ON THE UNIQUENESS OF D = 11 INTERACTIONS AMONG A GRAVITON, A MASSLESS GRAVITINO AND A THREE-FORM. II: THREE-FORM AND GRAVITINI

Transcript

1 International Journal of Modern Physics A Vol. 23, No c World Scientific Publishing Company ON THE UNIQUENESS OF D = INTERACTIONS AMONG A GRAVITON, A MASSLESS GRAVITINO AND A THREE-FORM. II: THREE-FORM AND GRAVITINI EUGEN-MIHĂIŢĂ CIOROIANU, EUGEN DIACONU and SILVIU CONSTANTIN SĂRARU Faculty of Physics, University of Craiova, 3 Al. I. Cuza Street, Craiova, , Romania manache@central.ucv.ro ediaconu@central.ucv.ro scsararu@central.ucv.ro Received 3 March 2007 The interactions that can be introduced between a massless Rarita Schwinger field and an Abelian three-form gauge field in space time dimensions are analyzed in the context of the deformation of the free solution of the master equation combined with local BRST cohomology. Under the hypotheses of smoothness of the interactions in the coupling constant, locality, Poincaré invariance, Lorentz covariance, and the presence of at most two derivatives in the Lagrangian of the interacting theory the same number of derivatives as in the free Lagrangian, we prove that there are neither cross-couplings nor self-interactions for the gravitino in D =. The only possible term that can be added to the deformed solution to the master equation is nothing but a generalized Chern Simons term for the three-form gauge field, which brings contributions to the deformed Lagrangian, but does not modify the original, Abelian gauge transformations. Keywords: BRST symmetry; BRST cohomology; consistent interactions. PACS number:.0.ef. Introduction It is known that the field content of D =, N = supergravity is remarkably simple; it consists of a graviton, a massless Majorana spin-3/2 field, and a threeform gauge field. The analysis of all possible interactions in D = related to this field content necessitates the study of cross-couplings involving each pair of these sorts of fields and then the construction of simultaneous interactions among all the three fields. With this purpose in mind, in Ref. we have obtained all consistent interactions that can be added to a free theory describing a massless spin-two field and an Abelian three-form gauge field in space time dimensions. Here, we develop the second step of our approach and analyze the consistent -dimensional interactions that can be introduced between a massless Rarita Schwinger field and 484

2 4842 E.-M. Cioroianu, E. Diaconu & S. C. Săraru an Abelian three-form gauge field. Our main result is that under the hypotheses of smoothness of the interactions in the coupling constant, locality, Poincaré invariance, Lorentz covariance, and the presence of at most two derivatives in the Lagrangian of the interacting theory the same number of derivatives as in the free Lagrangian there are neither cross-couplings nor self-interactions for the gravitino in D =. The only possible term that can be added to the deformed solution to the master equation is nothing but a generalized Chern Simons term for the three-form gauge field, which brings contributions to the deformed Lagrangian, but does not modify the original, Abelian gauge transformations. Our result does not contradict the presence in the Lagrangian of D =, N = SUGRA of a quartic vertex expressing self-interactions among the gravitini. We will see in Refs. 2 and 3 that this vertex, which appears at order two in the coupling constant, is due to the simultaneous presence of gravitini, three-form, and graviton. 2. Free Model: Lagrangian Formulation and BRST Symmetry Our starting point is represented by a free model, whose Lagrangian action is written like the sum between the standard action of an Abelian three-form gauge field and that of a massless Rarita Schwinger field in space time dimensions S0 L [A νρ, ψ ] = d x 2 4! F νρλf νρλ i 2 ψ γ νρ ν ψ ρ d x L0 A + Lψ 0, where F νρλ denotes the field strength of the three-form gauge field F νρλ = [ A νρλ]. We maintain the antisymmetrization convention explained in Ref. and work with that representation of the Clifford algebra γ γ ν + γ ν γ = 2σ ν for which all the γ matrices are purely imaginary. In addition, we take γ 0 to be Hermitian and antisymmetric and γ i i=,0 anti-hermitian and symmetric γ = γ, 3 γ = γ 0γ γ 0, = 0, 0. 4 The operations of Dirac and respectively Majorana conjugation are defined as usually via the relations where the charge conjugation matrix is ψ = ψ γ 0, 5 ψ c = Cψ, 6 C = γ 0. 7

