Current Exchanges and Unconstrained Higher Spins

A. Sagnotti (Scuola Normale Superiore; D. Francia (Chalmers U.; France); Goteborg; INFN; Italy); J. Mourad (APC; Pisa; Sweden); U. Paris VII

REVIEW

Current Exchanges and Unconstrained Higher Spins

D. Francia (Chalmers U., Goteborg, Sweden), J. Mourad (APC, U. Paris VII, France), A. Sagnotti (Scuola Normale Superiore, INFN

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Pisa, Italy)

High Energy Physics - Theory · Submitted 18 Jan 2007 · 35 references

Pith's verdict: A careful, useful correction paper: it shows the earlier curvature-based non-local equation fails the current-exchange test and builds the one that passes, though 'unique' is only unique inside a stated ansatz.

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arXiv IDhep-th/0701163 (version 2)

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CitationA. Sagnotti (Scuola Normale Superiore, D. Francia (Chalmers U., France), Goteborg, INFN, Italy), J. Mourad (APC, Pisa, Sweden), U. Paris VII. "Current Exchanges and Unconstrained Higher Spins." arXiv:hep-th/0701163 (2007). https://pith.science/paper/W6EO5IQY

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Worth your time if you work on higher-spin currents, less so otherwise. The real content is not the Lagrangian families — those are a cleanup of the authors' own [18] — but the negative result in Section 4.1: the Einstein-like tensor built from the iterated curvature operator F^(n) in [7] gives the wrong trace coefficients in the current exchange. The spin-3 comparison (4.15) vs (4.16), J·J − (3/D)J′·J′ against J·J − (3/(D−2))J′·J′, is a two-line check anyone can do, and it is convincing. The fix in Section 4.2 — a modified A_nl with coefficients a_k fixed by the Bianchi identity, plus the companion D_nl in (4.69) — is explicit and the s=3,4,5 cases (4.70)–(4.74) let you verify the general formula rather than take it on faith. The AdS extension in Section 2 is competently done and the invariant-tensor presentation (A, B, C and their fermionic analogues) is genuinely cleaner than the original derivation. The equivalence of constrained and unconstrained local formulations under current exchange, Sections 3.1–3.2, is the kind of thing that should have been written down years ago and now is. On soft spots: the abstract's 'unique set' [abstract:s4] is proved unique only within the no-eta ansatz (4.62). The paper says so in the text, so this is a wording problem rather than a hole in the math, but the abstract should be tightened. Second, the reduction to the non-Lagrangian equations (1.2)/(1.3) in the absence of sources is worked out only for s=4 (4.76)–(4.77); the general-s step is asserted. That is a real gap, small but real. Third, the off-shell continuation of singular non-local operators is acknowledged in 4.1 and is standard practice here, so I would not hold it against them, but a follow-up should say something about it. The heavy self-citation is not circularity — the new coefficients are derived, not fitted. Recommendation: send it out. It is a solid technical paper with a genuine correction at its center, and the referees should push on the uniqueness wording and the general-s reduction.

Referee report

Major objections (3)

[Abstract, sentence s4; Section 4.2, ansatz (4.62)] The abstract claims a 'unique set of non-local Lagrangian equations', but the uniqueness proof in Section 4.2 is confined to the ansatz (4.62), in which A_nl is a linear combination of F^(n+1) and its traces with no explicit metric (eta) terms. The paper states this restriction explicitly ('aside from the condition that no eta's be present in the linear combination (4.62)'), but does not argue that eta terms are excluded by the consistency conditions (4.35)-(4.36). If eta terms were admitted, the solution space could be larger and the 'unique set' characterization would fail. The explicit construction (4.67)-(4.69) and the s=3,4,5 checks are not affected by this gap, so the fix is to qualify the abstract and to state in Section 4.2 either why eta terms cannot enter or that uniqueness holds within the stated class.
[Section 4.2, end (eqs. 4.76-4.77)] The reduction of the source-free Lagrangian equation to the non-Lagrangian equations (1.2)/(1.3) of [7] is demonstrated only for s=4 (eqs. 4.76-4.77). For general s the paper asserts that A_nl=0 'can be shown to imply' (1.2)/(1.3) 'by making careful use of the Bianchi identities (4.60)', without giving the argument. Since this reduction is the stated consistency link between the new non-local equations and the earlier curvature-based formulation, either a general-s argument or an explicit statement that the reduction is checked only for low spins should be supplied.
[Section 4.1, off-shell continuation] The paper acknowledges that neither eta-bar nor the F^(n) are well defined on shell, and that the current-exchange comparison is performed off shell and only the final results are continued on shell. This is a standard technique in this literature and the authors flag it, but it is load-bearing for the equivalence claim: the non-local operators have poles, and it should be stated precisely which quantities are continued, in which order, and why the continuation is unambiguous for the amplitudes (3.46) and (3.68). A short paragraph making this explicit would strengthen the central comparison.

