Consistent interactions of Curtright fields

Consistent self-interactions of Curtright fields (Lorentz tensors with (2, 1) Young diagram index symmetry) are constructed in dimensions 5 and 7. Most of them modify the gauge transformations of the free theory but the commutator algebra of the deformed gauge transformations remains Abelian in all cases. All of these interactions contain terms cubic in the Curtright fields with four or five derivatives, which are reminiscent of Yang-Mills, Chapline-Manton, Freedman-Townsend and Chern-Simons interactions, respectively.


BRST differential
In our case the master action corresponding to the Lagrangian (1.2) can be taken as wherein S a µν and A a µν denote ghost fields, C a µ denote ghost-for-ghost fields, and T µν a , S µν a , A µν a denote the antifields for T a µν , S a µν and A a µν respectively (the antifields for C a µ are denoted C µ a ). The ghost fields and antifields have the index symmetries The total degree G of a total form ΩG = p ωG−p,p is the sum of the form-degree and the ghost number of its exterior forms ωG−p,p. A total form with total degree G is called a total G-form.

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wherein E a µν and E a µ are traces of a gauge invariant tensor E a µν στ : These tensors fulfill the identities For later purpose we also introduce the totally tracefree part W a µν στ of E a µν στ in dimensions D > 3: We remark that F µν σ , E a µν στ , W a µν στ , E a µν and E a µ are the counterparts of the linearized Levi-Civita-Christoffel connection, Riemann-Christoffel tensor, Weyl tensor, Ricci tensor und curvature scalar of general relativity, respectively. E a µν and E a µ vanish on-shell in the free theory, and E a µν στ equals W a µν στ on-shell in the free theory: wherein T µ a = T µν a ν (2.11) and ≈ denotes equality on-shell in the free theory (sT µν a is the Euler-Lagrange derivative of L (0) with respect to T a µν , i.e. the BRST-transformations sT µν a are the "left hand sides" of the equations of motion of the free theory).

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We arrive at the improved total form (3.4): (3.15) (s + d)Ω aµ D−3 = 0 is thus a direct consequence of (3.9). As Λ aµ D−3 is nontrivial in the cohomology of (s + d), Ω aµ D−3 is also nontrivial in that cohomology. Notice also that the exterior (D − 2)-form ω aµ and ∂ σ T σ b ) by subtracting a total form (s + d)η aµ −2,D−2 from Λ aµ D−3 , and afterwards also the strivial terms in the resultant redefined exterior (D − 1)-form and exterior D-form which however appears to be merely of academic interest and therefore is not done here (the exterior p-forms with p > D − 2 in Ω aµ D−3 anyway do not contribute to the deformations constructed below).
We remark that it is impossible to improve Λ aµ D−3 to an (s + d)-cocycle with a gauge invariant and x-independent exterior (D − 3)-form. Indeed, such an improvement would require the existence of x-independent exterior forms η aµ 0,D−4 and η aµ 1,D−5 that fulfill (3.10) but it can easily be shown that such forms do not exist. The improvement of Λ aµ D−3 thus necessarily depends explicitly on the coordinates x. Furthermore the improvement is crucial for the construction of consistent deformations involving Ω aµ D−3 , as will become clear below.
and Ω aµν D−2 in (3.5) can also be modified by removing terms that vanish on-shell in the free theory. In particular, using equation (2.10) one can write terms of Ω aµν 2 that vanish on-shell in the free theory as sη aµν −1,2 with an exterior 2-form η aµν −1,2 and redefine Ω aµν Furthermore one can write terms of ω aµν 0,D−2 that vanish on-shell in the free theory as sη aµν −1,D−2 with an exterior (D−2)-form η aµν −1,D−2 and redefine Ω aµν