3 On the Uniqueness of D = Interactions 4843 In what follows we use the notations γ k = Θ γ Θ γ Θ γ Θk, 8 k! Θ Σ k where Σ k represents the set of permutations of the numbers {, 2,..., k} and Θ is the signature of a given permutation Θ. We will need the Fierz identities specific to D = γ p γ ν νq = p+q 2k 2M δ [ν [ p δ ν2 p δ ν k p k+ γ ν k+ ν q] p k ], 9 where M = minp, q and also the development of a complex, spinor-like matrix N in terms of the basis {, γ, γ ν, γ νρ, γ νρλ, γ νρλσ } N = 32 5 kk /2 Trγ k Nγ k! k. 0 k=0 The theory described by action possesses an Abelian, off-shell, second-order reducible generating set of gauge transformations δ ε A νρ = [ ε νρ], δ ε ψ = ε. Related to the gauge parameters, ε ν are bosonic and completely antisymmetric and ε is a fermionic Majorana spinor. The fact that the gauge transformations of the three-form gauge field are off-shell, second-order reducible is treated in more detail in Ref.. In order to construct the BRST symmetry for we introduce the field, ghost, and antifield spectra Φ 0 = A νρ, ψ, Φ 0 = A νρ, ψ, 2 η = C ν, ξ, η = C ν, ξ, 3 η Γ2 = C, η Γ 2 = C, 4 η Γ3 = C, η Γ 3 = C. 5 The fermionic ghosts C ν correspond to the gauge parameters of the three-form, ε ν, the bosonic ghost ξ is associated with the gauge parameter ε, while the bosonic ghosts for ghosts η Γ2 and the fermionic ghost for ghost for ghost η Γ3 are due to the first- and respectively second-order reducibility of the gauge transformations from the three-form sector. The star variables represent the antifields of the corresponding fields/ghosts. The antifields of the Rarita Schwinger field are bosonic, purely imaginary spinors. Since the gauge generators of the free theory under study are field independent, it follows that the BRST differential decomposes again as in Ref. s = δ + γ, 6

4 4844 E.-M. Cioroianu, E. Diaconu & S. C. Săraru where δ represents the Koszul Tate differential and γ stands for the exterior derivative along the gauge orbits. More details of the various graduations of the BRST generators can be found in Ref.. In agreement with the standard rules of the BRST formalism, the degrees of the BRST generators are valued as aghφ 0 = aghη = aghη Γ2 = aghη Γ3 = 0, 7 aghφ 0 =, aghη = 2, aghη Γ 2 = 3, aghη Γ 3 = 4, 8 pghφ 0 = 0, pghη =, pghη Γ2 = 2, pghη Γ3 = 3, 9 pghφ 0 = pghη = pghη Γ 2 = pghη Γ 3 = The actions of the differentials δ and γ on the generators from the BRST complex are given by δa νρ = 3! λf νρλ, δψ = i ρ ψλ γ ρλ, 2 δc ν = 3 ρ A νρ, δξ = ψ, 22 δc = 2 ν C ν, δc = C, 23 δφ 0 = δη = δη Γ2 = δη Γ3 = 0, 24 γφ 0 = γη = γη Γ 2 = γη Γ 3 = 0, 25 γa νρ = [ C νρ], γψ = ξ, 26 γc ν = [ C ν], γξ = 0, 27 γc = C, γc = In this case the anticanonical action of the BRST symmetry, s =, S A,ψ, is realized via a solution to the master equation S A,ψ, S A,ψ = 0 that reads as S A,ψ = S0 L [A νρ, ψ ] + d x ψ ξ + A νρ [ C νρ] + C ν [ C ν] + C C Consistent Interactions between an Abelian Three-Form Gauge Field and a Rarita Schwinger Spinor The aim of this section is to investigate the cross-couplings that can be introduced between an Abelian three-form gauge field and a massless Rarita Schwinger field in D =. This matter is addressed, as in Ref., in the context of the antifield-brst deformation procedure. Very briefly, this means that we will associate with 29 a deformed solution S A,ψ S A,ψ = S A,ψ + λs A,ψ + λ 2 S A,ψ 2 + = S A,ψ + λ d D x a A,ψ + λ 2 d D x b A,ψ +, 30

5 On the Uniqueness of D = Interactions 4845 which is the BRST generator of the interacting theory, S A,ψ, S A,ψ = 0, such that the components of S A,ψ are restricted to satisfy the tower of equations: S A,ψ, S A,ψ = 0, 3 2 S A,ψ 2, S A,ψ + S A,ψ 3, S A,ψ + 2 S A,ψ, S A,ψ = 0, 32 S A,ψ, S A,ψ = 0, 33 S A,ψ, S A,ψ 2 = The interactions are obtained under the same assumptions like in Ref. : smoothness, locality, Lorentz covariance, Poincaré invariance, and preservation of the number of derivatives on each field derivative order assumption. The derivative order assumption means here that the following two requirements are simultaneously satisfied: i the derivative order of the equations of motion on each field is the same for the free and respectively for the interacting theory; ii the maximum number of derivatives in the interaction vertices is equal to two, i.e. the maximum number of derivatives from the free Lagrangian First-order deformation Initially, we construct the first-order deformation of the solution to the master equation, S A,ψ, as solution to Eq. 32. If we make the notation S A,ψ = d x a A,ψ, with a A,ψ a local function gha = 0, εa = 0, then 32 takes the local form sa A,ψ = m, 35 which shows that the nonintegrated density of the first-order deformation pertains to the local cohomology of the BRST differential in ghost number zero, a A,ψ H 0 s d. In order to analyze Eq. 32 we act like in Ref. : we develop a A,ψ according to the antighost number a A,ψ = I i=0 a A,ψ i, agh a A,ψ i = i 36 and obtain in the end that Eq. 35 becomes equivalent to the tower of equations δa A,ψ I δa A,ψ i γa A,ψ I I = m, 37 + γa A,ψ I = I m, 38 + γa A,ψ i = i m, i I, 39

