New higher-spin curvatures in flat space

It was shown that the Lie algebra underlying higher-spin holography admits a contraction including a Poincar\'e subalgebra in any space-time dimensions. The associated curvatures, however, do not reproduce upon linearisation those that are usually employed to formulate the equations of motion of free massless particles in Minkowski space. We show that, despite this mismatch, the new linearised curvatures can also be used to describe massless higher-spin fields. This suggests a new way to build interacting higher-spin gauge theories in Minkowski space that may admit a holographic description.


Introduction
The interactions of massless particles of spin greater than two, aka higher-spin particles, are strongly constrained by several no-go results, see e.g.[1] for a review.In spite of this, positive results accumulated over the years in an effort motivated, for instance, by the long-held conjecture that string theory might be a broken phase of a higher-spin gauge theory and, more recently, by applications in holography.We refer to [2,3] for reviews on these two research directions and to [4] for a recent status overview of higher-spin theories.In particular, non-linear equations of motion for massless higher-spin fields on constantcurvature backgrounds were built by Vasiliev and collaborators [5,6].Later on, these have been conjectured to provide the bulk duals of certain weakly-interacting conformal field theories within the AdS/CFT correspondence [7,8].These developments led to the common lore that higher-spin gauge theories do exist in the presence of a cosmological constant Λ , provided one is ready to accept some unconventional features, like e.g. an infinite spectrum of fields.
Vasiliev's equations and higher-spin holography rely upon an infinite-dimensional Lie algebra that is essentially unique when the dimension of space-time is bigger than three [9,10], modulo supersymmetric extensions and Chan-Paton factors [11][12][13].For instance, Vasiliev's equations are built using curvatures valued in this higher-spin algebra that we shall denote by hs D , with D the space-time dimension.This approach to the interactions of fields of arbitrary spin, often referred to as unfolded formulation, somehow extends to higher spins the Cartan formulation of general relativity.In one of the founding papers of the unfolded formulation, it was observed that the algebra hs 4 admits a contraction containing a Poincaré subalgebra [9].The result, however, was discarded as a candidate higher-spin algebra in Minkowski space because the associated linearised curvatures for spin s > 2 do not agree with those emerging from the first-order free action of [14].Equivalently, while the linearised curvatures of fields with s = 2 agree with the usual torsion and Riemann curvature, for s 3 they do not agree with the Λ → 0 limit of those entering the equations of motion for free massless particles on (Anti) de Sitter ((A)dS) space of [15].This observation was long considered as an additional no-go argument against higherspin interactions in Minkowski space: no appropriate symmetry algebra seemed to exist, at least for the same spectrum of fields as in Vasiliev's equations.This view was also supported by direct analyses of interactions within Fronsdal's metric-like formulation [16] in which a particle of spin s is described starting from a rank-s symmetric tensor.In this setup, various studies independently pointed out the inconsistency of the non-Abelian, two-derivative, minimal gravitational coupling of Fronsdal's gauge fields in flat space; see, e.g., [1,17] and references therein.As discussed in [1], Weinberg's famous low-energy theorem [18] as well as the generalised Weinberg-Witten theorem of [19] can also be reinterpreted in these terms.
On the other hand, in [20], a consistent non-Abelian cubic coupling between massless spin-s and spin-2 fields around Minkowski space containing a total of (2s − 2) derivatives was obtained in Fronsdal's formulation.Although it was shown to induce a consistent non-Abelian deformation of the free gauge algebra satisfying Jacobi identities [21], its analysis has not been pushed to next (quartic and higher) orders in the fields, which would be needed to complete an interacting theory.More recently, a complete interacting higherspin gauge theory on four-dimensional flat manifolds with Euclidean or split signature, dubbed chiral higher-spin gravity, has been built employing a different set of fields [22,23].The role of chiral higher-spin models in flat-space holography also begins to be explored [24][25][26].Besides, an analogue of the contraction of the algebra hs 4 discussed in [9] was recently defined in any space-time dimension [27].The contracted algebra, that we shall denote as ihs D , can also be obtained from the Poincaré algebra following a construction close to that relating its AdS ancestor hs D to the conformal algebra [27,28].
These indications naturally lead to reconsider the linearised curvatures of ihs D .In this note, we propose a new system of first-order equations of motion built upon them, that describes the free propagation of massless particles of arbitrary spin on Minkowski space.Our equations follow the same pattern as in the usual unfolded formulation: for any spin s we set to zero all corresponding curvatures but one, and impose that the latter is proportional to the spin-s Weyl tensor.We then prove that, even if our curvatures have a non-standard form, the resulting equations of motion are equivalent to the standard ones [15].
When expressed in terms of curvatures, the structure of our equations is the same as that of the customary free unfolded equations in AdS.Moreover, the full non-linear curvatures are a contraction of the AdS ones.This strongly suggests the option to deform our linear equations into an interacting theory following the path that led from [9] to [5,6] or, equivalently, the cohomological approach of [29,30].We defer a detailed analysis to future work, but we wish to stress that this programme is expected to provide, on the one hand, a model for higher-spin interactions in Minkowski space-time and, on the other hand, a simple and concrete model for flat-space holography (see, e.g., [31][32][33][34][35][36] for an overview of various approaches).The higher-spin algebra ihs D indeed also appears as a subalgebra of the higher symmetries of a Carrollian scalar living on null infinity [28].This is the analogue of a pillar of higher-spin holography: the symmetry algebra hs D is realised at the boundary of AdS as the algebra of higher symmetries of a free conformal scalar [37,38].All higher-spin dualities involve deformations of this basic setup preserving such symmetry, see e.g.[39] for a review focusing on this aspect.Any non-linear deformation of our free equations of motion will therefore provide a candidate gravitational dual of the simplest Carrollian field theory, thus fitting within the urgent quest for concrete dual pairs in flat-space holography, that is currently mainly driven by symmetry considerations.

