Cubic interactions of Maxwell-like higher spins

Journal of High Energy Physics, Apr 2017

We study the cubic vertices for Maxwell-like higher-spins in flat and (A)dS background spaces of any dimension. Reducibility of their free spectra implies that a single cubic vertex involving any three fields subsumes a number of couplings among different particles of various spins. The resulting vertices do not involve traces of the fields and in this sense are simpler than their Fronsdal counterparts. We propose an extension of both the free theory and of its cubic deformation to a more general class of partially reducible systems, that one can obtain from the original theory upon imposing trace constraints of various orders. The key to our results is a version of the Noether procedure allowing to systematically account for the deformations of the transversality conditions to be imposed on the gauge parameters at the free level.

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Cubic interactions of Maxwell-like higher spins

Received: November Cubic interactions of Maxwell-like higher spins Dario Francia 0 1 3 5 Gabriele Lo Monaco 0 1 3 4 5 Karapet Mkrtchyan 0 1 2 3 5 Open Access 0 1 3 5 c The Authors. 0 1 3 5 0 Piazza della Scienza 3 , I-20126 Milano , Italy 1 Piazza Fibonacci , 3, I-56126, Pisa , Italy 2 Max Planck Institut fur Gravitationsphysik 3 Piazza dei Cavalieri , 7 I-56126 Pisa , Italy 4 Dipartimento di Fisica, Universita di Pisa 5 Am Muhlenberg 1 , Potsdam 14476 , Germany We study the cubic vertices for Maxwell-like higher-spins in at and (A)dS background spaces of any dimension. Reducibility of their free spectra implies that a single cubic vertex involving any three elds subsumes a number of couplings among di erent particles of various spins. The resulting vertices do not involve traces of the elds and in this sense are simpler than their Fronsdal counterparts. We propose an extension of both the free theory and of its cubic deformation to a more general class of partially reducible systems, that one can obtain from the original theory upon imposing trace constraints of various orders. The key to our results is a version of the Noether procedure allowing to systematically account for the deformations of the transversality conditions to be imposed on the gauge parameters at the free level. cDipartimento di Fisica; Universita di Milano{Bicocca 1 Introduction Maxwell-like description of higher spins 3 On the Noether procedure for constrained gauge theories Generalities Trace constraints Divergence constraints 4 The cubic vertex Reduction to (A)dS covariant expressions: the 4 0 0 example 7.5 O -shell generating functions and Grassmann variables Completion of the vertex Conditions for gauge invariance Deformation of the transversality constraint Additional couplings and eld rede nitions 5 Partially reducible and irreducible cases 6 Inclusion of traces 7 (Anti) de Sitter cubic vertices Ambient space formulation The (A)dS cubic vertex Deformation of the constraint 8 Outlook A Notation and conventions A.1 Free elds A.2 Interacting elds B Some useful commutators C On the need for deforming the constraint D Spectra of partially reducible theories In this work we construct the cubic vertices on at and (A)dS backgrounds deforming the free Lagrangians for massless higher-spin elds proposed in [1]. Their equations of motion are based on a kinetic tensor retaining the same form as the Maxwell tensor for spin-one elds, gauge parameters where ' denotes a rank s tensor, and rely on an Abelian gauge symmetry with transverse M = 2' @@ ' = 0 ; ' = @ ; = 0 : They propagate a reducible spectrum of massless particles with spin s; s and it is mainly in this sense that this Maxwell-like description di ers from the Fronsdal one [2], whose equations describe the degrees of freedom of a single massless particle of spin s. As a consequence, a given cubic vertex involving a speci c triple of Maxwell-like tensors with ranks s1; s2 and s3, actually subsumes a number of cross-interactions among all the particles with di erent spins actually carried by each tensor. Cubic interactions involving massless particles of arbitrary spins have been investigated from several perspectives, starting from the light-cone results of the Goteborg group [3, 4], that also provided the very rst non-trivial instances of higher-spin interactions ever proposed. The systematics of covariant constructions, together with some explicit instances of couplings, were rst discussed in at space in [5, 6], while with the seminal work of Fradkin and Vasiliev [7] the relevance of (A)dS background was appreciated for the rst time. Subsequent extensive explorations have been performed, culminating in a classi cation of cubic vertices for massless symmetric higher-spin elds in arbitrary dimensions, both in the light-cone gauge [8, 9] and in covariant form [10] for the case of at backgrounds. For alternative perspectives and additional references, as well as for generalisations to (A)dS spaces, see [11{52]. There are three main aspects of our investigation where the di erences between Maxwell-like elds and Fronsdal elds are more remarkable: the structure of the cubic vertex for Maxwell-like elds is simpler than in the Fronsthe transversality conditions on the free gauge parameters are to be corrected by eld-dependent terms. This leads to modi cations of the Noether procedure where new equations have to be taken into account; in the Maxwell-like setting, we argue that the same Lagrangian can describe spectra of various degrees of complexity, upon imposing trace constraints of increasing strength on both elds and parameters. Our construction holds for all these cases, thus providing cubic interactions for putative complete theories with di erent particle DDD Since the detailed analysis of these issues is somewhat technical, in this introduction we aim to illustrate them in qualitative terms, thus providing a general summary of our Structure of Maxwell-like cubic vertices In the context of interactions of massless elds, at space is singled out among constantcurvature backgrounds. In at space, vertices containing a di erent number of derivatives are essentially independent, as far as gauge invariance is concerned. Therefore, on Minkowski backgrounds, the number of derivatives proves to be a useful guiding marker for classifying cubic interactions for irreducible massless (Fronsdal) elds. Reducible models are not di erent in this respect, so that we can study their cubic interactions assuming xed the overall number of derivatives. Thus, as far as its essential structure is concerned, any cubic vertex can be written as involving a total of n derivatives, acting in a prescribed way on the tensors 'i of rank L1 where no divergences or traces of the elds 'i are taken into account (the so-called transverse-traceless, or brie y TT, sector), to then proceed to include them in order to set up a proper scheme of cancellation of the gauge variation at each step. The possible types of terms are summarised in table 1, with reference to a basis of counterterms involving traces and de Donder tensors D := @ ' In the Fronsdal setting, in particular, typically all types of terms collected in table 1 actually enter the cubic vertex. (See e.g. [32, 35].) For Maxwell-like Lagrangians, di erently, due to the simplicity of the free kinetic term (1.1), one never needs to introduce traces in the procedure. Indeed, in the reducible framework that is of most interest for us, traces essentially represent independent elds that may or may not be considered in the construction. In a minimal scheme one can avoid introducing them altogether, reducing the types of terms to be taken into account just to the rst row of table 1, as summarised in table 2, with the \de Donder tensors" D here to be identi ed with divergences of the elds, D := @ '. Since at the TT-level there is no di erence between our construction and the corresponding analysis for Fronsdal elds, these vertices are found to admit a number of DDD derivatives bound to satisfy the usual double inequality [4, 8, 10] s1 + s2 + s3 2minfs1; s2; s3g +s1 + s2 + s3 : However, one has to remind that each Maxwell-like vertex provides a synthetic description of several cross-interactions involving low-spin particles. For the latter in general the total number of derivatives would exceed the bound, thus implying that in the full theory one should deal with the issue of clarifying the role of all those additional couplings. Cubic couplings for reducible systems were investigated from a di erent perspective in [11, 15, 17, 18]. Adapted Noether procedure and deformation of the constraints The second feature that we would like to stress concerns the need for deforming the transversality condition (1.2). Once all possible counterterms encoded in table 2 are considered, one needs to identify the deformation of the free gauge symmetry accounting for cubic-level gauge invariance. The possibility to drive the procedure to completion is tied to the form of the variation of the resulting cubic Lagrangian L1, that can be schematically written as