Minor objections (6)

[Eqs. (4.66) and (4.67)] The closed form (4.66) has the product starting at j=-1, while (4.67) has it starting at j=0 with the same prefactor (n+2)/(n-1). The two are inconsistent: (4.66) gives a_0=1 and a_1=n/(n+1), matching (4.64), whereas (4.67) as written gives a_0=(n+2)/(n-1). One of the two lower limits is a typo and should be corrected.
[Section 4, opening paragraph] The non-local analysis is restricted to bosonic fields ('we confine our attention, for brevity, to the case of bosonic fields, but similar results are expected to hold for fermionic fields'), while the abstract's claim about a unique set of non-local equations does not state this restriction. The abstract or the Section 4 introduction should make clear that the non-local uniqueness result is bosonic.
[Note Added] The Note Added states that the current exchange (3.46) and its fermionic counterpart 'determine the van Dam-Veltman-Zakharov discontinuity in flat space for this whole classes of (bosonic and fermionic) higher-spin fields'. This is a physics claim about the discontinuity that is not derived in the paper. Either a brief derivation or a softening of the claim would be appropriate.
[Eqs. (4.29) and (4.30)] The explicit s=4 comparisons give large coefficients for the (J'')^2 terms, and the reader is asked to trust that 'none of the two choices results in the correct current exchange'. A sentence indicating how these coefficients were checked, or a cross-check against (3.46), would make the comparison easier to verify.
[Section 3.3, eqs. (3.71)-(3.72)] The AdS fermionic current exchange is treated very briefly compared with the flat-space case. Some additional detail on how the modified conservation law (3.69) and the triple gamma-trace condition combine in (3.71)-(3.72) would help the reader.
[Section 2.2, eq. (2.62)] The paper notes that it is correcting 'a couple of misprints present in eq. (5.37) of [16]'. It would be helpful to state explicitly which terms are corrected, so that readers comparing with [16] can identify the changes.

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The paper

The paper claims

A conserved external current selects one particular non-local form of the free higher-spin field equation, singling it out from the other curvature-based candidates.

The reading

Free higher-spin gauge fields are usually described by Fronsdal's equations, which impose algebraic trace constraints on the field and on the gauge parameter. This paper works through the alternative, unconstrained descriptions, where a compensator field and a Lagrange multiplier replace the constraints, and it organises them into one-parameter families of Lagrangians that also extend to anti-de Sitter backgrounds. The decisive test is what happens when an external conserved current is coupled: the amplitude for exchanging a current with a current must count exactly the physical polarisations, no more and no fewer. The authors show that the constrained and unconstrained local formulations pass this test identically, but that the earlier non-local formulation built directly on higher-spin curvatures does not, and they then construct the non-local equations that do.

