Interactions of Multiple Spin-2 Fields Beyond Pairwise Couplings

PHYSICAL REVIEW LETTERS 122, 251101 (2019)

Interactions of Multiple Spin-2 Fields beyond Pairwise Couplings

S. F. Hassan1 and Angnis Schmidt-May2 1Department of Physics and The Oskar Klein Centre, Stockholm University, AlbaNova University Centre, SE-106 91 Stockholm, Sweden 2Max-Planck-Institut für Physik (Werner-Heisenberg-Institut), Föhringer Ring 6, 80805 Munich, Germany (Received 21 July 2018; revised manuscript received 16 April 2019; published 26 June 2019)

Thus far, all known ghost-free interactions of multiple spin-2 fields have involved at most pairwise couplings of the fields, which are direct generalizations of bimetric interactions. We present a class of spin-2 theories with genuine multifield interactions and explicitly demonstrate the absence of ghost instabilities. The construction involves integrating out a nondynamical field in a theory of spin-2 fields with only pairwise ghost-free interactions. The new multivierbein interactions generated are not always expressible in terms of the associated metrics.

DOI: 10.1103/PhysRevLett.122.251101

Introduction.—Interacting theories for multiple fields theory by simply adding copies of the bimetric potentials 1 2 ð Þ with spin 0, = , and 1 are well understood and realized Vbi g; f for pairs of metrics, but without forming loops I in nature via the standard model of particle physics, where [10]. An example for four metrics gμν would be ð 1 2Þþ ð 1 3Þþ ð 3 4Þ the multiplets and their mixings are crucial for the viability Vbi g ;g Vbi g ;g Vbi g ;g . So far, these pair- of the theory. In contrast, general relativity is the simplest wise couplings have been the only known ghost-free possible interacting theory of a single spin-2 field. It is a interactions of multiple spin-2 fields. fundamental question as to whether, in analogy with lower An important class of multiple spin-2 theories was spins, consistent theories of multiple spin-2 fields could constructed in Ref. [11] using antisymmetrized products exist. Such theories could have profound implications for of the vierbein fields which appeared to be ghost-free. the understanding of the gravitational force beyond general However, a more detailed analysis in Ref. [12] revealed that relativity, but they have not been easy to construct. generically such multivierbein interactions contained In a covariant setup, spin-2 fields have more components ghosts. It was argued that the ghost-free subset consisted than physically needed, and generic theories do not have only of models where the vierbeins could be traded off for enough symmetries and constraints to remove the unphys- the metrics by virtue of a vierbein symmetrization con- ical components. Some of these, if not eliminated, give rise dition, exactly as in bimetric theory. The only known to ghost instabilities, with an example being the Boulware- models of this type are the pairwise interactions mentioned Deser ghost [1] of a massive spin-2 field. A few years ago above and described in more detail in Ref. [13]. the ghost-free theory of two interacting spin-2 fields was Reference [14] reached a similar conclusion for multiple found [2], which also fulfills some other important con- spin-2 theories in three dimensions. sistency criteria [3,4]. This generalized previous work on a The question is whether there exist interactions between single massive spin-2 field in a fixed background [5–8]. multiple spin-2 fields beyond the pairwise ones, and thus The model is formulated in terms of two symmetric rank-2 beyond the class conjectured in Ref. [12].Hereweshowthat tensors (or “metrics”) gμν and fμν interacting through a this is indeed the case. These are genuine multiple spin-2 ð Þ— “ specific potential Vbi g; f hence the name ghost-free interactions in the sense that, similar to the gluon vertices in bimetric theory.” For a recent review see Ref. [9]. quantum chromodynamics, they lead to direct interactions Theories for more than two spin-2 fields are also strongly among more than two dynamical vierbein fields and cannot be restricted by the absence of ghosts. From the analysis of reduced to pairwise interactions through field redefinitions. Ref. [2], it is easy to see that certain ghost-free theories can Summary of results: In this work we derive a class of be constructed as straightforward extensions of bimetric ghost-free interactions for multiple spin-2 fields by integrating out a nondynamical field in a theory with ghost-free bimetric interactions. The result is an interaction term for A N vierbeins ðeIÞμ of the form Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Z N Further distribution of this work must maintain attribution to X ¼ − 4 4 βI ð Þ the author(s) and the published article’s title, journal citation, Smulti M d x det eI ; 1 and DOI. Funded by SCOAP3. I¼1