follows 2 are local operators depending on elds, gauge parameters and derivatives, whose form we compute explicitly. The peculiar aspects of our procedure are encoded in the last term in (1.5), proportional to the double divergence of the eld: although vanishing on the free mass shell, it cannot be locally related to the free equations of motion, and thus cannot be absorbed in the contribution to (1.5) proportional to M . For instance, for spin s = 2, M = and similarly for higher spins, with higher divergences of M to be involved. For this reason, following the steps of the usual Noether procedure, it is no longer clear, and in general it won't be true, that one can obtain from (1.5) the correction to the gauge transformation 1' in its standard local form. On the other hand, the variation of the free Lagrangian shows that the contributions in 2 can be compensated by a suitable deformation of the transversality constraint (1.2) of the form + O('; ) = 0 ; where in O('; ) all elds and gauge parameters may enter, a priori. For the spin 2 case, where (1.1) provides the linearised equations of traceful unimodular gravity [53], the corrections (1.8) are instrumental to reproduce the covariant form of the transversality condition, D interesting to notice that, in this special case, the Noether procedure at cubic order would actually not compel the introduction of corrections to (1.2). This shows, in our opinion, that including in the Noether procedure the possibility encoded in (1.8) in principle retains a deeper meaning than just allowing to enforce some algebraic cancellations. For higher spins, (1.8) provides an additional equation that enters the perturbative reconstruction of the gauge structure and that turns out to be necessary to the completion of the procedure. We judge that this option may be of more general interest. In this spirit, we shall present a systematic discussion of how to include in the Noether procedure perturbative corrections to possible constraints to be imposed on the free gauge symmetry. Partially reducible theories The Maxwell-like equations (1.1) propagate the maximal reducible unitary representation of the Lorentz group encoded in the symmetric tensor ', and one may wonder whether consistent truncations of the spectrum may be implemented. The degrees of freedom of the various particles propagating in (1.1) are essentially contained in the traces of the eld '.1 Thus, a natural guess is that partial truncations of the spectrum may be implemented by trace constraints of increasing strength. In this view we suggest that the projected equations of motion Mk = M + k @ @ '[k 1] = 0 ; where '[m] denotes th m th trace of ' while k is a coe cient ensuring that Mk be k traceless, should describe particles with spin s; s 2; : : : ; up to s that, on top of the transversality condition (1.2), the k th traces of the eld and of the gauge parameter be vanishing: '[k] = 0 ; [k] = 0 : In this view, the fully reducible theory described by (1.1) and its fully irreducible counterpart, obtained upon imposing tracelessness of both ' and [54], would represent just the extrema of a chain of theories describing spectra of decreasing complexity. Up to relatively simple modi cations, our construction of the cubic vertices apply to each of these options. Let us mention that models where each tensor carries the degrees of freedom of a number of particles that increases with its rank provide instances of higher-spin theories for which a full non-linear counterpart is not known. Indeed, in any putative Maxwell-like complete theory, that contains at least one copy of a Maxwell-like eld for each (even) rank, there would appear in nitely many massless particles for any given value of the spin, up to those that may be eliminated from the spectrum by conditions of the form (1.10). This is to be contrasted with the presently known Vasiliev's theories whose spectra involve at most a nite number of particles with the same spin [55{58]. 1An o -shell covariant separation of the various single-particle components would combine traces with multiple divergences. See section 4 of [1] for a detailed discussion of this point. Plan of the paper In section 2 we review the Maxwell-like theory of [1] and present its partially reducible generalisations. In section 3 we rephrase the Noether procedure in a way that allows to encompass the case of constrained gauge symmetries in a systematic fashion. The computation of the cubic vertex is presented in section 4. In particular, in section 4.1 we discuss the TT sector, while section 4.2 contains a detailed illustration of the subsequent steps for the fully reducible case. In section 5 we discuss how to adapt the construction of vertices to the full class of theories with partially reducible spectra, up to the fully irreducible one. We shall comment on the possibility of including traces in the vertices in section 6. Section 7 is devoted to the construction of (A)dS vertices. Here we switch to a di erent language, that of ambient space, that has the advantage of allowing to e ectively bypass the need for computing commutators of covariant derivatives, at the price of making the whole construction less explicit. In the Outlook we collect our nal comments, while four appendices contain technical remarks on our notation, on the absence of alternatives to the deformation of the constraint in our context, and on the spectra of a few selected partially reducible models. Maxwell-like description of higher spins In this section we illustrate the formulation of Maxwell-like higher-spin theories. We provide a review of the fully reducible model of [1] together with a proposal for partially reducible Maxwell-like theories, covering all possible unitary spectra encoded in principle in a symmetric tensor. The covariant description of massless representations of the Poincare group with a given nite spin (helicity) s is encoded in the Fierz system [59], 2' = 0 ; @ ' = 0 ; '0 = 0 ; 2 = 0 ; = 0 ; 0 = 0 ; and one can look for its possible o -shell completions under the requirements of locality, gauge invariance and, for simplicity, absence of auxiliary Once the d'Alembert variation of 2' requires indeed to combine it with the divergence of ' to form the tensor that in this sense provides the minimal building block of any gauge theory. Its gauge The remaining conditions in (2.1) may or may not be relaxed and the various options lead to theories with di erent particle contents and possibly di erent Lagrangian formulations. M = 2' M = As far as the structure of the kinetic tensor is concerned the simplest choice is to fully relax tracelessness of ' and while keeping the transversality condition on the latter, and thus consider the gauge-invariant Lagrangian with constrained gauge symmetry L0 = = 0 : Mk = M + Qik=1[D + 2(s @ @ '[k 1] = 0 : In appendix D we analyse in detail the spectra of a few non-trivial cases, providing quanencode the linearised version of unimodular gravity, with or without the additional scalar mode depending on whether the trace of the graviton is kept or discarded. (See e.g. [53] and references therein.) The unique local alternative to imposing the transversality condition (2.5) is to include the trace of ' in the kinetic operator, thus leading to the Fronsdal tensor '[k] = 0 ; [k] = 0 ; the strongest possible trace constraint that one can impose. In the latter case the theory can be interpreted as an allowed gauge xing of the Fronsdal Lagrangian and correspondingly the spectrum is known to collapse to the single massless particle of spin s [54]. On the basis of the fact that the physical polarisations of the fully reducible system are carried on-shell by the traces of ' [1], we argue that higher values of k in (2.6) should correspond to less severe truncations of the spectrum. In this view, for instance, the system 4 is expected to enter should lead to a reducible theory where only the lowest-spin particle, a scalar or a vector, gets eliminated from the spectrum. It is to be stressed that the Lagrangian always retains the same form (2.4), irrespective of the strength of the trace condition imposed in (2.6), while the equations of motion resulting after implementing (2.6) obtain as the k th traceless projection of (2.2) and look particles with spin s 2k, k = 0; : : : ; [ 2s ]. We put forward that the same Lagrangian (2.4) can be used to describe spectra of various degree of complexity if, on top of the transversality condition (2.5), one also requires both the gauge eld and the gauge parameter to be subject to trace conditions of varying F = 2' whose gauge invariance holds if 0 = 0 : condition (2.5) and the trace constraint (2.9) can be easily evaded by introducing additional, Stueckelberg compensator elds D and that transform as follows so as to guarantee unconstrained gauge invariance of the corresponding extended kinetic D = @ = 0 ; In the case of (2.13), upon introducing an additional auxiliary eld serving as a Lagrange multiplier for the double trace of ', one recovers the minimal unconstrained formulation of [60, 61]. Other unconstrained extensions of the