What carries the argument

The load-bearing object is the non-local Einstein-like tensor $G_{nl} = A_{nl} - 2^{-1} eta A'_{nl} + eta^2 B + ...$, where $A_{nl}$ is a fixed linear combination $A_{nl} = sum_k a_k (partial^{2k}/Box^k) F^{(n+1)[k]}$ of the curvature-built operator and its traces. The coefficients $a_k = (-1)^{k+1}(2k-1) [(n+2)/(n-1)] prod_j (n+j)/(n-j+1)$ are the unique solution of the recursion $a_{k+1} = -a_k (2k+1)(n+k)/[(2k-1)(n-k+1)]$ obtained by imposing $partial cdot A_{nl} - 2^{-1} partial A'_{nl} = 0$. The mechanism is that this single condition simultaneously delivers double-tracelessness and the compensator-like form $A_{nl} = F - 3 partial^3 alpha_{nl}$, so the divergence of the trace becomes a gradient and the equation $G_{nl} = J$ can be solved for $A_{nl}$ in the form that matches the local current exchange.

Extended reading notes

Core claim

The paper's central claim is that current exchange fixes the non-local form of the free higher-spin theory. In the local setting the authors show that, once the Lagrange multiplier is on shell, the gauge-invariant tensor $A$ built from the Fronsdal operator and the compensator is doubly traceless and obeys the same Bianchi identity as Fronsdal's $F$; the effective current $K = J - 2B$ (bosons) or $K = T^{(3)}_m J$ (fermions) then reproduces exactly the Fang-Fronsdal amplitude, so the constrained and unconstrained local theories exchange the same number of degrees of freedom. The non-local theory of the earlier curvature-based proposal fails here: its Einstein-like tensor $G^{(n)}$ satisfies a modified Bianchi identity with coefficient $1/(2n)$ instead of $1/2$, so its amplitude carries the coefficients $rho_n(D-2n,s)$ rather than $rho_n(D-2,s)$. For spin 3 the two answers are $J^2 - (3/D)(J')^2$ versus $J^2 - (3/(D-2))(J')^2$, a difference that shows up already in the trace terms. The authors then construct the corrected object: a tensor $A_{nl}$ formed from the curvature-based $F^{(n+1)}$ and its successive traces, with coefficients fixed by demanding the ordinary Bianchi identity, together with a $D_{nl}$ that makes the divergence of the trace a pure gradient, so that the resulting Einstein-like tensor is conserved and inverts to precisely the local amplitude.

Load-bearing premise

The uniqueness claim holds only inside the class of non-local tensors written as a linear combination of the curvature operator and its traces with no explicit metric factors, and the whole comparison is done off shell with operators that are singular on shell.

Editorial extensions

If this is right

For every spin $s$, the non-local Lagrangian equation that couples correctly to a conserved current is determined, removing an ambiguity left by the earlier curvature-based construction.
In the absence of sources the new equations reduce, via successive traces and the curvature Bianchi identities, to the non-Lagrangian equations $Box^{-p} partial cdot R^{[p]} = 0$ (odd $s = 2p+1$) and $Box^{-(p-1)} R^{[p]} = 0$ (even $s = 2p$), so the two non-local formulations are linked.
The equality of the local and non-local amplitudes means both describe the same physical polarisation count, an on-shell traceless symmetric tensor of $SO(D-2)$, despite their very different field content.
Because the amplitude depends explicitly on the spacetime dimension, comparing the $D$-dimensional result with the $(D+1)$-dimensional one gives the van Dam-Veltman-Zakharov discontinuity for these whole classes of bosonic and fermionic higher-spin fields.
The one-parameter families of local Lagrangians, and their AdS deformations, provide a template in which the compensator and multiplier can be gauge-fixed back to Fronsdal's constrained form.

Reading between the lines

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

The identification of a unique non-local equation is a statement about a restricted class: the paper assumes $A_{nl}$ contains no explicit metric factors, so a wider ansatz with $eta$-dependent terms might admit further solutions that also pass the current-exchange test.
Since the matching is performed off shell and continued on shell, the equivalence is really a statement about the off-shell continuation of singular pseudo-differential operators; a careful treatment of the singular locus could reveal where the two formulations part company.
The same current-exchange criterion could be applied to mixed-symmetry tensors, where the local minimal formulation is not yet known, and might serve as a substitute selector for the correct non-local equations there.
The dependence of the amplitude on $D$ suggests that the discontinuity statement is a sharp, checkable prediction: the trace and double-trace coefficients in the exchange amplitude are fixed numbers that any competing formulation must reproduce.