0031-9007=19=122(25)=251101(6) 251101-1 Published by the American Physical Society PHYSICAL REVIEW LETTERS 122, 251101 (2019) involving a mass scale M and arbitrary dimensionless These allow us to expresspffiffiffiffiffiffiffiffiffiffiffiffi the potential in terms of metrics coefficients βI, I ¼ 1; …; N . The kinetic terms of the −1 ¼ −1 ð Þ¼ pbyffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi replacing u eI f0 fI in addition to det u vierbein fields have the standard Einstein-Hilbert form. j det f0j. It is then straightforward to verify that the For restricted vierbein configurations the multivierbein arguments for the absence of a ghost in bimetric theory [2] vertex can be expressed in terms of metrics, but this is extend to this case. not always possible. The interactions can involve up to four Multiple spin-2 action: Let us set the coupling m0 ¼ 0 to different vierbeins in each term and are therefore more A freeze the kinematics of the vierbein uμ . We can then general than the pairwise couplings known so far. A eliminate uμ algebraically using its equation of motion to Generating new interactions.—The starting point is a A A obtain an action for the remaining N vierbeins (note that theory for ðN þ 1Þ vierbeins, uμ and ðe Þ , I ¼ 1; …; N , I μ “ ” → ∞ with ghost-free bimetric interactions. Let us denote the massive gravity limit of m0 gives a similar A 0 A B I starting action, but with a fixed background uμ which the corresponding metrics by fμν ¼ uμ ηABuν and fμν ¼ A B cannot be varied and eliminated; see Ref. [16]). Indeed, ðeIÞμ ηABðeIÞν . The action has the following structure: varying the action (2) for m0 ¼ 0 with respect to the inverse μ XN vierbein uA gives an equation with the unique solution S½u; e ¼ S ½fJþS ½u; e : ð2Þ I EH int I N ¼0 3 X J A A uμ ¼ − ðeIÞμ : ð7Þ β0 It includes the Einstein-Hilbert kinetic terms I¼1 Z A pffiffiffiffiffi Using this to eliminate uμ in Eq. (4) gives the new ½ J¼ 2 4 J ð JÞ ð Þ SEH f mJ d x f R f ; 3 interactions for the remaining N vierbeins, Z ðN þ 1Þ ½ ¼− 4 4 ð þ þþ Þ ð Þ where mJ are the Planck masses. Sint contains the Sint eI M d x det e1 e2 eN ; 8 A simplest ghost-free bimetric interactions between uμ and A each one of the ðeIÞμ , 4 4 −3 with M ≡ −54m β0 . These are not reducible to linear combinations of the pairwise bimetric potentials. That the Z XN 4 4 I −1 starting action (2) was ghost-free does not by itself S ½u; e ¼−2m d x det u β0 þ β Trðu e Þ : int I I guarantee the absence of ghosts in the final theory since I¼1 the procedure changes the number of dynamical and ð Þ 4 nondynamical fields. However, we will explicitly confirm

−1 μ the absence of ghosts below. ð Þ¼ ð ÞA A In addition to the traces Tr u eI uA eI μ , we could also The N symmetrization constraints (6), with uμ given by include the remaining ghost-free bimetric interactions, but Eq. (7), provide a set of constraint equations for the new we have chosen not to do so. For brevity, we set βI ¼ 1 for multiple spin-2 theory. Equivalently, one can derive these I I ¼ 1; …; N ,byscalingeI → eI=β and redefining the from the multiple spin-2 action (8) using Eq. (5). In matrix Planck masses mI in Eq. (3) accordingly. notation these new Lorentz constraints are Lorentz constraints: Each vierbein contains six Lorentz parameters that drop out of the corresponding metric and XN XN Tη ¼ Tη ¼ 1 … N ð Þ hence appear in the action (2) only through the potential eJ eI eI eJ;J; ; : 9 I¼1 I¼1 terms in Sint. Since these are nondynamical, their equations of motion are constraints. Specifically, these are the Note that if we sum Eq. (9) over J, the resulting equation is antisymmetric parts of the equations of motion for ðe ÞA I μ identically satisfied. Hence, there are N − 1 independent [11,15], precisely six equations per vierbein, matrix relations among the antisymmetric parts of the ð ÞAη ð ÞB δ δ combinations eJ μ AB eI ν . These can be used to elimi- Sint AB ν Sint AB μ η ðeIÞ − η ðeIÞ ¼ 0: ð5Þ nate 6ðN − 1Þ nondynamical components of the vierbeins. δðe ÞA B δðe ÞA B I μ I ν Another six components are removed by the overall