Consistent deformations in first order formulation
To explore whether or not the consistent first order deformations derived in the previous section exist to all orders we employ the first order formulation [9] of the free theory. The classical fields of that formulation are denoted ϕ a µν and B a µν σ whose Lorentz indices have the permutation symmetries We take as Lagrangian of the first order formulation The B-fields are auxiliary fields which can be eliminated using the algebraic solution of their equations of motion. Elimination of the B-fields reproduces the Lagrangian (1.2) (up to a total divergence ∂ µ R µ ) with the definitions 5 The ghost fields of the first order formulation of the free theory are denoted D a µν andĤ a µν = H a [µν ] , the ghost-for-ghost fields again C a µ , and the antifields again with a and indices corresponding to the indices of the respective field. These fields and antifields have the following ghost numbers, antifield numbers, Graßmann parities and BRST transformations JHEP06(2020)012 (corresponding to the master actionŜ We also note that D a µν = S a µν + 3A a µν , i.e. S a µν = D a (µν) and A a µν = 1 3 D a [µν] . We now introduce the following total 1-forms and 2-forms analogously to (3.1): and the following total (D − 3)-forms analogously to (3.4): whereinŴ a ν σµτ is defined analogously to W a ν σµτ in (2.7), withÊ in place of E. The total forms (5.7) and (5. with H aµν and Ω aµν 1 as in (3.1). Hence, in the first order formulation of the free theory the total 1-formΩ aµν are (s + d)-cocycles: e.g., one haŝ This implies indeed that Ω CM 5 , Ω CM 7 , Ω FT 5 and Ω CS 5 are equivalent in H(s + d) toΩ CM 5 , Ω CM 7 ,Ω FT 5 andΩ CS 5 , respectively, and that the defomations of the free theory which arise from these (s + d)-cocycles are equivalent as well, respectively.

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Now, the first order deformationsŜ (1) which arise from the solutionsΩ CM 5 ,Ω CM 7 ,Ω FT 5 andΩ CS 5 of (1.5) fulfill (Ŝ (1) ,Ŝ (1) ) = 0 simply because the exterior D-forms present in these solutions do not depend on the fields ϕ, and the only antifields on which these exterior D-forms depend are the antifields ϕ of ϕ (of course, Ω CS 5 andΩ CS 5 do not depend on antifields at all and therefore it is actually not necessary to substituteΩ CS 5 for Ω CS 5 in order to get (S (1) , S (1) ) = 0 for this deformation by itself; however this changes when one considers linear combinations of Ω CM 5 , Ω FT 5 and Ω CS 5 ). Hence, these first order deformationŝ S (1) provide in fact already a complete deformationŜ =Ŝ (0) + gŜ (1) of the master action S (0) of the first order formulation of the free theory. This implies that the first order deformations arising from the solutions (4.4), (4.5), (4.7) and (4.8) of (1.5) indeed exist to all orders and the complete deformations in the second order formulation of the free theory with Lagrangian (1.2) can be obtained fromŜ by eliminating the auxiliary fields B (e.g., perturbatively). It should also be noticed that this reasoning does not only apply to the Chapline-Manton, Freedman-Townsend and Chern-Simons type solutions in D = 5 individually but also to any linear combination thereof.
The author has not found an analogous line of reasoning for the Yang-Mills type deformations yet. The reason is that it does not appear straightforward to find B-dependent total formsΩ analogous to (5.7) and (5.8) for the Yang-Mills type deformations which allow a reasoning similar to comment (ii) in section 4.

Conclusion
The first order deformations L (1) of the Lagrangian (1.2) that arise from the solutions of (1. Here we assumed that the flat metric has signature (−, +, +, +, +). Other conventions can result in a minus sign in (6.6) and a plus sign in L N in (6.7). We remark that all results presented in this work are actually valid also for non-Minkowskian metrics, with possible reversed signs in (6.6) and in L (1) N in (6.7).