Original abstract

The (Fang-)Fronsdal formulation for free fully symmetric (spinor-) tensors rests on (gamma-)trace constraints on gauge fields and parameters. When these are relaxed, glimpses of the underlying geometry emerge: the field equations extend to non-local expressions involving the higher-spin curvatures, and with only a pair of additional fields an equivalent ``minimal'' local formulation is also possible. In this paper we complete the discussion of the ``minimal'' formulation for fully symmetric (spinor-) tensors, constructing one-parameter families of Lagrangians and extending them to (A)dS backgrounds. We then turn on external currents, that in this setting are subject to conventional conservation laws and, by a close scrutiny of current exchanges in the various formulations, we clarify the precise link between the local and non-local versions of the theory. To this end, we first show the equivalence of the constrained and unconstrained local formulations, and then identify a unique set of non-local Lagrangian equations which behave in exactly the same fashion in current exchanges.

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What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.13 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.17 · compares with prior results 0.78 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

The (Fang-)Fronsdal formulation for free fully symmetric (spinor-) tensors rests on (gamma-)trace constraints on gauge fields and parameters. When these are relaxed, glimpses of the underlying geometry emerge: the field equations extend to non-local expressions involving the higher-spin curvatures, and with only a pair of additional fields an equivalent ``minimal'' local formulation is also possible. In this paper we complete the discussion of the ``minimal'' formulation for fully symmetric (spinor-) tensors, constructing one-parameter families of Lagrangians and extending them to (A)dS backgrounds. We then turn on external currents, that in this setting are subject to conventional conservation laws and, by a close scrutiny of current exchanges in the various formulations, we clarify the precise link between the local and non-local versions of the theory. To this end, we first show the equivalence of the constrained and unconstrained local formulations, and then identify a unique set of non-local Lagrangian equations which behave in exactly the same fashion in current exchanges.

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	The (Fang-)Fronsdal formulation for free fully symmetric (spinor-) tensors rests on (gamma-)trace constraints on gauge fields and parameters.	background 1.00	0.05	0.03	asserted 1.00
s1	When these are relaxed, glimpses of the underlying geometry emerge: the field equations extend to non-local expressions involving the higher-spin curvatures, and with only a pair of additional fields an equivalent ``minimal'' local formulation is also possible.	background 0.88	0.25	0.09	hedged possibility 0.52
s2	In this paper we complete the discussion of the ``minimal'' formulation for fully symmetric (spinor-) tensors, constructing one-parameter families of Lagrangians and extending them to (A)dS backgrounds.	contribution 1.00	0.95	0.21	asserted 0.99
s3	We then turn on external currents, that in this setting are subject to conventional conservation laws and, by a close scrutiny of current exchanges in the various formulations, we clarify the precise link between the local and non-local versions of the theory.	contribution 0.95	0.93	0.21	asserted 1.00
s4	To this end, we first show the equivalence of the constrained and unconstrained local formulations, and then identify a unique set of non-local Lagrangian equations which behave in exactly the same fashion in current exchanges.	method 0.67	0.95	0.39	asserted 1.00

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

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References

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High Energy String Theory and the Celestial Sphere
hep-th 2025-04 conditional novelty 6.0 of 10

The high-energy string amplitude and the celestial sphere amplitude share the same saddle point, and their subleading 1/alpha' and 1/beta corrections match order by order.
The tensionless path from closed to open strings
hep-th 2019-05 conditional novelty 6.0 of 10

The tensionless limit of the closed bosonic string vacuum equals a Neumann boundary state, so closed strings become open strings as tension goes to zero.
Spinning AdS Loop Diagrams: Two Point Functions
hep-th 2017-08 accept novelty 6.0 of 10