A Lorentz invariance of the action. This leaves us with The corresponding equation for uμ is a linear combination 16N − 6N ¼ 10N independent components, which is of Eq. (5) due to the overall Lorentz invariance of the the same as the number of components in N symmetric action. For the potential in Eq. (4), the Lorentz constraints rank-2 tensor fields. uniquely imply the following symmetrization conditions, Direct ghost proof.—In the following we will use 3 þ 1 metric variables to demonstrate the existence of the con- Aη ð ÞB ¼ Aη ð ÞB ¼ 1 … N ð Þ uμ AB eI ν uν AB eI μ;I; ; : 6 straints that remove the Boulware-Deser ghosts from the

251101-2 PHYSICAL REVIEW LETTERS 122, 251101 (2019) physical spectrum. This choice of variables is convenient linearity. The Einstein-Hilbert terms are also linear in these for isolating the nondynamical fields in the Einstein-Hilbert variables; hence, the entire action has the form actions, but their use requires some justification. In general, N Lorentzian metrics may not admit compatible notions of XN Z 3 þ 1 ¼ 4 ½ðπ Þi ð _ Þa − C − Ci ð Þ space and time and compatible decompositions. S d x I a EI i NI I NIi I : 12 ¼1 However, the vierbeins eI are not fully independent since I the parent theory gives rise to constraints (6). These, as 3 þ 1 ðπ Þi shown in Ref. [3], ensure that simultaneous decom- Here, I a are the canonical momenta conjugate to the ð Þ ð Þa positions exist for the pair u; eI for each I. Geometrically, EI i , and all other fields are nondynamical. The functions C Ci π a the null cone of each eI shares some common timelike and I and I contain the EI, the I, and the Lorentz fields, vI spacelike directions with the null cone of u [3]. Therefore, and ΩI, but not the lapses and shifts. We note that thus far in with further mild restrictions on the ranges of metric this section the considerations, including the form of the variables, there always exist large classes of configurations action in Eq. (12), are not restricted to the model (8) but where all eI share a common spatial hypersuface with u also apply to the general class of multivielbein theories and, hence, admit simultaneous 3 þ 1 decompositions. In introduced in Ref. [11]. However, the difference is crucial the new multivierbein theory, the null cones are further for the argument that follows. correlated by Eq. (7) for u. The ghost proof below assumes ð Þa The dynamical fields EI i contain the ghost modes. such configurations admitting simultaneous 3 þ 1 decom- These must be eliminated by the constraints that arise from positions. It may be possible to further generalize it to a the equations of motion for the NI and NiI, that is, CI ¼ 0 covariant analysis along the lines of Refs. [17,18]. and Ci ¼ 0, as well as from Eq. (5) for the Lorentz fields, A I We decompose a vierbein eμ into a gauge-fixed vierbein and the gauge-fixing conditions for general covariance. In A ΛA Eμ , rotated