Fronsdal theory with bigger eld content (and only two-derivative kinetic tensors) were proposed in [62{65]. The unconstrained completion of M provided by (2.12), on the other hand, may be further supthe triplet Lagrangian emerging from free tensionless strings, as rst shown in [66, 67] and further elaborated upon in [68{71], whose spectrum corresponds to that of the fully reducible Maxwell-like theory with no trace conditions imposed. The tensors (2.2) and (2.8) somehow provide higher-spin counterparts of the kinetic tensors of the Maxwell and of the (linearised) Einstein theories, respectively. The idea that the spin-one and the spin-two models may have di erent higher-spin incarnations was rst put forward in [72, 73] and further elaborated upon in [74{76]. Reducible higher-spin models were also given a free frame-like formulation on at and (A)dS backgrounds in [77], while related constructions stemming from a worldline perspective were presented in [78, 79]. Our goal is to study the interactions among elds whose free Lagrangian is (2.4). Since the full e ect of possible trace constraints would be to give rise to projected equations of motion while leaving the free Lagrangian untouched, the structure of the corresponding cubic vertex turns out to be universal for the whole class of partially reducible models, with di erences emerging only at the level of their gauge deformations. From our perspective one relevant issue concerns the role of the transversality condition (2.5) in the deformation procedure. As we shall see in section 4, in general we are able to nd a local solution for the cubic vertices only upon allowing for corrections to (2.5). (See also appendix C.) Thus, in order to better frame the details of our computations, in the next section we illustrate the general framework for adapting the Noether procedure to the case of constrained gauge symmetries. The basic assumption underlying the Noether procedure2 [6] is that both the action functional S['] and its gauge invariances ' admit a perturbative expansion in powers of S['] = S0['] + gS1['] + g2S2['] : : : ; ' = 0' + g 1' + g2 2' : : : ; On the Noether procedure for constrained gauge theories number of elds: k' where in this section ' (and later ) represents a collective symbol denoting all types of elds (and parameters) entering the theory. The coupling g plays the role of a counting parameter for the powers of elds to be added to the corresponding zero-th order quantity. S0['] is the free action depending quadratically on the elds, while Sk['] involves k additional powers of the elds: Sk['] 'k+2. Similarly, 0' represents the Abelian gauge the transformations are linear in the gauge parameter (since in the present context we always consider in nitesimal gauge transformations) and depend in principle on an increasing Asking for perturbative gauge invariance of the action leads to the Noether system of 0S0['] = 0 ; 1S0['] + 0S1['] = 0 ; 2S0['] + 1S1['] + 0S2['] = 0 ; whose solutions provide in principle the possible interaction vertices compatible with gauge invariance, while also determining the corresponding deformations, Abelian or non-Abelian, of the free gauge symmetry. The procedure just outlined applies to the case of unconstrained gauge symmetries or even when constraints are present that, anyway, do not get deformed themselves. On general grounds, however, one may consider the possibility that the free gauge parameters are subject to given o -shell conditions implemented through the action of some (linear) operator O in the schematic form concrete examples being provided by the transversality and trace conditions (2.5) and (2.9), respectively. What we would like to stress is that the constraints (3.3) may themselves receive perturbative corrections in increasing powers of the elds: + gO1( ; ') + g2O2( ; '2) + : : : = 0 : 2The Noether procedure has been widely used in recent years in the higher-spin literature. Here we quote the work that, to our knowledge, rst implemented it to the purpose of investigating interactions among higher-spin massless elds. = 0 ; In a minimal scheme where one does not consider the option of trading (3.3) for the inclusion of auxiliary elds, in order to properly take the corrections encoded in (3.4) into account one has to modify the Noether equations (3.2). In particular, from this perspective, the terms encoding the deformation of the gauge symmetry order by order, denoted with k', would admit themselves a perturbative expansion in powers of ', due to their implicit dependence on ' hidden in because of (3.4). In order to display the e ective dependence of the various terms by powers of the elds we shall make use of the following notation: k(l)' := O('k+l) k' = k(0)' + k(1)' + k(2)' + : : : ; Correspondingly, the terms entering the Noether system (3.2) get modi ed as follows o( ; ') : S = o( ; '2) : S = o( ; '3) : S = For instance, in the case under scrutiny in this paper, both 0(0)' and 0(1)' are given by the Abelian transformation @ . On the other hand, while the parameter in 0(0)' solves + gO1( ; ') = 0; to rst order in ', in case a non-trivial correction term O1( ; ') is considered. Similarly for the higher-order terms. Let us recall that here denotes collectively all the gauge parameters entering the procedure and in general the correction terms Ok( ; 'k) may depend linearly on all of them. As usual, from (3.6) one can rst determine the cubic vertices L1 on the free mass shell, i.e. for those con gurations that vanish when L'0 = 0 holds. At this stage the corrections encoded in (3.4) do not play any special role. Once L1 is found one should proceed to collect all terms in S that are quadratic in ' and that vanish on the free mass shell, and arrange them as in the second equation of (3.6) so as to compute 1'. The latter in general, as one can see in (3.6), comprises two contributions, whose splitting may well be not uniquely determined. It seems to us that two situations are possible: (1) in some cases the nature of the constraints (3.3) may be such that it is always possible to reabsorb any correction encoded in (3.4) in a suitable local deformation of ' while keeping (3.3) unchanged; (2) in other cases corrections of the form (3.4) and correspondingly modi ed Noether equations (3.6) may be unavoidable in order to grant for the existence of a local solution to the deformation procedure. If option (1) occurs, then the general system (3.6) encoding the corrections (3.4) can be equivalently traded for the more customary Noether system (3.2), with constraints on the gauge parameters kept in their original \free" form (3.3). However, we would like to stress that even in these cases taking the deformations into account can be relevant. They may provide some algebraic simpli cations, to begin with, but more important than that, as suggested by the perturbative reconstruction of unimodular gravity (see section 4), they may be connected with the underlying geometry of the theory. On the other hand, whenever (2) holds, addressing the procedure in its generalised form (3.6) becomes unescapable, as far as one wishes to keep manifest locality at the perturbative level. While the procedure just outlined applies in principle to all sort of constraints possibly imposed on gauge parameters o shell, in the following we shall comment more in detail on the two cases that are directly relevant for higher spins, from our perspective. Trace constraints Algebraic constraints, like the trace conditions (2.9) of the Fronsdal theory, are usually regarded as harmless from the perspective of the deformation procedure, since on Minkowski background there appears to be no need to deform them in order to get the corresponding covariant form of the rst-order deformation [10, 16, 29]. From our perspective, this feature of the Fronsdal theory may be envisaged considering the variation of the Fronsdal Lagrangian for spin s = 3, L = from which it is manifest that deformations of the trace conditions may only a ect terms that are locally proportional to the free equations of motion and that, as such, can be equivalently included in a local rede nition of 1(0)'. For these reasons, it seems plausible to us that, more generally, the option (1) may more easily refer to constraints (3.3) that are algebraic in nature. However, a few remarks are in order: to begin with, it has to be stressed that the cubic deformation of the spin-three Fronsdal theory on (A)dS background computed in [20] was indeed found to require a deformation of the trace constraint on , taking into account perturbative corrections to the background metric tensor, used to compute the trace, via additional dynamical spin-two contributions. From this perspective, similar considerations should apply to the double-trace conditions to be enforced on the Fronsdal gauge elds o shell.3 3It is to be mentioned that this option does not manifest itself in the frame-like formulation, where trace conditions are encoded in the choice of the frame elds and of the generalised spin connections as taking values in irreps of the orthogonal group in the tangent space. (See