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Notice that the first order deformations (6.1) and (6.3) exist for any number of Curtright fields (and in particular for only one Curtright field), whereas the first order deformations (6.4) and (6.5) require at least two Curtright fields, and the first order deformations (6.2) and (6.6) require at least three Curtright fields because of equations (4.3), (4.6) and (4.9). Furthermore notice that all the above first order deformations are Lorentz invariant, in spite of the explicit x-dependence of the deformations (6.3), (6.4) and (6.5). 8 This explicit x-dependence results from the fact that a gauge invariant improvement of the conserved exterior (D − 3) form λ aµ 0,D−3 in (3.8) necessarily depends explicitly on the coordinates x, cf. remark (i) in section 3. The deformations (6.3), (6.4) and (6.5) are thus Lorentz invariant but appear to be variant under standard spacetime translations. The deformations (6.1), (6.2) and (6.6) are Poincaré invariant.
Notice also that all the above first order deformations are cubic in the Curtright fields and that the deformations (6.1) and (6.2) contain four derivatives of the Curtright fields (terms ∂ 2 T ∂T ∂T ) whereas the deformations (6.3)-(6.6) contain five derivatives of the Curtright fields (terms ∂ 2 T ∂ 2 T ∂T ), respectively. Furthermore, all deformations (6.1)-(6.5) are accompanied by deformations of the gauge transformations of the free theory. The first order deformations of these gauge transformations are obtained from the corresponding solutions of (1.5) given in section 4, more precisely from the terms with antifield number 1 in the exterior D-forms of these solutions. We leave it to the interested reader to write out these deformations of the gauge transformations explicitly. The commutator algebra of the first order deformed gauge transformations remains Abelian in all cases, however. This corresponds to the fact that the exterior D-forms of the solutions of (1.5) given in section 4 do not contain terms with antifield number exceeding 1.
The deformations derived here are thus compatible with the results of [10,11] where it was shown that Poincaré invariant first order consistent deformations of the free theory that modify nontrivially the gauge transformations leave the commutator algebra of the deformed gauge transformations Abelian on-shell, and that there are actually no nontrivial consistent deformations of this type containing at most three derivatives of the Curtright fields. In fact it can easily be shown that x-independent and Lorentz invariant nontrivial consistent deformations that are strictly invariant under the gauge transformations of the free theory and contain at most four derivatives do not exist either. Indeed, according to the results of [10,11] such deformations can be taken to be quadratic in the tensors E aµν στ but all such quadratic terms actually vanish on-shell up to a total divergence because of (2.5)-(2.9) and are thus trivial deformations of the Lagrangian (1.2). Therefore it seems that the above deformations might actually provide the simplest possible Lorentz invariant nontrivial deformations of the free theory in dimensions D = 5 and D = 7 at first order.
As shown in section 5 the above first order deformations (6.3)-(6.6) can in fact be extended to all orders, most readily using the first order formulation of the theory. Fur-JHEP06(2020)012 thermore in D = 5 any linear combination of the deformations (6.3), (6.5) and (6.6) can be extended to all orders. Whether or not the first order deformations (6.1) and (6.2) can be extended to higher orders is left open here.
We also remark that in all above first order deformations the tensors E aµν στ can be replaced by the traceless tensors W aµν στ (2.7) and vice versa because of E aµν στ ≈ W aµν στ , see also remark (iii) in section 3 (such replacements provide equivalent deformations and modify the deformed gauge transformations).
The author admits that he has no complete proof yet that the above deformations are really nontrivial. Therefore some (or all) of these deformations may actually turn out to be trivial. The proof of nontriviality is hampered by the possible explicit x-dependence of the terms (forms) that may make the deformations trivial. The author plans to investigate this issue, and whether or not the first order deformations (6.1) and (6.2) can be extended to higher orders in a future work (unless someone else does the job). 9 However, the similarity of (6.1)-(6.6) to Yang-Mills [12], Chapline-Manton [13], Freedman-Townsend [14] and Chern-Simons [15] interactions, respectively, in combination with some BRST-cohomological considerations, suggests the nontriviality of the deformations.
Let me therefore briefly comment on similarities (and differences) of the deformations (6.1)-(6.6) to Yang-Mills, Chapline-Manton, Freedman-Townsend and Chern-Simons interactions. To that end standard p-form gauge potentials are denoted A a p = 1 p! A a µ 1 ...µp dx µ 1 . . . dx µp , the corresponding field strength (p + 1)-forms F a p+1 = dA a p and the Hodge dualsF a D−p−1 of the field strength forms.