by a Lorentz transformation B, particular it is necessary that, after solving for the Lorentz fields, CI ¼ 0 become equations for the EI and πI alone, A ¼ ΛA B ð Þ eμ BEμ ; 10 remaining independent of the lapses. Then they can eliminate the ghost fields in favor of the remaining and use the following 3 þ 1 parametrization [11,12,19]: dynamical variables. The difficulty is that for the general multivierbein interactions of Ref. [11] which also have the N 0 A form of Eq. (12), the Lorentz constraints (5) render the Eμ ¼ ; a j a Ω C Ej N Ei spatial rotations I, and hence the I, dependent on the NI [12]. Then CI ¼ 0 can be solved for the NI rather than γ vc 10 ΛA ¼ : ð11Þ eliminating the ghost fields which will remain in the B a δa þ 1 a 0 ðΩÞc v c 1þγ v vc b spectrum. We now show that the multivierbein interaction in Eq. (8) circumvents this problem. i a Here N is the lapse, N is the shift vector, and Ei is a gauge- First, ðN − 1Þ combinations of the N vector equations fixed spatial vierbein with six independent components. Ci ¼ 0 ðN − 1Þ I can be used to determine the boost vectors The Lorentz transformation has been further decomposed a að π ΩÞ Ci Ci vI as vI E; ; . The unused combination of the I, say , into spatial rotations parametrized by an SOð3Þ matrix Ω are the three constraints of the spatial diffeomorphisms. To (containing three parameters), and Lorentz boosts para- determine the rotation matrices ΩI, consider the Lorentz metrized by thepffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi three-dimensional (rescaled) boost vector constraints that, for the model (8), give the ðN − 1Þ a γ ≡ 1 þ a v , with vav . independent matrix equations in Eq. (9). Their spatial parts ðN − 1Þ ð ÞA In the action, we write of the vierbeins eI μ are the 3ðN − 1Þ equations using the 3 þ 1 parametrization given above. The overall Lorentz invariance of the action allows us to take the last XN vierbein to be of the same form but with va ¼ 0 and Ω ¼ 1. ð ÞAη ð ÞB ¼ 0 ¼ 1 … N − 1 ð Þ eI ½i AB eJ j ;J; ; : 13 Since the Lorentz parameters do not show up in the kinetic I¼1 terms, they appear without derivatives in the action. It was shown in Ref. [20] that the Lorentz constraints (5) are ð ÞA Crucially, the eI i do not contain the lapses or the shifts equivalent to the equations of motion of the Lorentz since, from Eq. (10), it follows that parameters. It is easy to show that the potential (8) is linear in all Ωb c i vb cEi lapses NI and shifts NI before integrating out the Lorentz eA ¼ : ð14Þ a Ω ð Þ i ðδa þ 1 a ÞΩb c fields vI and I: The potential is det u , where the matrix u b 1þγ v vb cEi i is given by Eq. (7). The NI and NI appear only in the first A column u0 . Since detðuÞ is antisymmetric in the columns of Therefore, the constraints (13), along with the solutions for A— a 3ðN − 1Þ u, each term in it contains only one factor of u0 hence the vI , determine the parameters of the rotation