A spectral decomposition of spinning propagators reduces AdS one-loop two-point Witten diagrams to Mellin-Barnes spectral integrals times boundary conformal integrals, and shows that bubble diagrams alone miss the ano...
Spinning Witten Diagrams
hep-th 2017-02 conditional novelty 6.0 of 10

A systematic framework computes the conformal partial wave expansion of spinning Witten diagrams and reconstructs all cubic couplings of symmetric fields in the dual anti-de Sitter theory.
Cubic interactions of Maxwell-like higher spins
hep-th 2016-11 conditional novelty 6.0 of 10

Cubic interaction vertices for Maxwell-like higher-spin fields are constructed in flat and (Anti) de Sitter space and shown to be trace-free and simpler than their Fronsdal counterparts.
Continuous spin gauge field in (A)dS space
hep-th 2016-10 conditional novelty 6.0 of 10

A gauge-invariant Lagrangian formulation of continuous spin fields in (anti-)de Sitter space is constructed, with a BRST-invariant gauge-fixed version whose partition function equals one.
Tensionless Strings from Worldsheet Symmetries
hep-th 2015-07 conditional novelty 6.0 of 10

Tensionless closed strings have a vacuum that is a Bogoliubov-squeezed state of ordinary string oscillators, and their physical states are not mass eigenstates.
Towards holographic higher-spin interactions: Four-point functions and higher-spin exchange
hep-th 2014-11 accept novelty 6.0 of 10

Complete off-shell massless higher-spin AdS propagators are derived in three gauges, and the arbitrary-spin s-channel exchange between scalars is expressed as a boundary conformal partial wave expansion.
Towards metric-like higher-spin gauge theories in three dimensions
hep-th 2012-08 conditional novelty 6.0 of 10

The quadratic-order metric-like action for a spin-3 field coupled to 3D gravity is identified as the counterpart of the SL(3,R) x SL(3,R) Chern-Simons theory, yielding a corrected area law for higher-spin black holes.
On the cubic interactions of massive and partially-massless higher spins in (A)dS
hep-th 2012-03 accept novelty 6.0 of 10

The traceless-transverse parts of all consistent cubic vertices for massive and partially-massless higher-spin fields in (A)dS are solved as generating functions, with a new condition for partially-massless highest-de...
General trilinear interaction for arbitrary even higher spin gauge fields
hep-th 2010-03 conditional novelty 6.0 of 10

A closed-form Noether-procedure solution gives all cubic interactions of three arbitrary higher-spin gauge fields in flat space, with a classification theorem tying the derivative count to the spin ratios.
Boundary Carroll CFTs: SUSY and Superstrings
hep-th 2025-08 conditional novelty 5.0 of 10

Two boundary superconformal Carroll algebras are constructed, and the homogeneous one is shown to be the worldsheet symmetry algebra of a newly constructed open null superstring.
A time-like window into tensionless worldsheets
hep-th 2024-12 reject novelty 5.0 of 10

Kasner worldsheets near their null horizons reproduce the tensionless, Carrollian string physics previously found for Rindler worldsheets, with a Bogoliubov-map identification that depends on an arbitrary regularizati...
Cubic Interaction for Higher Spins in $AdS_{d+1}$ space in the explicit covariant form
hep-th 2019-08 conditional novelty 5.0 of 10

The authors solve the recurrence relations that finalize the radial pullback of the main cubic higher spin vertex from flat d+2 space to AdS_{d+1}, trace and divergence corrections included.
Fermionic continuous spin gauge field in (A)dS space
hep-th 2017-03 conditional novelty 5.0 of 10

A gauge-invariant Lagrangian for fermionic continuous spin fields in (A)dS space is constructed, and its partition function is shown to equal one.
Higher-spin massless S-matrices in four-dimensions
hep-th 2013-11 accept novelty 5.0 of 10

Pole-counting on four-particle amplitudes built from classified three-point vertices leaves only general relativity, Yang-Mills, phi-cubed, and supersymmetric theories as consistent massless interactions in four dimensions.
Cubic interactions of massless higher spins in (A)dS: metric-like approach
hep-th 2011-10 conditional novelty 5.0 of 10