Higher-spin extension of the Poincaré algebra
The higher-spin algebra ihs D can be obtained as an İnönü-Wigner contraction of the algebra hs D underlying Vasiliev's equations in AdS space [27].The latter can be built, e.g., by evaluating the universal enveloping algebra of so(2, D − 1) on Dirac's singleton module [38,[40][41][42].Its generators can be collected in irreducible and traceless tensors M a(s),b(t) with s 0 and 0 t s, where the shorthand a(s) denotes a set of s symmetrised indices.They thus satisfy η cd M a(s−2)cd,b(t) = 0 and M a(s),ab(t−1) = 0, where repeated indices denote a symmetrisation with strength one, and correspond to representations of the Lorentz algebra labelled by two-row Young tableaux.Their commutators take the form with (s 1 + s 2 + s 3 ) mod 2 = 1 and (t 1 + t 2 + t 3 ) mod 2 = 1.For s = 1 one recovers the so(2, D − 1) conformal algebra and explicit structure constants can be found in [42].
The generators M a(s),b(t) with s − t even thus form a subalgebra and one can rescale the others as In the limit ǫ → 0 all commutators involving only generators with s − t odd vanish, while the others remain untouched.The so(2, D − 1) subalgebra contracts into a iso(1, D − 1) subalgebra and one obtains a non-Abelian higher-spin extension of the Poincaré algebra.We chose to present this algebra as a contraction of the AdS higher-spin algebra hs D , although one can also build it as a quotient of the universal enveloping algebra of the Poincaré algebra iso(1, D − 1) [27,28].One can also prove that, in a generic space-time dimension D > 3, ihs D is the only algebra with the same set of generators as hs D that can be built with this procedure [27].When D = 3, the option to build sensible higherspin algebras in Minkowski space from contractions of the AdS ones was already observed in [43] (see also [44][45][46]27]), and Vasiliev-like equations of motion were proposed in [47].