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N matrices as ΩIðE; πÞ. This insures that CI depend only on X ¼ 0 ¼ 1 … N ð Þ the EI and πI, as desired. AIJ ;I; ; : 17 Finally, the 0i components of the Lorentz constraints J¼1 provide 3ðN − 1Þ equations linear in the lapses and shifts. Hence, the antisymmetric parts drop out of Eq. (15).For These determine ðN − 1Þ of the shift vectors Ni as linear I any pair of vierbeins, if we set A ¼ 0, then one could functions of the N and of the N th shift vector, say, Ni . IJ I N replace e−1e ¼ðg−1g Þ−1=2, where g ¼ eTηe . Then a The linearity ensures that on using these solutions, the I J I J I I I i covariant formulation in terms of the metrics would be action is linear in NI and N . At this stage, the only other N possible if A1 ¼ 0 for all J. But the constraints (17) that π J variables in the action are the EI and the I. arise in the theory are weaker. Hence, although the A drop C IJ Of the remaining constraints, one combination of the I, out of the action, the theory has no equivalent metric C Ci say, , together with the unused , form a set of four first formulation. However, metric formulations exist under class constraints associated with general covariance, as in mild restrictions. N − 1 C Ref. [21]. The remaining combinations of the I are Metric formulation for three fields: We now consider _ second class constraints. Their preservation in time, CI ¼ 0, metric representations for the case N ¼ 3, following an gives another set of N − 1 constraints. Although we have analysis carried out in Ref. [20]. Let us introduce three not proven this here, their existence can be argued in the ð ¯ ÞA Lorentz matrices LI B as Stückelberg fields for the three parent theory (2), where the calculations are very similar to A vierbeins ðeIÞμ , that is, we write every vierbein as a Lorentz the bimetric case explicitly analyzed in Ref. [21]. A rotation of a gauge-fixed vierbein ðe¯ Þμ , The degree of freedom count is now easy. The fields I ðE Þa and the momenta ðπ Þi contain 2 × 6N ¼ 12N I i I a ðe ÞA ¼ðL¯ ÞAðe¯ ÞB: ð18Þ phase space variables, including the N ghost fields and I μ I B I μ their N conjugate momenta. The N − 1 second class Defining ðL Þν ¼ðe¯ Þν ðL¯ ÞCðe¯ ÞB, we can also write constraints and their associated time preservation condi- I μ I C I B I μ tions eliminate 2ðN − 1Þ ghost fields and ghost momenta. ð ÞA ¼ð¯ ÞAð Þν ð Þ The four first class constraints and the associated symmetry eI μ eI ν LI μ: 19 eliminate eight phase space variables, including the last pair of ghost variables, just as in general relativity. The physical Let us choose the gauge-fixed vierbeins such that phase space thus consists of 12N − ð2N þ 6Þ¼10N − 6 ð¯ ÞAη ð¯ ÞB ¼ð¯ ÞAη ð¯ ÞB ð Þ variables, corresponding to one massless field (with four eI μ AB eJ ν eI ν AB eJ μ 20 phase space modes) and ðN − 1Þ massive fields of spin-2 ¼ 1 (each with ten phase space modes). for I;J , 2, 3. General vierbeins may not be gauge fixed Existence of metric formulations.—Ghost-free multivier- in this way [22]. Such gauge fixing is possible only for bein theories with only pairwise interactions can be restricted configurations for which the null cones associ- expressed in terms of the corresponding metrics, although ated with the I and J vierbein intersect [3]. Note that this is some extra information contained in the vierbeins is a stronger requirement than the existence of simultaneous 3 þ 1 retained (corresponding to the choices of square-root decompositions assumed for the ghost analysis. Now matrices that appear in the theory). Here we explore the the Lorentz constraints (9) provide the following two existence of a similar metric formulation for the non- independent sets of equations for the LI which determine pairwise interaction in Eq. (8). We show that the additional two of the three Lorentz matrices: vierbein information cannot be encoded as simple require- T ¯Tη¯ − ð1 ↔ 2Þ¼ T ¯Tη¯ − ð1 ↔ 3Þ ments on familiar matrix functions unless additional con- L1 e1 e2L2 L3 e3 e1L1 ; T T T T straints are imposed. L2 e¯2 ηe¯3L3 − ð2 ↔ 3Þ¼L3 e¯3 ηe¯1L1 − ð1 ↔ 3Þ: ð21Þ By extracting a factor of det e1 and using gμν ≡ ð ÞAη ð ÞB −1T −1 e1 μ AB e1 ν , the interaction becomes Multiplying the first by L1 from the left and L1 from the right and the second by L−1T from the left and L−1 from the Z XN 3 3 pffiffiffi −1 ¼ S ¼ −M4 d4x g det 1 þ g−1 eTηe : ð15Þ right, it is obvious that the covariant solution is L1L2 int 1 J −1 ¼ −1 ¼ 1 ¼ ¼ J¼2 L1L3 L3L2 , which implies L1 L2 L3. Then the multiple spin-2 potential becomes Defining the antisymmetric matrices AIJ as 4 1 V ¼ M det ðe¯1 þ e¯2 þ e¯3Þð22Þ ≡ ð Tη − Tη Þ ð Þ AIJ 2 eI eJ eJ eI ; 16 P since the undetermined overall Lorentz matrix drops out the action depends on the linear combination J¼1A1J. due to Lorentz invariance. We can extract a factor of det e1 Also, in terms of these, the Lorentz constraints (9) read to obtain

