · 0.99 · paraphrased: the licence does not allow quoting

By the same authors

Wrote its own curvature in this paper's hyperform notation, giving equations equivalent to the Fierz-Pauli equation for the spin-2 field.

Off-Shell Hodge Dualities in Linearised Gravity and E11 (2012) (same authors) · 0.82 · paraphrased: the licence does not allow quoting

How the literature uses it

78 arXiv papers cite this in 176 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 107 sentences without a confident label

Builds on it 3 papers

Shortly after [ref] appeared, the Francia-Sagnotti results were instead extended to arbitrary mixed-symmetry massless Bose fields in [ref] , and more recently also to the case of massive fully symmetric Bose and Fermi fields [ref] .

Metric-like Lagrangian Formulations for Higher-Spin Fields of Mixed Symmetry · Introduction · 0.89
It generalizes to $(A)dS_D$ and to higher spin the complexes of [ref] .

Manifest Duality for Partially Massless Higher Spins · Differential complex · 0.93
I will content myself with stating that one can extend to this case the non–local equations [ref] [ref] and, up to some subtleties that are discussed in detail in [ref] , the compensator formulation as well.

Notes on Strings and Higher Spins · Introduction · 0.82

Uses its method or tool 13 papers

They are determined via the generalised Poincaré lemma [ref] from the conditions

Higher dualities in $E_{11}$ exceptional field theory · Summary of results · 0.99
They strongly rely on the generalized Poincaré lemmas of [ref] for tensors of mixed Young symmetry.

Prepotentials for linearized supergravity · Conformal geometry · 0.91
Since its curl on both groups of indices vanishes, the generalized Poincaré lemma of [ref] implies that the tensor $S_de^m$ can be written as [equation] for some $ρ$.

The Action of the (Free) $\mathcal{N} = (3,1)$ Theory in Six Spacetime Dimensions · Gauge completeness · 0.98

Compares with it 5 papers

Third, the twisted-duality relation [ref] exactly reproduces, in the limit where the cosmological constant goes to zero, the twisted-duality relations given by Hull in [ref] for linearised gravity in flat spacetime, see also section 4 of [ref] .

Spin-2 twisted duality in (A)dS · Massless twisted duality · 1.00
The constrained formulation involves the minimal field content leading to a local covariant description, but does not link directly to the non-local geometrical description [ref] involving the curvatures of de Wit and Freedman [ref] or their mixed-symmetry generalizations [ref] .

Metric-like Lagrangian Formulations for Higher-Spin Fields of Mixed Symmetry · Conclusions · 0.81
The same conclusion was reached by [ref] via a slightly different chain of reasoning.[See [ref] for discussions of the duality-invariant formulation of linearized Einstein gravity.] As in the scalar case, the anomaly appearing in [ref] at coincident points (which we have not written explicitly) is necessary to fully fix the normalization of the two-point function.

Impossible Symmetries and Conformal Gravity · Electric-magnetic duality · 0.92

Cites it as support 1 paper

As demonstrated in [ref] , this form of field equation is precisely what one should have for a mixed-symmetric gauge field $A_ab,cd$ that propagates non-trivially in four dimensional Minkowski spacetime.

Higher dualisations of linearised gravity and the $A_1^{+++}$ algebra · The first higher dual graviton in four dimensions · 0.97

Cites it as background 30 papers

Vasiliev's construction of non-linear equations of motion for higher-spin gauge fields [ref] represents a generalisation of the “frame-like” formulation of Einstein's theory. at the full interacting level, but it can exhibited at least for the linear theory, where it already displays a sensible amount of non-trivial features [ref] .

Geometric Lagrangians for massive higher-spin fields · Introduction · 0.81
Within the approach of [ref] fields and gauge parameters are not subjected to any trace constraints.

Frame-like Actions for Massless Mixed-Symmetry Fields in Minkowski space · Introduction · 0.96
Recently the Fronsdal constraints have been overcome in a particular simple framework by Francia and Sagnotti in [ref] and, for mixed symmetry fields, in [ref] .