All parity-invariant cubic interactions of massless higher-spin gauge fields in (A)dS are given in closed form using traceless-transverse constraints and an ambient flat-space embedding.
On the Tensionless Limit of String theory, Off - Shell Higher Spin Interaction Vertices and BCFW Recursion Relations
hep-th 2010-09 conditional novelty 5.0 of 10

Off-shell cubic and quartic higher-spin vertices are constructed via BRST, and a BCFW consistency test is argued to fail for two candidate couplings, suggesting missing non-local states.
On higher spin interactions with matter
hep-th 2009-03 conditional novelty 5.0 of 10

Summing the exchange of an infinite tower of higher-spin gauge fields between scalar particles yields a closed-form amplitude that can fall off exponentially at high energy for suitable couplings.
Elementary particles with continuous spin
hep-th 2017-08 conditional novelty 3.0 of 10

Continuous-spin particles, massless with an unbounded tower of helicities, now have gauge-invariant free actions on flat and anti-de Sitter space that reduce to Fronsdal's when the spin parameter goes to zero.
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...
Basic introduction to higher-spin theories
hep-th 2022-06 accept novelty 2.0 of 10

A structured lecture-note introduction to massless higher-spin theory, covering free field representation theory, the Fronsdal action in flat and AdS space, higher-spin algebras, the Noether procedure, no-go theorems,...
Elements of Vasiliev theory
hep-th 2014-01 accept novelty 2.0 of 10

A self-contained review that walks from free Fronsdal fields through the unfolded formalism and the four-dimensional spinor dictionary to the nonlinear Vasiliev higher-spin equations.
Gauge Invariant Lagrangians for Free and Interacting Higher Spin Fields. A Review of the BRST formulation
hep-th 2008-05 accept novelty 2.0 of 10

The paper reviews how nilpotent BRST charges generate gauge-invariant Lagrangians for higher-spin fields and how cubic interactions are built from BRST-invariant vertices.
Snowmass White Paper: Higher Spin Gravity and Higher Spin Symmetry
hep-th 2022-05 unverdicted novelty 1.0 of 10

A community survey argues that infinite-dimensional higher spin symmetry may make quantum gravity tractable, and lays out the open problems and mathematical tools of the program.

How later papers used it

63 later papers cite it

Evidence: substantial. 48 of the citing papers share no author with it, and 8 of those carry a confident label from Pith's citation classifier.

Pith read 23 of the 267 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

1 later paper by other authors is listed below: 1 built on it. 1 more paper by the same authors is listed: 1 built on it and 1 used its method or tool. It appears under more than one label.

Builds on it 2 of 9 papers with this label · 1 by the same authors

Constructed an expression for the current exchanges for any spin-s fields, extending this paper's previous result to mixed symmetry fields and non-transverse external currents.

Higher Spins and String Interactions (2010) · 0.99 · paraphrased: the licence does not allow quoting

By the same authors

Generalized this paper's previous massive flat-space results to the case of (A)dS backgrounds.

(A)dS exchanges and partially-massless higher spins (2008) (same authors) · 1.00 · paraphrased: the licence does not allow quoting

Uses its method or tool 1 of 6 papers with this label · 1 by the same authors

By the same authors

Proceeded as in this paper to turn the field equation with an external current in (d+1)-dimensional Minkowski space into another form.

(A)dS exchanges and partially-massless higher spins (2008) (same authors) · 0.91 · paraphrased: the licence does not allow quoting

How the literature uses it

59 arXiv papers cite this in 260 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 · 172 sentences without a confident label

Contests it 1 paper

The explicit dependence of $_$ on the curvatures has been given in [ref] and looksWe correct here a misprint in the corresponding equation ($4.67$) in [ref] . _ = - 3 ^ 3 _ = ∑_k=0^n+1 (-1)^k+1 (2 k - 1) { n + 2n - 1 ∏_j=-1^k-1 n + jn - j + 1 } ^ 2k^ k _n+1^ [k] , with $_ n+1$ defined in ([ref]).