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4 −1 −1 N V ¼ M detðe1Þ det ð1 þ e¯1 e¯2 þ e¯1 e¯3Þ: ð23Þ X βIJKLϵ A ∧ B ∧ C ∧ D ð Þ ABCDeI eJ eK eL 27 I;J;K;L¼1 Let us introduce the metrics gμν, fμν, and hμν, for the vierbeins e¯1, e¯2, and e¯3, respectively. Because of the symmetrization constraint (20),wehave with βIJKL ¼ βIβJβKβL. Such interactions with general qffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffi βIJKL were proposed in Ref. [11], where the N ¼ 2 case −1 −1 −1 −1 e¯1 e¯2 ¼ g f; e¯1 e¯3 ¼ g h; ð24Þ reproduces the bimetric theory [2]. However, the only ghost- free cases known thus far are the pairwise bimetric interactions [12]. Given the interactions in Eq. (27) with arbitrary Then the multiple spin-2 potential can be written in terms of βIJKL, a direct analysis that identifies the ghost-free cases is metrics as thus far not known. But the construction presented here qffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffi pffiffiffi generates a class of genuinely multivierbein ghost-free V ¼ M4 g det ð1 þ g−1f þ g−1hÞ: ð25Þ interactions with up to four different vierbeins in one vertex. This easily extends to D space-time dimensions, in which case the vertices would contain up to D different fields. Note We could have chosen to extract the determinant of a that these vertices do not contain closed loops of purely vierbein other than e1 and correspondingly obtain pairwise interactions, which are not ghost-free [12],even pffiffiffi qffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffi though they may appear ghost-free in a certain decoupling V ¼ M4 f det 1 þ f−1g þ f−1h limit [23]. pffiffiffi qffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffi The class of theories obtained here can be further ¼ M4 h det 1 þ h−1g þ h−1f : ð26Þ generalized. First, note that the interactions (1) cannot simply be added to a theory involving the previously known ghost- a free interactions of the N vierbeins ðeIÞμ. Such a setup would A Since the parent theory with nondynamical uμ possesses a correspond to forbidden loop couplings in the parent theory a metric formulation, we also could have tried to integrate out with vierbein uμ. However, compatible interactions can be 0 ¼ Aη B the metric fμν uμ ABuν directly. However, to do this in a constructed by extending the parent theory by additional way consistent with the vierbein conditions, we would have allowed pairwise couplings, as will be discussed in Ref. [24]. had to assume the existence of two additional square-root The new terms couple any of the vierbeins eI to an additional 0 matrices, just as in the above considerations. set of N vierbeins vK and include the following: (1) One of N 3 The analysis partially generalizes to > . Now the the eI can interact with the vK through the standard pairwise gauge choices (20) cannot be made for all vierbeins since interactions;P thatP is, the potential can have the form N ðN − 1Þ 2 ∼ ð Þþ ð Þ these are = conditions, while there are at most V det IeI KVbi e1;vK .(2)OneoftheeI can N Lorentz transformations to implement the symmetriza- interact with the vK through a determinantP interaction;P i.e., ∼ ð Þþ ð þ Þ tions. The remaining conditions may be imposed by hand the potential could be V det IeI det e1 KvK . on the e¯I, but this is not necessary. To find a metric It is straightforward to introduce a standard coupling to a formulation similar to, say, Eq. (25), it is enough to impose matter via any of the dynamical vierbeins ðeIÞμ, as this will ¯ A ¯ B Eq. (20) only on the ðe1Þμ ηABðeJÞν , which gives N − 1 not influence the procedure of integrating out the non- a a gauge conditions. Then, with appropriate restrictions and dynamical vierbein uμ. On the other hand, if uμ couples to ð Þa on choosing a similar solution for the LI, one obtains an matter, then the multiple spin-2 theory for the eI μ will expression which is a direct generalization of Eq. (25) to N have matter interactions that are heavily modified [25].At metrics. low energies these reduce to the matter coupling suggested Finally, it is important to note that, while the vierbeins in Refs. [26,27]. were restricted by hand to obtain a metric formulation, such The interactions for ðN þ 1Þ vierbeins that we started restrictions are inbuilt in the final multimetric theory. with in Eq. (4) were not of the most general ghost-free Hence, the resulting multimetric theories can be considered form. It would be interesting to extend our setup to more in their own right, independent of the starting vierbein general interactions and obtain possibly ghost-free multiple formulations. spin-2 theories that are different from the one studied here. Discussion.—To summarize, we have constructed non- It is not obvious that the algebraic equations for the trivial interactions of multiple spin-2 fields, beyond the nondynamical vierbein can be solved covariantly for more known pairwise potentials, and have demonstrated the general parameter choices in the action. In any case, if the existence of constraints that eliminate the extra ghost modes. linear relation in Eq. (7) is lost, the resulting interactions A The interactions are given in terms of the N vierbeins ðeIÞμ in may not have the general form in Eq. (27). It is important to Eq. (1). On expressing the determinant in terms of the wedge find out if such ghost-free multivierbein interactions, A products of the one-forms eI , one gets beyond the classes discussed here, could exist.

