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, 7 and 9. 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.


Introduction
This work concerns consistent interactions of Curtright fields [1]. Curtright fields are Lorentz tensors T a µν̺ with Lorentz indices µ,ν,̺ having the permutation symmetries 1 T a µν̺ = −T a νµ̺ , T a [µν̺] = 0. (1.1) The additional index a is no Lorentz index but only enumerates the Curtright fields, i.e. we examine also models with more than one Curtright field. The Lagrangian that we use for free (non-interacting) Curtright fields is wherein F a µν̺σ = ∂ µ T a ν̺σ + ∂ ν T a ̺µσ + ∂ ̺ T a µνσ , F a µν = F a µν̺ ̺ (1. 3) and Lorentz indices are lowered and raised with a flat metric η µν and its inverse η µν . Curtright fields are particularly interesting in D = 5 dimensions because there a Curtright field is the elementary field (counterpart of the metric field) in a dual formulation of linearized general relativity [2,3].
(1.4) implies descent equations sω 1,D−1 + dω 2,D−2 = 0, sω 2,D−2 + dω 3,D−3 = 0 etc. with increasing ghost number and decreasing form-degree that can be compactly written as (see section 9 of [7] and section 3 of [8] for reviews) wherein Ω D is a "total form" with "total degree" 2 D, and m is some form-degree at which the descent equations terminate (the value of m varies from case to case).

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 ⋆µν̺ a , S ⋆µν a , A ⋆µν a denote the antifields for T a µν̺ , 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 fields, antifields, spacetime coordinates x µ and differentials dx µ have the following ghost numbers (gh), antifield numbers (af), Graßmann parities (| |) and BRST transformations (s): µν̺ and E a µ are traces of a gauge invariant tensor E a µν̺στ : These tensors fulfill the identities For later purpose we also introduce the totally tracefree part W a µν̺στ of E a µν̺στ in dimensions D > 3: We remark that F µν̺σ , E a µν̺στ , W a µν̺στ , E a µν̺ 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 µν̺ and E a µ vanish on-shell in the free theory, and E a µν̺στ equals W a µν̺στ 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 ⋆µν̺ a is the Euler-Lagrange derivative of L (0) with respect to T a µν̺ , i.e. the BRST-transformations sT ⋆µν̺ a are the "left hand sides" of the equations of motion of the free theory).

2
: The forms defined in equations (3.1) fulfill and total (D − 2)-forms Ω aµν̺ D−2 : The forms defined in equations (3.4) and (3.5) fulfill Comments: (i) The total (D −3)-forms Ω aµ D−3 defined in equations (3.4) derive from the following simpler total (D − 3)-forms Λ aµ D−3 : with ω aµ −3,D and ω aµ −2,D−1 as in equations (3.4). Using table (2.2) it can be readily checked that the total forms Λ aµ D−3 are (s + d)cocycles: Furthermore it can readily be shown that Λ aµ D−3 is no (s + d)-coboundary. Indeed,   already a gauge invariant improvement of λ aµ 0,D−3 but we proceed one step further and remove also all terms from the exterior (D−3)-form (3.13) which vanish on-shell in the free theory. Using equations (2.8)-(2.10) one finds that these terms are the BRST-transformation of the following exterior (D − 3)-form η aµ −1,D−3 : (3.14) We arrive at the improved total form (3.4): 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 s-trivial terms in the resultant redefined exterior (D − 1)-form and exterior Dform 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−3 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 aµν̺ D−2 in (3.5) can also be improved so as not to contain terms that vanish on-shell in the free theory. Using equation (2.10) one can write the terms of Ω aµν̺ 2 that vanish on-shell in the free theory as sη aµν̺ −1,2 with an exterior 2-form η aµν̺ −1,2 and redefine Ω aµν̺ Furthermore one can write the terms of ω aµν̺ 0,D−2 that vanish on-shell in the free theory as sη aµν̺ −1,D−2 with an exterior (D − 2)-form η aµν̺ −1,D−2 and redefine Ω aµν̺ and Ω aµν̺ D−2 can be constructed likewise (and equivalently) with the redefined total forms.
Comments:  With this multi-index notation the total forms Ω YM D in equations (4.1)-(4.3) can all be written as and one obtains, using (s + d)Ω (which holds owing to (3.3)): where we used thatΩ if f abc = f (abc) for k = 2m and f abc = f [abc] for k = 2m + 1. We remark that (4.13) actually vanishes for k > 4 because in dimensions D = 2k + 1 > 9 there is no way to contract the nine free Lorentz indices of Ω aµ 1 µ 2 µ 3 2k−1 Ω bµ 4 µ 5 µ 6 1 Ω cµ 7 µ 8 µ 9 1 in a Lorentz invariant way. For the same reason there is no Ω YM D in even dimensions D. (iii) Using the same multi-index notation as above, one can construct further Chern-Simons type solutions of (1.5) in odd dimensions: We remark that the Chern-Simons type solution (4.9) can be written in this form.