Geometric Lagrangians for massive higher-spin fields · Geometric Lagrangians · contest 0.91

Builds on it 9 papers

In this work we would like to show that the geometric Lagrangians proposed in [ref] admit relatively simple quadratic deformations, so as to provide a consistent description of the corresponding massive theory.

Geometric Lagrangians for massive higher-spin fields · Introduction · 0.98
In this paper we have generalized the previous massive flat-space results of [ref] to the case of $(A)dS$ backgrounds.

(A)dS exchanges and partially-massless higher spins · Discussion · 1.00
The main purpose of this paper is thus to extend the minimal “metric-like” formulation of unconstrained higher spins of [ref] to the case of mixed symmetry.

Unconstrained Higher Spins of Mixed Symmetry. I. Bose Fields · Introduction · 1.00

Uses its method or tool 6 papers

As explained in [ref] we can compute the current-current interaction between those currents using a projector of the form[ The operator $P$ is polynomial of powers of $η_μν$ and $Π_μν= η_μν- p_μp̅_ν- p_νp̅_μ$ where $p^2=p̅^2=0, p·p̅=1$, which guarantees that only physical degrees of freedom are exchanged between external currents.

Current Exchanges for Reducible Higher Spin Multiplets and Gauge Fixing · The Current-Current Interaction · 0.98
In an arbitrary coordinate system, its equation of motion in the unconstrained formalism of [ref] reads

(A)dS exchanges and partially-massless higher spins · Foliating a flat $(d+1)$-dimensional space time by $dS$ sections · 0.80
To do that, we shall follow the main lines of the strategy adopted by Francia and Sagnotti to build their local unconstrained formulation for fully symmetric fields [ref] .

Metric-like Lagrangian Formulations for Higher-Spin Fields of Mixed Symmetry · Unconstrained theory · 0.93

Compares with it 8 papers

The expression [ref] is exactly the current exchange obtained by the authors of [ref] .

Continuous-spin field propagator and interaction with matter · Euclidean analytic continuation · 0.92
(3.67) of [ref] , but with $L^2 →- L^2$, since here we are discussing the $dS$ case),

(A)dS exchanges and partially-massless higher spins · Free massive $dS$ fields · 0.82
This result, with respect to the one presented in [ref] , looks more closely related to the underlying geometric description of [ref] , which, in a sense, should provide its ultimate meaning.

Geometric Lagrangians for massive higher-spin fields · Introduction · 0.98

Cites it as support 5 papers

A crucial observation in order to corroborate this hypothesis, and to generalise the result to all spins, is related to the analysis of the current exchange in the presence of weak external sources performed in [ref] .

Geometric Lagrangians for massive higher-spin fields · Geometric Lagrangians · 0.96
As shown in [ref] , this is indeed true, and moreover there is a unique choice to this effect.

Notes on Strings and Higher Spins · Introduction · 0.99
This would not modify the current-current interactions because the residue of the propagator is automatically double-traceless, as pointed out in [ref] .

On higher spin interactions with matter · Conclusion and discussion of results · 0.97

Cites it as background 26 papers

In [ref] a systematic analysis of the interaction between unconstrained higher-spin fields and external sources, both in flat and in (A)dS backgrounds, was proposed.

Geometric massive higher spins and current exchanges · Introduction and summary · 0.83
The construction of geometric theories for higher-spin fields proposed in [ref] , and reviewed in Sections $2.1.2$ and $2.1.3$ does not produce a unique answer, in the sense that infinitely many gauge-invariant Lagrangians are actually available, whose corresponding free equations can all be shown to imply the (Fang-)Fronsdal ones, after a suitable, partial gauge-fixing is performed.

Geometric Lagrangians for massive higher-spin fields · Setting of the problem · 0.91
The response of unconstrained higher-spin gauge fields to external currents was investigated systematically in [ref] .

Continuous-spin field propagator and interaction with matter · Current exchange · 0.95

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.

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