e.g. [57, 58].) Similar metric-deformations of the trace constraint were also considered in a threedimensional setup in [80], for the case of a Fronsdal spin-three eld coupled to gravity, motivated by the need to maintain manifest di eomorphism invariance. In [81] and [82] the analysis was repeated from a frame-like perspective and extended to include spin-four cubic self-interactions together with 3 4 vertices. Interestingly, there it was found that deriving the metric-like vertices from the frame formulation tions of the trace constraints also involving higher-spin elds. Indeed, it is only on (A)dS backgrounds that cubic couplings of the form 2 entail a minimal coupling vertex. In this sense (at least for a covariant and local theory) only in (A)dS can one interpret the gauge deformation on the spin-two side really as the rst non-linear contribution to full di eomorphism invariance. In the at case, on the other hand, the allowed cubic couplings are non-minimal, and thus one should not necessarily expect to be forced to treat h uctuation of the metric tensor from the very beginning. In other words, it may not be obvious a priori that the Minkowski metric be corrected by powers of the rank-two tensor h . These general considerations notwithstanding, we think that there are reasons to expect the general status of trace conditions to be more complicated, both for (A)dS and for Minkowski backgrounds. To begin with, in the same sense as the deformation of the Fronsdal trace constraint found in (A)dS could be expected on account of the existence of spin-two higher-spin covariance of the full theory leads us to expect that in general the full deformation of the trace conditions should involve also higher spins and not just be determined by a metric correction to the trace, consistently with the three-dimensional ndings of [82]. From our vantage point, as summarised in our comment to (3.8), the expectation is that those deformations may well be reabsorbed by a local eld rede nition (at the cubic level at least this option should manifest itself both on at and on (A)dS backgrounds), but theory from its complete frame-like counterpart. For similar reasons, one may wonder whether the absence of deformation for the Fronsdal trace constraint on Minkowski background may be a spurious e ect of performing the analysis only up to the cubic level. Indeed, from the perspective of the unconstrained absence of Noether corrections for the gauge transformation of the compensator eld there introduced (see (2.13)). Although not impossible in principle we have no reason to expect it in general. In addition, the case of spin s 4 appears to be more involved than the example of spin 3 that we referred to in (3.8), due to the interplay between presence or absence of the double trace constraint and actual form of the corresponding free equations of motion. Finally, as already mentioned (see also section 4.5), in the context of unimodular gravity knowledge of the full non-linear theory explicitly indicates that the transversality constraint on the parameter has to be eventually covariantised, = 0 = 0 ; although at cubic level in Minkowski space one nds that this option is not forced upon by consistency of the Noether procedure. Divergence constraints As we shall see in the next section, option (2) is realised by the Maxwell-like systems described by (2.4) and (2.5). Since we are going to discuss this point in detail, here we shall limit ourselves to make just a few remarks. We could not nd in general a local solution to the deformation procedure without taking the corrections to the constraints (2.5) into account. As we discuss in appendix C, our conclusion is that this feature is intrinsic to the systems under consideration: starting with the free Maxwell-like Lagrangians (2.4) it is unavoidable to implement the deformation of the transversality condition (2.5) in order to grant for local cubic-level gauge invariance. The main underlying reason is that, for Maxwell-like systems, there are eld con gurations that vanish on the free mass shell in force of conditions that are non-locally proportional to the equations of motion. These aspects will be also discussed in section 4. It should be mentioned, however, that an alternative option for Maxwell-like theories would be to solve for the transversality condition (2.5) in terms of unconstrained gauge parameters, to then proceed with the implementation of the Nother method in the standard fashion. The general solution to (2.5) was computed in [83] and takes the form with (0) taking values in the irrep of GL(D) corresponding to a rectangular tableau with In this parametrisation the gauge symmetry is reducible, the pattern of gauge-for-gauge transformations being encoded in the following chain of reducibility conditions = @ with k = 0; 2, where the symmetries of the parameters (k) are encoded in the However, the customary system of Noether equations (3.2) applied to this high-derivative and reducible gauge symmetry would eventually result to be far more complicated to deal with than the equivalent one (3.6), where the constraint is not solved for and its corrections are taken into account. Moreover, while the two options for 1::: s , either satisfying (2.5) or being solved for as in (3.10), are certainly locally equivalent, it may expected in general that they may di er when global issues are considered. In addition, we see no reasons in principle why it should be always possible to solve any constrained gauge symmetry in terms of local unconstrained alternative parametrisations. In the absence of such solutions, once again, it seems to us that the only options would be either to look for suitable sets of auxiliary elds entering the theory as substitutes of the condition (3.3), or to implement the Noether procedure as encoded in (3.6). The cubic vertex to compute three types of terms: the cubic vertex L1; In this section we construct the cubic vertex for Maxwell-like elds in at space-time. Our goal is to solve the Noether system (3.6) to rst-order in the parameter g, which requires the explicit deformation of the gauge transformation 1(0)'; the rst correction to the transversality constraint: @ + gO1( ; ') = 0. In this section we brie y recall how to determine the general form of the transverse-traceless part of the cubic vertex. At this level there are no di erences with the case of Fronsdal elds and for this reason we shall only sketch the general structure of the computation. For more details see [10], whose conventions we follow here, together with appendices A and B. We aim to construct the cubic interactions among three gauge elds of arbitrary spins, '1 := '(s1)(x1; a); '2 := '(s2)(x2; b); '3 := '(s3)(x3; c) ; 4Actually, as observed in [83], it is always possible to parametrise the gauge symmetry of a theory (at the linear level, at least) in in nitely many di erent, yet equivalent ways. Even for ordinary, spin 1 Maxwell elds one may write the scalar gauge parameter as a contracted, multiple divergence of a symmetric tensor of arbitrary rank, A = @ @ 1 1::: s , obtaining in this way a rather unusual-looking high-derivative and reducible parametrization of the gauge symmetry of a free massless vector eld. For the Maxwell-like case this observation also shows that, contrarily to standard lore, a fully unconstrained, local description of massless metric-like higher spins is possible without invoking auxiliary elds in the construction. starting from the so-called TT sector of the vertex, including no divergences nor traces, that we write in the schematic form T T = D T (Q; n)'1'2'3 ; T (Q; n) := T (Q12; Q23; Q31; n1; n2; n3) is the operator performing all the contractions of indices and containing the derivatives assumed to enter the vertex. Its general form is where n is the total number of derivatives, while the various coe cients are linked by the following relations n1 + n2 + n3 = n ; Q12 + Q31 + n1 = s1 ; Q12 + Q23 + n2 = s2 ; Q23 + Q31 + n3 = s3 : Vertices that di er by total derivatives are to be regarded as equivalent, and it is convenient to establish a convention allowing to systematically deal with integration by parts. Our de nition (4.3) for T (Q; n) corresponds to the cyclic ansatz, de ned such that a derivative operator gets always contracted with the eld that precedes it cyclically, i.e. elds and derivatives always appear in the following order: The space-time arguments of each eld are taken to be di erent in order to simplify the various manipulations. dD := dDx2dDx3 D(x1 is the integration measure, containing the delta-functions that eventually compute the various elds at coinciding points, thus ensuring locality of the result. KfQ;ng := KfQ12;Q23;Q31;n1;n2;n3g are the relative coe cients among the various terms of the vertex and represent the unknowns of our initial problem. Due to the relations (4.4) they e ectively depend only on the Qij 's. (Or on the ni's.) Correspondingly, the sum in (4.2) e ectively runs only over one set of independent options. Let us observe that in the cyclic ansatz d'Alembertian operators acting on elds are automatically