New equations of motion in Minkowski space
We now consider a one-form taking values in the Lie-algebra ihs D , and its Yang-Mills curvature, The one-forms ω a and ω a,b correspond, respectively, to the space-time vielbein and spin connection.
We wish to build equations of motion describing free massless particles using the linearisation of the curvatures F a(s),b(t) around the Minkowski background, linearisation that we shall denote as F a(s),b(t) .We thus split the vielbein as ω a = h a + e a and, for simplicity, we choose Cartesian coordinates so that the background vielbein reads h µ a = δ µ a and the background spin connection vanishes.The following discussion can be extended to arbitrary coordinates by introducing a flat background Lorentz connection, but working in Cartesian coordinates makes some arguments more transparent.
The linearised curvatures only depend on the commutators between the higher-spin generators M a(s),b(t) and those of the Poincaré subalgebra.With our choice of coordinates, they read for s − t even, (5a) where braces denote a two-row Young projection together with a traceless projection, so that the corresponding term shares the same symmetry properties as the others: where we recall that repeated indices denote a symmetrisation with strength one, e.g.A a B a := 1 2 (A a 1 B a 2 + A a 2 B a 1 ).Notice that the second term on the r.h.s. of (5b) is absent for t = 0, when this value is allowed by the parity condition.The curvature with t = s, instead, always fits in the class (5a).The linearised curvatures (5) are invariant under the gauge transformations for s − t even, (7a) To describe a particle with spin s we propose to impose the equations of motion where C a(s),b(s) is a gauge-invariant and Lorentz-irreducible tensor.The integrability of eq.(8b) also imposes a tower of Bianchi identities on this zero-form, leading to its interpretation as Weyl tensor, see e.g.[48,49].For s = 2, eqs.( 8) are the linearised vacuum Einstein equations, where the vanishing of the Ricci tensor is reformulated by equating the Riemann curvature with the Weyl tensor.In the following, we prove that they describe the free propagation of a massless particle of spin s by showing that they are equivalent to the Lopatin-Vasiliev equations of motion on Minkowski space [15]; see also [50][51][52].
Alternatively, one can obtain eqs.( 8) by rescaling ω a(s−1),b(s−2n) → ǫ ω a(s−1),b(s−2n) in the equations of motion of [15] and sending ǫ → 0 while keeping their dependence on the cosmological constant Λ fixed.The latter can then be absorbed in a redefinition of the connections ω a(s−1),b(s−2n−1) .If one instead keeps ǫ fixed while sending Λ → 0, one obtains a different limit, displayed in eqs.( 22) below, which is what we refer to as the Lopatin-Vasiliev equations on Minkowski space.

Equivalence with the Lopatin-Vasiliev equations
To prove that eqs.( 8) are equivalent to the Lopatin-Vasiliev equations [15] and, therefore, to the Fronsdal equation in Minkowski space [16], we begin with the instructive spin-three example.We then extend the proof to any spin.

The spin-three example
For a spin-three particle eqs.(8) read where, for clarity, we renamed the fields ω ab → e ab and ω ab,cd → X ab,cd and we stressed that F ab and F ab,c play the role of torsions, while F ab,cd plays that of a curvature.These equations are invariant under δX ab,cd = dρ ab,cd .(10c) Thanks to the Poincaré lemma, eq.(9a) implies that e ab is pure gauge.We can thus set it to zero using the gauge symmetry generated by ξ ab .In this gauge, eqs.(9b) and (9c) take the same form as the (Λ → 0 limit) of the corresponding Lopatin-Vasiliev equations.To show that these two equations suffice to describe a massless spin-three particle, it is enough to notice that, in the gauge e ab = 0, eq.(9b) implies This is the case because X ab,cd is symmetric in its last two indices and with our choice for the background vielbein, dh a = 0.The Poincaré lemma then allows one to introduce the one-form ẽab such that This relation is valid for all ω ab,c , in particular for a pure-gauge infinitesimal configuration δω ab,c for which we denote the corresponding r.h.s. of the previous equation by dδẽ ab .The configuration δẽ ab is then seen to be identically equal to (recall that we used ξ ab to fix the gauge e ab = 0, so that ξab is a new gauge parameter that does not affect ω ab,c ).
The fields ẽab , ω ab,c and X ab,cd manifestly satisfy the Λ → 0 limit of the Lopatin-Vasiliev equations, that is with the gauge symmetries For the reader's convenience, we recall that once the form ( 14) of the equations of motion is reached, one can use the parameter λ ab,c to gauge away the corresponding component of h µ c ẽµ ab , so as to recover the Fronsdal field ϕ µνρ = h (µ a h ν b ẽρ) ab .The torsion constraint (14a) then allows one to express ω ab,c in terms of the first derivative of ϕ µνρ , except for a pure-gauge component which is gauged away using ρ ab,cd .The torsion constraint (14b) plays a double role: some of its irreducible components only involve ω ab,c and impose Fronsdal's equation on ϕ µνρ , while the others express X ab,cd in terms of the first derivatives of ω ab,c and, eventually, in terms of two derivatives of the Fronsdal field.The last equation (14c) then expresses the Weyl tensor in terms of third derivatives of ϕ µνρ .For a more detailed review of this mechanism see, e.g., [51].