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 µν̺ and B a µν̺σ 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 µν̺ =Ĥ a [µν̺] , 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 (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 ν̺σµτ is defined analogously to W a ν̺σµτ in (2.7), withÊ in place of E. The total forms (5.7) and (5.8) with H aµν̺ and Ω aµν̺ 1 as in (3.1). Hence, in the first order formulation of the free theory the total 1-formΩ ′ aµν̺ 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. 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 substitutê Ω 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 deformationsŜ (1) provide in fact already a complete deformation S =Ŝ (0) + gŜ (1) of the master actionŜ (0) of the first order formulation of the free theory. This implies that the first order deformations arising from the solutions (4.5), (4.6), (4.8) and (4.9) 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 (1.5) given in section 4 in the respective dimensions D = 5, 7, 9 are obtained from the antifield independent parts L (1) d D x of the exterior D-forms of these solutions. The first order deformations L Here we assumed that the flat metric has signature (−, +, +, +, +). Other conventions can result in a minus sign in (6.7) and a plus sign in L N in (6.8). We remark that all results presented in this work are actually valid also for non-Minkowskian metrics, with possible reversed signs in (6.7) and in L Notice that the first order deformations (6.1), (6.3) and (6.4) exist for any number of Curtright fields (and in particular for only one Curtright field), whereas the first order deformations (6.5) and (6.6) require at least two Curtright fields, and the first order deformations (6.2) and (6.7) require at least three Curtright fields because of equations (4.4), (4.7) and (4.10). Furthermore notice that all the above first order deformations are Lorentz invariant, in spite of the explicit x-dependence of the deformations (6.4), (6.5) and (6.6). 8 This explicit Notice also that all the above first order deformations are cubic in the Curtright fields and that the deformations (6.1)-(6.3) contain four derivatives of the Curtright fields (terms ∂ 2 T ∂T ∂T ) whereas the deformations (6.4)-(6.7) contain five derivatives of the Curtright fields (terms ∂ 2 T ∂ 2 T ∂T ), respectively. Furthermore, all deformations (6.1)-(6.6) 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 consistent deformations that do not deform nontrivially 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 aµν̺στ but all such quadratic terms actually vanish on-shell up to a total divergence because of (2.5)-(2.9). 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, 7, 9 at first order.
As shown in section 5 the above first order deformations (6.4)-(6.7) can in fact be extended to all orders, most readily using the first order formulation of the theory. Furthermore in D = 5 any linear combination of the deformations (6.4), (6.6) and (6.7) can be extended to all orders. Whether or not the first order deformations (6.1)-(6.3) can be extended to higher orders is left open here.
We also remark that in all above first order deformations the tensors E aµν̺στ can be replaced by the traceless tensors W aµν̺στ (2.7) and vice versa because of E aµν̺στ ≈ W aµν̺στ , 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)-(6.3) can be extended to higher orders in a future work (unless someone else does the job). However, the similarity of (6.1)-(6.7) 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)-(6.7) 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 duals of the field strength formsF a D−p−1 . Yang-Mills interactions in D dimensions areF a D−2 A b 1 A c 1 f abc . This is analogous to (4.1)-(4.3) with Ω a··· 1 of (3.1) corresponding to A a 1 , and Ω a··· D−2 of (3.5) corresponding toF a D−2 . 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 potentials A a 1 areF a D−3 F b 2 A c 1 e abc . This is analogous to (4.5) and (4.6) with Ω a··· 1 of (3.1) corresponding to A a 1 , Ω a··· 2 of (3.1) corresponding to F a 2 , and Ω a· D−3 of (3.4) corresponding toF a D−3 . Cubic Freedman-Townsend interactions in 5 dimensions areF a 1F b 1 A c 3 d abc . This is analogous to (4.8) with Ω a· 2 of (3.4) corresponding toF a 1 , and Ω a··· 1 of (3.1) corresponding to A a 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 F a 2 F b 2 A c 1 c abc . This is analogous to (4.9) with Ω a··· 1 of (3.1) corresponding to A a 1 , and Ω a··· 2 of (3.1) corresponding to F a 2 . The difference of the deformations (6.1)-(6.7) as compared to standard Yang-Mills, Chapline-Manton, Freedman-Townsend and Chern-Simons interactions results on the one 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 (6.4) and (6.5) 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 (4.5) and (4.6). 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 (6.4) (for one Curtright field) when the Lorentz structure of the fields is taken into account. 9 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 N = −A µ j µ , j µ = ǫ 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 = ǫ ν 1 ...ν 5 W aν 3 ν 4 ν 5 σ̺ x ̺ (6.8) 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 aµν̺ 2 Ω b 2µν̺ Ω 1 k ab of (1.5) wherein k ab = k ba are constant symmetric coefficients and Ω aµν̺ 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)-(6.3) and may be constructible analogously to (4.1)-(4.3) 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.