excluded, consistently with the fact that, at cubic level, they can always be reabsorbed (up to terms proportional to traces and divergences of the elds) by a eld rede nition of the free Lagrangian of the schematic form ' ! ' + ''.5 The coe cients KfQ;ng can be xed, up to an overall constant, by requiring that the variation of (4.2) under the gauge transformation of each eld, '(s)(x; a) = sa @ (s 1)(x; a) vanishes, up to contributions that vanish when the free equations of motion hold, terms containing traces or divergences of the elds, as well as contributions proportional either to 2 , whose cancellation requires the introduction of counterterms proportional to traces or divergences of the elds. The result is KfQ;ng = Q12!Q23!Q31! where the overall constant k is expected to be xed by the next order in the Noether procedure. (See [84, 85] for earlier results in the light-cone formulation, and [86] for a related discussion from a holographic perspective.) A simple consequence of (4.7), as brie y explained in [10], is that the number of derivatives n in the vertex cannot be lower than s1 + s2 + s3 2 minfs1; s2; s3g. To see this we rst recall that the sum of the Q's is xed by the spins of the elds involved in the vertex and by the number of derivatives, since (4.4) implies s1 + s2 + s3 = n + 2(Q12 + Q23 + Q31) : Now, from (4.7) we deduce that, for gauge invariance to hold, the sum in (4.2) has to run over all non-negative values of the Q's, satisfying (4.4). In particular, there are nonvanishing coe cients in (4.7) with one of the Q's equal to zero. Let us take for example Q12 = 0. Obviously, Q23 + Q31 s3, from which it follows, that Q12 + Q23 + Q31 to hold for all values of the Q's, otherwise one of the terms in the ansatz (4.2) will have negative powers of contractions. Similarly, one can show that Q12 + Q23 + Q31 i = 1; 2; 3. This in turn implies that n = s1 + s2 + s3 2(Q12 + Q23 + Q31) s1 + s2 + s3 2 minfs1; s2; s3g ; consistently with the light-cone analysis of [84, 85]. 5This observation alone explains the upper bound on the total number of derivatives that can appear in the cubic vertex: since any number of derivatives higher than the sum of the spins would generate d'Alembertian operators, upon integration by parts. One should also keep in mind, however, that eld rede nitions will propagate their e ects to higher orders, and it may be not obvious a priori that a simpli cation at the cubic level would correspond to an overall simpli cation of the full theory. All in all, making a choice about rede nitions amounts to choosing a basis of elds, and the general convenience of a given option may be not at all apparent at the cubic level, or at any given The ensuing steps in the calculation depend on the gauge variation of the TT part of the vertex, that we write schematically as follows: T T = s1s2Q12T (Q12 s2s3Q23T (Q23 s3s1Q31T (Q31 where in particular all arguments in T (Q; n) that are not explicitly displayed are meant to Our goal is to compensate (4.10) by means of suitable counterterms and, if needed, by also reconsidering the transversality condition (2.5). We shall illustrate the corresponding results in the next two sections. Completion of the vertex The full cubic vertex will turn out to have the following schematic form, L1 = L1 where in particular the various counterterms will involve up to three divergences (here denoted with D) but no traces of '. The gauge variation of the TT part of the vertex, computed in (4.10), suggests a systematic way of introducing these counterterms. Here we shall illustrate the main steps of the computation, also addressing the reader to appendix B for further technical details. In order to select possible counterterms to be added to (4.2) we focus rst on the terms in (4.10) containing d'Alembertian operators acting on gauge parameters. (Those propor 2' can be traded for free equations of motion and enter the deformation of the gauge symmetry.). Given that at the free level we see that terms of the type 2 '' are to be compensated by monomials of the form D''. The explicit computation xes the corresponding coe cients as follows: L1;D = X KfQ;ng 1)D1'2'3+ Clearly, the cancellation mechanism has to work in the same fashion for the three elds (whose ranks may well be not all distinct), so that in the following we can limit ourselves to illustrating the procedure with reference to one of them, say '1. D = 2 ; that we denote with 0'1 := 0'1, is The gauge variation of L1T T + L1;D with respect to free Abelian transformation of '1, Kn1 1n1T (n1 Kn3 1n3T (n3 Kn3 1n2T (n2 From the second line of (4.14) , in particular, one can read the structure of the counterterms containing D1D2'3 and D1'2D3 needed for the second step. Taking the variations with respect to '2 and '3 into account one nds eventually L1;DD = X KfQ;ng The overall gauge variation of the cubic vertex to this order contains, besides terms proportional to the free equations of motion, the following terms: X KfQ;ng X KfQ;ng X KfQ;ng n3n2Q12T (n3 In order to compensate the contribution containing last term in our chain of compensations L1;DDD = X KfQ;ng 2 DD in (4.16) we introduce the construction of the Maxwell-like cubic vertex (4.11), via (4.2), (4.7), (4.13), (4.15) and (4.17). Before moving to analysing the gauge structure of the theory, let us pause to comment on our result in comparison with related literature. Cubic interactions for reducible higher-spin systems have been investigated using BRST techniques in [15, 17, 19, 31], in the so-called triplet formulation [66{71] related to the free tensionless limit of string theory and involving additional, unphysical elds. Our focus is on reducible models involving physical tensors only, those containing the propagating polarisations of each particle. Our main motivation is essentially one of simplicity, in the following sense: focusing on physical tensors we manage to keep the resulting action as simple as possible, since already at the free level the additional elds of the triplet description make the theory more involved, the di erence being quite signi cant if one compares the corresponding free Lagrangians for mixed-symmetry elds [1, 70]. It was not guaranteed by any general arguments that the simplicity of the free action would be kept at the cubic level, but our results con rm it. The absence of auxiliary elds implies that from our vertices we can better read the physical couplings, while at the technical level it dispenses us with the need to compute the deformation of the gauge symmetry for a wider class of tensors. Moreover, the fully reducible Maxwell-like theory appears to admit an easy generalisation to a whole hierarchy of reducible theories with spectra of decreasing complexity, as we illustrate in section 5. All of these theories are described by the same free Lagrangian and correspondingly deform with the same cubic vertices, the di erences among them being visible only at the level of the gauge deformation, where the structure of the free equations of motion becomes relevant. The choice of working with the minimal Lagrangian (2.4), however, brings in a novelty with respect to the standard implementation of the Noether procedure. As anticipated in the section 3, compensating the variation of (4.11) will require to rethink the conditions that make the free Lagrangian gauge invariant, and speci cally to introduce eld-dependent corrections to the transversality condition (2.5). Conditions for gauge invariance Let us compute once more the variation of (4.11), now collecting all terms that we could not compensate otherwise. In order to properly discuss and appreciate the outcome at this level we consider the full variation of L1, under the simultaneous transformation of the three elds. Schematically, L1 = (2i) are operators depending on all the elds and all the parameters other than 'i and i, whose explicit form is the following: (11)M1 = + encoding the full set of transverse gauge transformations, (U; X) = (U @X )E(U; X) ; (@U @X )E(U; X) = 0 ; while also satisfying appropriate homogeneity and tangentiality conditions U @U )E(U; X) = 0 ; (X @U )E(U; X) = 0 : The (A)dS cubic vertex with the on-shell gauge invariance: We want to write the most general cubic deformation of the free action that is consistent L1 = D C(@i; @Ui ) 1 2 3 ; General arguments of symmetry and covariance lead us to state that the vertex function can be chosen to depend on the following six operators Yi = @Ui @i+1 ; Zi = @Ui 1 @Ui+1 ; i = 1; 2; 3 ; divergence and trace operators: Thus, one looks for the most general operator C(Yi; Zi) satisfying: (@Ui @Xi ) = Di ; (@Ui @Ui ) = Ti : D C(Yi; Zi) 1 2 3 For instance, varying with respect to 1LT T = D C(Yi; Zi)(U1@1)E1 2 3 = D [C(Yi; Zi); (U1@1)]E1 2 3 : Following the same procedure outlined in appendix B, we can rearrange the commutator in (7.14) in such a way to recognise the part containing just TT operators. The vanishing of this traceless-transverse sector imposes a di erential constraint on the function C, Y3@Y3 )@Y1 g C(Yi; Zi) = 0 ; whose general solution reads [38]: In particular, the function K(Yi; G) is arbitrary