Yang-Mills interactions in
This is analogous to (4.1) and (4.2) with Ω a··· D−2 corresponding toF a D−2 , and Ω b··· 1 and Ω c··· 1 corresponding to A b 1 and A c 1 , respectively. I stress that the terminology "Yang-Mills type interactions" used in the present work only relates to this structure of the interactions and not to the commutator algebra of the deformed gauge transformations (i.e. it is not related to the question whether or not this algebra is Abelian).
Cubic Chapline-Manton interactions in D dimensions with two 1-form gauge fields arē F a D−3 F b 2 A c 1 e abc . This is analogous to (4.4) and (4.5) with Ω a· D−3 corresponding toF a D−3 , Ω b··· 2 corresponding to F b 2 , and Ω c··· 1 corresponding to A c 1 . Cubic Freedman-Townsend interactions in 5 dimensions areF a 1F b 1 A c 3 d abc . This is analogous to (4.7) with Ω a· 2 and Ω b· 2 corresponding toF a 1 andF b 1 , and Ω c··· 1 corresponding to A c 3 . The correspondence here does not match the form-degrees and total degrees but concerns the structureFF A.
Cubic Chern-Simons interactions in 5 dimensions are A a 1 F b 2 F c 2 c abc . This is analogous to (4.8) with Ω a··· 1 corresponding to A a 1 , and Ω b··· 2 and Ω c··· 2 corresponding to F b 2 and F c 2 . The difference of the deformations (6.1)-(6.6) as compared to standard Yang-Mills, Chapline-Manton, Freedman-Townsend and Chern-Simons interactions results on the one 9 In particular there seems to be no obvious counterpart of the Yang-Mills type self-interactions (6.1) and (6.2) in previous works, such as [19] which aimed to classify cubic consistent interactions of "mixed symmetry" and higher spin fields rather completely. Therefore it seems to be worthwhile to check whether or not especially the self-interactions (6.1) and (6.2) are trivial, and if they turn out to be nontrivial, to clarify their relation to results of previous works.

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hand from the additional Lorentz indices of the Ω's as compared to standard p-form gauge potentials A p and, on the other hand, from the fact that the action L (0) d D x does not correspond to the standard Maxwell type action for free p-form gauge potentials A p containing terms F p+1FD−p−1 .
As far as the author knows the self-interactions of Curtright fields obtained in this paper have not been disclosed anywhere else in the literature so far. Nevertheless, self-interactions of "mixed symmetry gauge fields" similar to the Chapline-Manton type interactions (6.3) and (6.4) have been found in [11]. They are disclosed under item (iv) in section 8.1 of the arXiv-version of [11]. The self-interactions disclosed there also depend explicitly on the coordinates x and have a structure analogous to the Chapline-Manton type interactions (6.3) and (6.4). In the particular case (p, q) = (2, 1) (corresponding to a Curtright field) and s = 1 (using the notation of [11]) the interactions given there will very likely in D = 5 provide a self-interaction of a Curtright field equivalent to the Chapline-Manton type interaction (6.3) (for one Curtright field) when the Lorentz structure of the fields is taken into account. 10 Let me finally remark that it is quite straightforward to construct interactions of Curtright fields with other fields in appropriate dimensions similar to the above self-interactions using the approach of the present paper. For instance, similarly to equation (4.7) one easily constructs solutions Ω N 5 of equation (1.5) in D = 5 which provide first order deformations L (1) N of the Lagrangian from the total (D − 3)-forms (3.4) for D = 5 and the total 1-form Ω 1 = C + A µ dx µ which is the sum of a standard Abelian 1-form gauge potential A µ dx µ and the corresponding ghost field C: D = 5 : Ω N 5 = Ω aµ 2 Ω b 2µ Ω 1 g ab , Ω 1 = C + A µ dx µ , L N = −A µ j µ , j µ = µν 1 ν 2 1 2W a ν 1 ν 2 σW b 1 2 σ g ab ,W a ν 1 ν 2 σ = ν 1 ...ν 5 W aν 3 ν 4 ν 5 σ x (6.7) wherein g ab = g ba are constant symmetric coefficients and L N is a Noether coupling of the gauge field A µ and an ("improved") Noether current j µ of the free theory (∂ µ j µ ≈ 0). Analogously one constructs in D = 5 Chern-Simons type interactions of Curtright fields and a standard Abelian 1-form gauge potential from the solution Ω aµν 2 Ω b 2µν Ω 1 k ab of (1.5) wherein k ab = k ba are constant symmetric coefficients and Ω a··· 2 and Ω b··· 2 are the 2-forms of (3.1). Cubic interactions ∂T ∂T ∂ 2 h of a Curtright field T with a symmetric 2-tensor field h µν = h νµ representing the metric field of linearized general relativity were obtained in section 5 of [16] (see equation (5.14) there). These interactions are reminiscent of the Yang-Mills type self-interactions (6.1) and (6.2) and may be constructible analogously to (4.1) and (4.2) using a total curvature (D − 2)-form for the h-field in place of Ω aµ 1 µ 2 µ 3 D−2 . This indicates that the approach used here may also be useful for the construction of consistent interactions of other "mixed symmetry" or higher spin fields. Cubic interactions of various fields of that type in various dimensions, both in flat space and in anti-de Sitter space,