Higher Spins and String Interactions · Introduction · 0.80

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]

C. Fronsdal, Phys. Rev. D18 (1978) 3624; J. Fang and C. Fronsdal Phys. Rev. D18 (1978) 3630; T. Curtright, Phys. Lett. B85 (1979) 219

View on Pith 1978

[2] [2]

J. M. F. Labastida, Nucl. Phys. B322 (1989) 185

View on Pith 1989

[3] [3]

C. Burdik, A. Pashnev and M. Tsulaia, Mod. Phys. Lett. A 16 (2001) 731 hep-th/0101201; Nucl. Phys. Proc. Suppl. 102 (2001) 285 hep-th/0103143

View on Pith Pith review arXiv 2001

[4] [4]

D. Francia and A. Sagnotti, hep-th/0207002. 43

View on Pith Pith review arXiv

[5] [5]

J. A. Nieto, Phys. Lett. A 262 (1999) 274 hep-th/9910049; U. Ellwanger, hep-th/0201163; H. Casini, R. Montemayor and L. F. Urrutia, hep-th/0206129

View on Pith Pith review arXiv 1999

[6] [6]

C. M. Hull, JHEP 09 (2001) 027, hep-th/0107149

View on Pith Pith review arXiv 2001

[7] [7]

C. M. Hull, Nucl. Phys. B583 (2000) 237–259, hep-th/0004195; Class. Quant. Grav. 18 (2001) 3233–3240, hep-th/0011171; JHEP 12 (2000) 007, hep-th/0011215

View on Pith Pith review arXiv 2000

[8] [8]

J. M. F. Labastida and T. R. Morris, Phys. Lett. B180 (1986) 101

View on Pith 1986

[9] [9]

P.J. Olver, “Diﬀerential hyperforms I,” Univ. of Minnesota repor t 82-101

View on Pith

[10] [10]

P.J. Olver in S. Koh, Invariant theory , (Springer-Verlag, 1987) 62-80

View on Pith 1987

[11] [11]

M. Dubois-Violette and M. Henneaux, Lett. Math. Phys. 49 (1999) 245–252, math.qa/9907135

View on Pith arXiv 1999

[12] [12]

M. Dubois-Violette and M. Henneaux, math.qa/0110088

View on Pith arXiv

[13] [13]

M. Fierz and W. Pauli, Proc. Roy. Soc. Lond. A173 (1939) 211–232

View on Pith 1939

[14] [14]

N. Boulanger, T. Damour, L. Gualtieri, and M. Henneaux, Nucl. Phys. B597 (2001) 127–171, hep-th/0007220

View on Pith Pith review arXiv 2001

[15] [15]

R. Feynman, F. Morinigo, and W. Wagner, Feynman Lectures on Gravitation . (Addison-Wesley, 1995)

View on Pith 1995

[16] [16]

S. Deser, Gen. Rel. Grav. 1 (1970) 9–18; D. G. Boulware and S. Deser, Ann. Phys. 89 (1975) 193

View on Pith 1970

[17] [17]

T. Curtright, Phys. Lett. B165 (1985) 304

View on Pith 1985

[18] [18]

C. S. Aulakh, I. G. Koh, and S. Ouvry, Phys. Lett. B173 (1986) 284

View on Pith 1986

[19] [19]

R. Troncoso and J. Zanelli, Phys. Rev. D58 (1998) 101703, hep-th/9710180; Int. J. Theor. Phys. 38 (1999) 1181–1206, hep-th/9807029; M. Banados, Phys. Rev. Lett. 88 (2002) 031301, hep-th/0107214

View on Pith Pith review arXiv 1998

[20] [20]

X. Bekaert, N. Boulanger and M. Henneaux, Phys. Rev. D67 (2003) 044010, hep-th/0210278

View on Pith Pith review arXiv 2003

[21] [21]

M. Dubois-Violette, math.qa/0005256

View on Pith arXiv

[22] [22]

X. Bekaert, “Spins ´ elev´ es et complexes g´ en´ eralis´ es,” to appear in the proceedings of the “Rencontres Math´ ematiques de Glanon (6` eme ´ edition)”, Glanon (France) 1-5 july 2002. 44

View on Pith 2002