and admits the Taylor expansion C(Yi; Zi) = e F (Zi)K(Yi; G)jG G(Y;Z) ; G(Yi; Zi) = Y1Z1 + Y2Z2 + Y3Z3 ; K(Yi; G) = gs1;s2;s3;nY1s1 nY s2 nY s3 nGn ; 2 3 whose coe cients gs1;s2;s3;n that are nothing but the coupling constants of the vertices. Thus, to any choice of K it corresponds a choice of the coupling constants,11 while the parameter n counts the number of derivatives in each vertex as follows #@ = s1 + s2 + s3 D [D1@Y1 + D2@Y2 + D3@Y3 ] C(Yi; Zi) (a1; x1) (a2; x2) (a3; x3) : We can observe that, since the Y 0s and G are di erential operators, perturbative locality s1 + s2 + s3 s1 + s2 + s3 ; thus implying the usual at-space bound. Finally, we observe that the function e F (Z) produces low-derivative terms and takes into account part of the low-derivative dimensiondependent tail typical of the (A)dS vertices. The variation of the transverse and traceless part of the Lagrangian produces some terms containing next-order operators 0LT T = As usual, we can now rearrange the 2-terms in such a way to extract a part proportional to the equations of motion together with a part depending on the divergence of the elds: 0LT T = The last term suggests the form of the D-counterterm to be added to the TT Lagrangian LD = L1 = 12 0L1 = Noether procedure. Proceeding along the same chain of compensations implemented in section 4.2 one can reach the full o -shell completion of the vertex DiDj @Yi @Yj + D1D2D3@Y1 @Y2 @Y3 A C(Yi; Zi) 1 2 3 : Deformation of the constraint As we observed in section 4.4, the Maxwell-like Noether procedure entails a subtlety, i.e. the variation of the Lagrangian (7.23) vanishes only up to some weird contributions of 11As a consequence, one can expect that the form of the function K be xed at the quartic order in the where the presence of the double divergences points to the same mechanism of deformation of the constraint that we have already discussed in the at case. In fact we can observe that, relaxing the transversality constraint on the parameter, the variation of the free Lagrangian reads: 0L0 = from which it is apparent that we can exactly compensate the (D22 + T2)-terms appearing in 0L1 by means of an appropriate deformation of the transversality constraint. Reduction to (A)dS covariant expressions: the 4 From the ambient-space vertex it is possible to recover an explicit (A)dS-covariant expression using the appropriate dictionary. The master eld must be reduced to its (A)dS form, where u lives on the tangent space and ea is the (A)dS vielbein. Moreover the role of the partial derivatives @U is taken by the generalised covariant derivatives in RD can be rearranged in such a way to extract a radial component where r is the usual covariant derivative on (A)dS. The coordinates XM of the embedding ' = s=1 1 D := r X = RX^ ; X2 = R so that in particular the metric assumes the following form ds2amb = MN dXM dXN = dR2 + g (x)dx dx = e~Re~R + abe~ae~b ; where x now labels (A)dS coordinates while g is the AdS metric, related to the embedding @XM (x) @XM (x) In this frame, the components of the ambient vielbein read e~RR(R; x) = 1; e~R(R; x) = 0; e~aR(R; x) = 0; e~a (R; x) = U M = e~RM (R; x)v + e~aM (R; x)ua = X^ M (x)v + L In the same fashion, the tangent vectors split in a radial component and d (A)dS compoGiven this decomposition, we rearrange the ambient vertex operators as follows: @XM = X^ M @R + @UM = X^M @v + L while the measure has to be understood as where the delta function enforces the elds to have support on the (A)dS hyper-surface in RD. Finally, we mention that the master eld depends homogeneously on R (R; x; v; u) = h = u @u 2 is the homogeneity factor. As a concrete illustration we specialise (7.23) to the case of a 4 where Q denotes the spin-four eld and i are the scalar elds. Making use of the relations (7.33) together with some suitable tricks, after some algebra one nds that the S4 0 0 = g4 0 0 where in particular we can appreciate the appearance of a contribution in the double trace of the spin-four eld, that would not be present in the irreducible (nor in the Fronsdal case), and thus represents a peculiarity of the reducible formulation. O -shell generating functions and Grassmann variables We would like to notice here some interesting aspects of the o -shell cubic vertices, i.e. vertices involving elds that are not subject to any conditions like transversality and tracelessness, nor involve partially gauge xed elds. For the Fronsdal elds, the o -shell generating functions were found to have a simple form [32, 35]. It was noticed in [32], that certain deformations of the TT vertex involving Grassmann variables allow to immediately write down the generating functions for o -shell vertices. The origin of (BRST-like) Grassmann structure appearing in these generating functions is not well understood. It turns out that the Maxwell-like vertices follow the same pattern and thus their o -shell formulation can be achieved in a similar way. Remarkably, the procedure for elds requires a deformation only involving the derivative operators Yi but not the contractions Zi. This is related to the fact that traces are not appearing in the Maxwell-like vertices. Indeed in [32, 35] the contractions comprise Grassmann deformations involving traces of the elds. One can check easily, that the o -shell vertices (7.23) can be written in the following form L1 = where C(Y; Z) is the function (7.16), that de nes the TT vertices, while Y^i = Yi + D ^ = i3=1d id iePi3=1 i i : provide suitable deformations of the operators Yi and of the integration measure, involving eld. As shown in [32] a similar procedure works for both the anti-symmetric ansatz and the cyclic one. Since there is no conceptual di erence between di erent ansatze, we expect it to work for Maxwell-like elds on at and (A)dS backgrounds too. In fact, based on this evidence one can also conjecture, that the Fronsdal o -shell vertices in (A)dS can be obtained through the same procedure as in [32] starting from the TT vertices of [38], and can be simply with the only modi cations compared to (7.38) being that Di is to be interpreted as the full de Donder tensor, while L1F = Z^i = Zi + In particular, in the de Donder gauge we recover the gauge- xed result of [86]. In this work we constructed cubic interactions for massless elds of any spin described at the free level by the Maxwell-like Lagrangians (2.4). This formulation is more exible than the Fronsdal one, in that it can accommodate all possible unitary spectra of particles whose degrees of freedom are carried by symmetric tensors: from the maximally reducible representation associated to traceful elds and parameters to the single-particle description for the fully traceless case, all intermediate cases appear to be admissible and can be conveniently described in our approach, with relatively minor modi cations in passing from one case to the other. Our construction covers indeed all possible spectra for the case of at-space cubic vertices, while also extending to the case of (A)dS backgrounds. One peculiarity of the Maxwell-like theories, as we stressed, is the spectrum that they would describe in a putative complete non-linear construction. As we saw, from the perspective of the solution to the Noether conditions reducibility of the spectra does not appear to play any signi cant role. In this sense, one would expect that the quartic-level analysis reproduce the main features of the Fronsdal case, including in particular the need for in nitely-many tensors of unbounded ranks. Assuming that a complete theory exist, possibly on (A)dS, involving for simplicity one copy for each even-rank Maxwell-like tensor, the resulting particle spectrum in the fully reducible case would be of the type illustrated in table 3 (where on each column one can read the particle content associated to the tensor of the corresponding rank): with in nitely-many particles present for any given spin, although with seemingly decreasing multiplicity as the spin increases. Let us stress that there would be only one \graviton" in the spectrum, i.e. only one minimally coupled spin 2 particle, all the other spin 2 particles present in the spectrum only entering non-minimal couplings. Other particle contents can be realised upon enforcing trace constraints of various sorts. Many of them would give rise anyway to spectra involving in nitely-many particles with given spins, whenever the reducibility conditions be chosen so as to allow for a number of particles that increases with the rank of each tensor. For instance one may think of truncating away all the scalars (probably with the exception of the one carried by the rank 0 tensor) or all the spin 2 particles (probably with the exception of the one carried by the rank 2 tensor), and so forth. Moreover, in a theory with in nitely-many Maxwell-like elds the very notion of \multiplicity" of a given spin would be no longer obviously de ned, and the picture sketched above may turn out to be even misleading, to some extent. For instance, Let us consider one copy of each spin described by reducible Maxwell-like elds. As already noticed, at rst sight