Arbitrary spin
For an arbitrary value s of the spin, the equations F a(s−1),b(t) = 0 with s − t − 1 even and greater than zero imply that the fields ω a(s−1),b(t) are pure gauge thanks to the Poincaré lemma.As such, they can be set to zero using the gauge variations (7a): In this gauge, most of the other torsion-like equations become closure conditions too: This shows that the fields ω a(s−1),b(s−2n) with n 2 can also be eliminated using the parameters λ a(s−1),b(s−2n) with n 2. Eventually, one is left with the equations The first one implies and, thanks to the Poincaré lemma, −h c ∧ ω a(s−1),b(s−3)c = dω a(s−1),b(s−3) .
The same argument can be repeated using generic coordinates on Minkowski spacetime: the main modification is that the exterior derivative has to be traded for the background Lorentz-covariant derivative ∇.This does not affect the previous analysis because we only used that the latter annihilates the background vielbein, ∇h a = 0, and it is nilpotent.

Discussion
We proposed new first-order equations of motion for free massless particles of arbitrary spin in Minkowski space, built upon the linearised curvatures of the flat-space higherspin algebra ihs D introduced in [27,28].If the set of fields that we use in our equations is the same as in Lopatin-Vasiliev's equations on (A)dS background [15], that are the free equations of motion on top of which Vasiliev's unfolded formulation for interacting higher-spin fields is constructed, we stress that their precise expressions differ from the zero cosmological constant limit of the equations in [15].In spite of this difference, that a priori could prevent one from eliminating some auxiliary fields, we showed that our equations nevertheless propagate the correct degrees of freedom for a massless field in Minkowski space-time of dimension D 4. This is so because all fields ω a(s−1),b(t) with t s − 3 are actually pure gauge, while the field equations involving ω a(s−1),b(s−1) , which encode the degrees of freedom via the Weyl tensor, take the same form in both systems of equations.
As a result, the non-linear curvatures of the non-Abelian, flat-space higher-spin algebra ihs D of [27] can be considered as the basic building blocks to construct an interacting higher-spin gauge theory in Minkowski space in the unfolded formalism, along the purely algebraic lines of [5,6] or of its reformulation in [29,30].The fact that the flat-space higher-spin algebra ihs D possesses an Abelian ideal, contrary to the AdS algebra hs D , suggests even more freedom in introducing interactions via the cohomological approach of [29,30].
Another remark supporting the proposal to build a non-linear theory based on the algebra ihs D is that such a theory should include the (2s −2)-derivative coupling of a massless spin-s field to gravity of [53,20].Indeed, in [20] it was shown that this cubic vertex induces a non-Abelian deformation of the free gauge algebra, leading to a contribution proportional to the translation generator P a in the commutator M a(s−1),b(s−1) , M c(s−1),d(s−2) .As explained in Section 2, the latter commutator is unaffected by the contraction leading from hs D to ihs D , which does contain a contribution proportional to P a .Moreover, as discussed in [20], the (2s − 2)-derivative vertex is the one that possesses the highest number of derivatives among those that constitute the Fradkin-Vasiliev gravitational coupling in AdS [54], that has later been reproduced within the unfolded formulation.Only this top vertex survives the flat limit that coincides with the high-energy limit of the Fradkin-Vasiliev action, thereby evading the low-energy no-go results [18,19].
Finally, let us stress that our result suggests the option to define an interacting dual of a Carrollian scalar on null infinity, with similar features to the bulk models entering higher-spin holography [3].The Carrollian approach to flat-space holography, see e.g.[35,36], appears particularly well suited to this proposal, which relies on the observation that a Carrollian scalar admits a ihs D symmetry algebra [28]; see also [55] for similar bulk/boundary realisations of higher-spin symmetries in D = 3.