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Our couplings are the first instance of ghost-free spin-2 [13] O. Baldacchino and A. Schmidt-May, Structures interactions where the vierbein formulation admits more in multiple spin-2 interactions, J. Phys. A 50, 175401 general configurations than the associated, more restrictive (2017). metric formulation. Hence, it is interesting to directly [14] H. R. Afshar, E. A. Bergshoeff, and W. Merbis, Interacting investigate the associated multimetric theories, without spin-2 fields in three dimensions, J. High Energy Phys. 01 (2015) 040. recourse to the vierbein formulation. One expects that [15] B. Zumino, Effective Lagrangians and broken symmetries, the extra restrictions on the metrics, already encoded in the in Lectures on Elementary Particles and Quantum Field multimetric interactions [3], would lead to better causal Theory, Vol. 2, edited by S. Deser (MIT Press, Cambridge, properties for the theory. MA, 1970), pp. 437–500. [16] S. F. Hassan, A. Schmidt-May, and M. von Strauss, The work of A. S. M. is supported by a grant from the Particular solutions in bimetric theory and their implica- Max Planck Society. S. F. H. acknowledges support from tions, Int. J. Mod. Phys. D 23, 1443002 (2014). the Swedish Research Council. [17] C. Deffayet, J. Mourad, and G. Zahariade, Covariant constraints in ghost free massive gravity, J. Cosmol. Astropart. Phys. 01 (2013) 032. [1] D. G. Boulware and S. Deser, Can gravitation have a finite [18] L. Bernard, C. Deffayet, A. Schmidt-May, and M. von range?, Phys. Rev. D 6, 3368 (1972). Strauss, Linear spin-2 fields in most general backgrounds, [2] S. F. Hassan and R. A. Rosen, Bimetric gravity from ghost- Phys. Rev. D 93, 084020 (2016). free massive gravity, J. High Energy Phys. 02 (2012) 126. [19] S. F. Hassan, M. Kocic, and A. Schmidt-May, Absence of [3] S. F. Hassan and M. Kocic, On the local structure of ghost in a new bimetric-matter coupling, arXiv:1409.1909. spacetime in ghost-free bimetric theory and massive gravity, [20] S. F. Hassan, A. Schmidt-May, and M. von Strauss, Metric J. High Energy Phys. 05 (2018) 099. formulation of ghost-free multivielbein theory, arXiv: [4] M. Kocic, Causal propagation of constraints in bimetric 1204.5202. relativity in standard 3 þ 1 form, arXiv:1804.03659. [21] S. F. Hassan and A. Lundkvist, Analysis of constraints and [5] C. de Rham, G. Gabadadze, and A. J. Tolley, Resummation their algebra in bimetric theory, J. High Energy Phys. 08 of Massive Gravity, Phys. Rev. Lett. 106, 231101 (2011). (2018) 182. [6] S. F. Hassan and R. A. Rosen, On non-linear actions for [22] C. Deffayet, J. Mourad, and G. Zahariade, A note on “ ” massive gravity, J. High Energy Phys. 07 (2011) 009. symmetric vielbeins in bimetric, massive, perturbative [7] S. F. Hassan and R. A. Rosen, Resolving the Ghost Problem and non perturbative gravities, J. High Energy Phys. 03 in Nonlinear Massive Gravity, Phys. Rev. Lett. 108, 041101 (2013) 086. (2012). [23] J. H. C. Scargill, J. Noller, and P. G. Ferreira, Cycles of [8] S. F. Hassan, R. A. Rosen, and A. Schmidt-May, Ghost-free interactions in multi-gravity theories, J. High Energy Phys. massive gravity with a general reference metric, J. High 12 (2014) 160. Energy Phys. 02 (2012) 026. [24] S. F. Hassan and A. Schmidt-May (to be published). [9] A. Schmidt-May and M. von Strauss, Recent developments [25] M. Lüben and A. Schmidt-May, Ghost-free completion of in bimetric theory, J. Phys. A 49, 183001 (2016). an effective matter coupling in bimetric theory, Fortschr. [10] K. Nomura and J. Soda, When is multimetric gravity ghost- Phys. 66, 1800031 (2018). free?, Phys. Rev. D 86, 084052 (2012). [26] J. Noller and S. Melville, The coupling to matter in massive, [11] K. Hinterbichler and R. A. Rosen, Interacting spin-2 fields, bi- and multi-gravity, J. Cosmol. Astropart. Phys. 01 (2015) J. High Energy Phys. 07 (2012) 047. 003. [12] C. de Rham and A. J. Tolley, Vielbein to the rescue? [27] K. Hinterbichler and R. A. Rosen, Note on ghost-free matter Breaking the symmetric vielbein condition in massive couplings in massive gravity and multigravity, Phys. Rev. D gravity and multigravity, Phys. Rev. D 92, 024024 (2015). 92, 024030 (2015).

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