it may seem that the would-be theory of fully reducible Maxwell-like higher spins would contain in its spectrum more spin s1 elds than spin s2 elds for s1 < s2. This might not necessarily be the case, though. In gure 1 below,12 we show how could a theory with equal number of particles of any spin be organized into Maxwell-like multiplets. For the case of a countable in nity of \Regge trajectories", which contain one massless particle of each integer spin, we arrive at a spectrum that could be described by one copy of fully reducible eld of each spin. Now, imagine that the number of Regge trajectories is nite, say M . Then, we can still use the same Maxwell-like elds, all of which are subject to M -th traceless constraint, as shown in the gure 2. Complete, fully non-linear higherspin theories with such kind of spectra have not been constructed so far, to the best of our knowledge, nor is it obvious that they should exist at all. The study of the global symmetries of the corresponding models, as well as the investigation about the existence of prospective holographic counterparts, provide conceivable complementary guiding lines along which to address the question about their existence. In particular, in its conventional formulation, Vasiliev theory and its coloured generalisations involve nitely many elds of any given spin and thus correspond to the case of 12We use the notation V asd for the Vasiliev algebra in (A)dSd which, is also known as Eastwood-Vasiliev algebra and denoted in literature as hso[d 1; 2] for negative cosmological constant. We do not need to specify the sign of the cosmological constant, therefore here V asd is some real form of the complex algebra hs(sod+1), studied in [95]. theory with symmetry algebra U(1) V asd. N numerates Regge trajectories, starting with the \Regge trajectory", while the lines connect the particles that combine into reducible representations described by Maxwell-like elds. Regge trajectories and symmetry algebra U(2) V asd. Blue dots with coordinates (s; n) correspond to massless states of spin s in n-th \Regge trajectory", while the lines connect the particles that combine into reducible representations described by Maxwell-like elds with fourth trace constraint. a nite number of Regge trajectories, whose spectra could be reproduced in terms of irreducible or partially reducible Maxwell-like elds. Let us also notice that the Maxwell-like description implements a grading between even and odd spins, therefore the spectra of all known coloured Vasiliev theories, based on the algebras discussed in [96, 97], can still be matched by Maxwell-like spectra. It is worth recalling, however, that the fully reducible Maxwell-like theories describe free tensionless strings; from this perspective it is far from being apparent which possible truncations may or may not be allowed when interactions are turned on while the tension is still kept to zero. The Regge trajectories in String Theory contain massive states, and the collapse to zero mass of the full spectrum, including mixed-symmetry particles, is described in [70]. In the simplest case, the rst Regge trajectory contains massive states corresponding to symmetric elds of arbitrary rank. In the massless limit each spin s massive particle decomposes into a tower of helicity states, from 0 to s. One can actually think of this spectrum as composed of two towers of reducible massless elds, that can be Higgsed in the following way: a spin s Maxwell-like eld from one tower eats a spin (s 1) Maxwell-like eld from the other tower and becomes massive. The fact that traces do not appear in the at-space Maxwell-like action might turn out to carry a relevant piece of information about the full non-linear theory. It was recently argued that the remarkable properties of gravitational amplitudes allowing to interpret them as squares of gauge theory amplitudes might be made manifest at the level of the action [98]. An important condition for this to be possible is that the action does not contain traces of the eld. This is automatically satis ed for Maxwell-like elds of any spin without any gauge xing. We are grateful to A. Bengtsson, P. Benincasa, D. Blas, N. Boulanger, A. Campoleoni, E. Conde, E. Joung, R. Metsaev, S.J. Rey, A. Sagnotti, P. Sundell, M. Taronna, M. Tsulaia and Yu. Zinoviev for useful discussions and comments. The work of D. F. was supported in part by Scuola Normale Superiore and INFN (I.S. Ste ). The work of K. M. was partly supported by the BK21 Plus Program funded by the Ministry of Education (MOE, Korea) and National Research Foundation of Korea (NRF) and by Alexander von Humboldt foundation. Both D.F. and K.M. were also supported in part by the Munich Institute for Astroand Particle Physics (MIAPP) of the DFG cluster of excellence \Origin and Structure of the Universe". K.M. would like to thank Scuola Normale Superiore and INFN Sezione di Pisa for the kind hospitality extended to him during the nal stages of this work. Notation and conventions For the case of free elds we exploit a notation where all indices are omitted. Thus a rank-s symmetric tensor ' 1::: s is denoted simply as ', while '0 denotes the Lorentz trace of ' ' stands for its divergence. Products of di erent tensors are symmetrised with the minimal number of terms and with no weight factors. In particular the Abelian gauge gauge parameter. For more details about the combinatorics the reader can consult e.g. [94]. In order to deal with cubic and higher-order vertices one needs a notation allowing to keep track of the various possible types of contractions [10]. For a rank s tensor ' 1::: s we exploit s powers of an auxiliary commuting vector a of the tangent space at the base-point x to de ne '(s)(x; a) = ' 1 s (x)a 1 Derivations with respect to x or to a are distinguished as follows: := @ ; := @a : Correspondingly, all the relevant operations to be performed on tensor elds, involving either space-time derivatives or contraction and symmetrization of indices, are implemented in terms of di erential operators. We collect the main ones in the following list, where we also display the proper conversion factors needed to compare with our index-free convention for free elds: a @'(s)(x; a) = (@')(s+1)(x; a) ; (a @)2'(s)(x; a) = @ @a'(s)(x; a) = s(@ ')(s 1)(x; a) ; 2a'(s)(x; a) = s(s 1)('0)(s 2)(x; a) ; a2'(s)(x; a) = In this notation, for instance, the Maxwell-like and Fronsdal tensors take the form Quadratic scalars can be computed by means of the contraction operator M (s)(x; a) = f2 (s)(x; a) = a @@ @a + (a @)22a '(s)(x; a) : a := (s!)2 i=1 so that, for instance = '(s)(x; a) a (s)(x; a) ; LM = LF = 'M = 2 F 0 = Some useful commutators In order to solve the Noether procedure at the cubic level we need an explicit expression for commutators of the following type [T (n1; n2; n3jQ12; Q23; Q31); (a @1)] ; where T is the operator de ned in (4.3), that we report here for convenience: from which one can read the operators that can give a contribution to (B.1). We shall use demonstrate by induction the following relation [An; B] = nAn 1[A; B] : Using (B.3) we can reduce the various non trivial commutators in (B.1) to the following [(@a @2)n1; (a @1)] = @1 @2 ; [(@a @b)Q12; (a @1)] = (@b @3)n2 [(@c @a)Q31; (a @1)] = @c @1 ; from which we can evaluate (B.1) [T; (a @1)] = + Q31T (n3 + 1jQ31 Q12T (n2 + 1jQ12 Q12T (Q12 1)(@b @2) + n1T (n1 and similarly for [T; (b @2)] and [T; (c @3)]. On the need for deforming the constraint One can argue that the setup in this work is limited in that we do not include more than one divergence of any given eld in the cubic ansatz. Despite the fact that double (and higher) divergences are vanishing on the free mass shell, they are non-locally related to the equations of motion, and thus they cannot be removed by local eld rede nitions. Therefore, the most general ansatz for the cubic vertex should involve higher divergences of the elds and allow for a richer parameter space for the cubic vertex, thus potentially altering our conclusion about the need to deform the constraints on the gauge parameters. Now we will try to solve Noether procedure for a general ansatz involving higher divergences of the elds, Lm1;m2;m3 = Cm;1;;m2;m3T ( ; ; jm1; m2; m3)'s1(a1)'s2(a2)'s3(a3) ; (C.1) L3 = + + =n where the m's can take values higher than one, and in general the variation of the (C.1) should be proportional to the free equations of motion and to their divergences. First we compute the divergences of the equations of motion for any eld. The kinetic tensor in the language of the oscillators a reads as in (A.8): M ('(a)) = (2 (a @)(@ @a))'(a) : Now we compute the n-th divergence of the equation of motion: (@ @a)nM ('(a)) = [(1 (a @)(@ @a)n+1]'(a) : The general solution should be gauge invariant with respect to the gauge variation of any of the elds. We are looking for a universal solution to the Noether equation, which should be symmetric with respect to the three elds included. We therefore check the gauge invariance with respect to the variation of only one of the elds, namely '(s1). Taking into account that the terms with higher than one divergences on the eld '(s1) do not contribute to the gauge variation of cubic Lagrangian due to transversality constraint,13 the solution of the Noether equation ensuring gauge invariance with respect to the gauge transformation of the eld '(s1) can give information only about the coe cients C0;;m;2;m3 ; C1;;m;2;m3 . From the symmetry of the general solution with respect to three elds, we should be able to recover also other coe cients di erent from zero, if any. The variation of the Lagrangian (C.1) gives (the commutators of the new T -operator can be computed as in appendix B): 0L0;m2;m3 = + + =n 1)j0; m2 + 1; m3) 1)j0; m2; m3) s1 1(a)'s2 (b)'s3 (c); + + =n Now let us introduce two groups of compensating terms: 0L1;m2;m3 = C1;;m;2;m3 T ( ; ; j0; m2; m3)21 s1 1(a1)'s2 (a2)'s3 (a3) : (C.5) 0;m2;m3 = 0;m2;m3 = s1 1(a)'s2 (b)'s3 (c); s1 1(a1)'s2 (a2)'s3 (a3) ; B0;;m;2;m3 T^( ; ; j0; m2; m3)(a2 @2)(@a2 @2)m2 (@a3 @3)m3 ; D0;;m;2;m3 T^( ; ; j0; m2; m3)(a3 @3)(@a2 @2)m2 (@a3 @3)m3 ; (@a2 @a3 ) (@a3 @a1 ) (@a1 @a2 ) ; We should add and subtract terms (C.6) and (C.7) to the (C.4), and arrange the coe cients B and D to satisfy equations, which ensure 3 = O((@2 @a2 )nM ('s2 (a2)); (@3 @a3 )kM ('s3 (a3))) ; n = 0; 1; : : : ; s2; k = 0; 1; : : : ; s3 : 13If we allow deformation of the transversality constraint, the variation of the terms with more than one divergence on the eld '(s1) contribute to orders higher than the cubic one. In any case we can drop them while discussing the cubic order only. 14Using T^ we just normal order (a2 @2); (a3 @3) with the divergences (@a2 @2); (@a3 @3). terms that we add are to be combined with the terms that include d'Alembertian operator on the elds, to form equations of motion and their divergences. The tricky point here is that these terms themselves give rise to d'Alembertian operators, so they give backreaction that is not possible to take into account order-by-order in general, but is possible to handle altogether simultaneously. This is the reason why we add and subtract corrections to all orders in divergences at the same time, and then get the following recursion relations on the coe cients: + 1)D0;;m;2;m3 = B0;;m;2+1;m3 (1 + 1)B0;;m;2;m3 = D0;;m;2;m3+1(1 m2) ; (C.11) m3) ; (C.12) ( + 1)[C0;;m;2 +11;m3 + C0;;m;2;+m13 ] + ( + 1)[C0;;m+2;1m;3 1 +( + 1)[B0;+m12;;m; 3 1 + B0;+m12;;m; 3 ] + ( + 1)[D0;+m12; ;1;m3 ( + 1)B0;;m;2;+m13 + ( + 1)D0;;m+2;1m; 3 = 0 ; + 1)D0;;m;2;m3 = It is possible to check that the following choice for the coe cients C, B and D C0;;m;2;m3 = B0;;m;2;m3 = D0;;m;2;m3 = C1;;m;2;m3 = solves the system (C.11){(C.14). This implies that it is always possible to compensate at least the deformation of one constraint adding the appropriate higher divergence tail. However, since the solution is not cyclic-symmetric with respect to the indices mi, it cannot compensate the deformation of the constraints stemming from the variation of L with respect to the second and third elds, i.e. the corresponding Lagrangian can be invariant with respect to gauge transformations of the elds '(s2) and '(s3), only with non-vanishing deformations of transversality constraints for these elds. To summarise, we found that the introduction of couplings proportional to higher divergences via the solution (C.15) does not dispense with the need to deform (2.5). Let us stress, however, that we are not proving the solution (C.15) to be unique, therefore the proof of the need for deformation is not yet complete. If we were able to nd a set of coe cients C, B and D solving the system (C.11){(C.14) and cyclic-symmetric in the mi indices, then we could add the corresponding higher-divergence tail so as to render L gauge invariant without any deformation of the constraints. We studied this possibility in one explicit case, the 3 1 vertex: using Mathematica it is possible to see that the system (C.11){(C.14) does not admit any solution simultaneously compensating the deformations of the spin-2 and spin-3 constraints, thus allowing to conclude that corrections of the form (3.4) are necessary in the most general case.15 Spectra of partially reducible theories In this section we analyse the spectra of three Maxwell-like theories on which we impose trace constraints of order higher than one. We wish to show that, as argued in section 5, the lower-spin particles present in the fully reducible theory get truncated away from the spectrum. We shall resort to a light-cone analysis of the components, always assuming to choose a frame s.t. p+ 6= 0. Rank four: elimination of the scalar mode Let us consider rst the case of a rank-four tensor on which we impose a double-trace constraint. The corresponding theory is then characterised as follows: M2 = M + '00 = 0 ; = 0 : @ @ '0 = 0 ; From the divergence of the equations of motion we get and thus, for normalisable modes M2 = = 0 ; @ @ '0 = 0 : The trace of (D.3) allows to set to zero rst @ @ '0 and then, as a consequence, the whole of @ @ '. At this point we can set to zero the divergence of ' by a partial gauge xing, @ ' = 2 @ ' = 0 ; since both @ ' and are transverse, rank-three tensors, thus obtaining the reduced Fierz 2' = 0 ; @ ' = 0 ; '00 = 0 ; 2 = 0 ; = 0 ; that indeed describes two massless particles, one of spin 4 and one of spin 2. 15In some special cases, the deformation can be avoided because the derivatives arrange in such a way to give rise to the equations of motion; for instance, in the spin-2 case the term @@ @ h M may appear, or in the spin-3 case | the term @ @ @ ' M 0. However, this kind of mechanism strictly depends on the vertex under investigation and does not manifests itself in the general case. ve: elimination of the vector mode. A slightly more complicated case obtains by considering a rank- ve tensor subject to a double-trace condition. In this case, since we are aiming to truncate the vector of the Maxwell-like multiplet we should make sure that this is done in a gauge-invariant way and for this reason we need to also constrain the gauge parameter to be doubly traceless: M2 = M + '00 = 0 ; = 0; 00 = 0 : (D + 4)(D + 2) @ @ '0 = 0 ; Now the divergence of the equations of motion is not a pure gradient, and in order to investigate the corresponding cancellations we resort to a light-cone analysis in momentum space, assuming to choose a frame s.t. p+ 6= 0: p M5 = (D + 4)(D + 2) p p p '0 = 0 : The component analysis provides, to begin with, the following outcome: p p '+++ = 0 p p '++i = 0 p p '+ij = p p '+i = 4 := (D+4)(D+2) . In particular, from the relation putting together with (D.10) and (D.11) we obtain p p '0+ = p p '0+ = 0 = 0; p p '+ij = 0 : Continuing the analysis, from which, since p p 'ijk = we obtain, putting together (D.16), (D.15), (D.14) and (D.12) p p '0i = p p '0i = 0 p p 'ijk = 0; p p '+ i = 0 : For the remaining components we get p p p '0 = 0 ; p+ p p p '0 = 0 ; + 2 pp+i p p '0 = 0 ; + 6 pp+ p p '0 = 0 : In the rst two conditions we can substitute Tracing the second equation and solving the system with the rst one obtains p p p '0 = 2p+p p '+ p p '0 = = 0 ; p p 'kk = 0 ; from which all leftover components of the double divergence of ' can be seen to vanish. Altogether one nds = 0 : The gauge dependent part of the divergence, in its turn, can be set to zero by a partial gauge xing as in (D.4), observing that in the present case both @ ' and and doubly traceless. In this way we prove equivalence of (D.6) and (D.5), where now all propagates only the particles of spin 5 and spin 3, with the vector eliminated from the Rank six: elimination of the spin-0 and of the spin-2 modes. As our last example we would like to consider a case where trace constraints are supposed to cut half of the spectrum of the original theory. To this end we impose on a rank-six tensor the following set of conditions: M2 = M + '00 = 0 ; = 0; 00 = 0 : (D + 6)(D + 4) @ @ '0 = 0 ; Once again, the main issue at stake is to show that the equations of motion are strong enough to impose the vanishing of the double divergence. As discussed above, the transverse part of the divergence can be eliminated by a partial gauge xing. Thus, we consider the strongest condition emerging on @ @ ', enforced by the divergence of the equations of p M2 = (D + 6)(D + 4) p p p '0 = 0 : p p '++++ = 0 p p '+++i = 0 4)p p '++ i = p p '++ij = ij ( 2p p '+++ p p '+ijk = (ij ( 2p p 'k)++ where in particular, tracing over the transverse indices in (D.30) and solving together with (D.29) one obtains p p '+++ = 0 and p p '++kk = 0 and, inserting back in (D.30), p p '++ i and p p '+ijk. The following equations form another coupled system whose solution requires to also take the double trace condition into account: = 4'++ = 0: To begin with, one can combine the trace of (D.34) with (D.33) and (D.36) to get p traceless part (in the transverse indices) of p p '+ ij . To check that it vanishes, one has to combine (D.34) with the trace of (D.35) and again (D.36), to eventually recover components. 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Cubic interactions of Maxwell-like higher spins, Journal of High Energy Physics, 2017, DOI: 10.1007/JHEP04(2017)068