6 4846 E.-M. Cioroianu, E. Diaconu & S. C. Săraru where, moreover, Eq. 37 can be replaced in strictly positive antighost numbers by γa A,ψ I = 0, I > The nontriviality of the first-order deformation a A,ψ is thus translated at its highest antighost number component into the requirement that a A,ψ I H I γ, where H I γ denotes the cohomology of the exterior longitudinal derivative γ in pure ghost number equal to I. So, in order to solve Eq. 35 we need to compute the cohomology of γ, Hγ, and, as it will be made clear below, also the local cohomology of δ in pure ghost number zero, Hδ d. Using the results on the cohomology of the exterior longitudinal differential for an Abelian three-form gauge field computed in Ref. as well as definitions 25 28, we can state that the most general solution to 40 can be written, up to γ-exact contributions, as a A,ψ I = α I [Fνρλ ], [ [ ψ ν] ], [χ ] ω I C, ξ, 4 where χ = { } Φ 0, η, ηγ 2, ηγ 3 and ω I denotes the elements with pure ghost number I of a basis in the space of polynomials in the corresponding ghosts. The objects α I with agh α I = I are nontrivial elements of H 0 γ, known as invariant polynomials. They are in fact polynomials in the antifields χ, in the field strength of the three-form F νρλ, in the antisymmetrized first-order derivatives of the Rarita Schwinger fields [ ψ ν] as well as in their subsequent derivatives. Just like in Ref., it can be shown that a necessary condition for the existence of nontrivial solutions a I is that the invariant polynomials α I are nontrivial objects from the local cohomology of the Koszul Tate differential Hδ d in antighost number I > 0 and in pure ghost number zero. Using the fact that the Cauchy order of the free theory under study is equal to four together with the general result according to which the local cohomology of the Koszul Tate differential in pure ghost number zero is trivial in antighost numbers strictly greater than its Cauchy order, we can state that H J δ d = 0 for all J > On the other hand, it can be shown that any invariant polynomial α J that is trivial in H J δ d with J 4 can be taken to be trivial also in the invariant characteristic cohomology in antighost number J, HJ invδ d: α J = δ b J+ + c J, agh α J = J 4 α J = δ β J+ + γ J, 43 with both β J+ and γ J invariant polynomials. Results 42 and 43 yield the conclusion that H inv J δ d = 0 for all J > It can be shown that the spaces H J δ d J 2 and HJ invδ d J 2 are spanned by H 4 δ d, H inv 4 δ d : C, 45 H 3 δ d, H inv 3 δ d : C, 46 H 2 δ d, H inv 2 δ d : C ν, ξ. 47

7 On the Uniqueness of D = Interactions 4847 These results on Hδ d and H inv δ d in strictly positive antighost numbers are important because they allow the elimination of all pieces with I > 4 from 36. In the case I = 4 the nonintegrated density of the first-order deformation a A,ψ, 36, becomes a A,ψ = a A,ψ 0 + a A,ψ + a A,ψ 2 + a A,ψ 3 + a A,ψ We can further decompose a A,ψ in a natural manner, as a sum between three kinds of deformations a A,ψ = a A + a A ψ + a ψ, 49 where a A contains only BRST generators from the Abelian three-form sector, a A ψ describes the cross-interactions between the two theories, and a ψ is responsible for the Rarita Schwinger self-interactions. The component a ψ can be shown to take the same form like in the case D = 4 see Ref. 4 and satisfies individually an equation of the type 35. It admits a decomposition of the form where a ψ = a ψ 0 + aψ, 50 a ψ = imψ γ ξ, a ψ 0 = 9 2 m ψ γ ν ψ ν, 5 with m an arbitrary, real constant. Since a A ψ mixes the variables from the threeform and the Rarita Schwinger sectors and a A depends only on the BRST generators from the three-form sector, it follows that a A ψ and a A are subject to two separate equations sa A = m A, 52 sa A ψ = m A ψ. 53 The nontrivial solution a A to 52 has been discussed in Ref. and reduces to a A = qε A 2 3 F 4 7 F 8, 54 with q an arbitrary, real constant. Let us analyze now the solutions to Eq. 53. In agreement with the previous results on H inv δ d, we can always take the decomposition of a A ψ along the antighost number to stop at antighost number equal to four a A ψ = a A ψ 0 + a A ψ + a A ψ 2 + a A ψ 3 + a A ψ 4, 55 such that 53 becomes equivalent with the tower of equations δa A ψ I γa A ψ 4 = 0, 56 + γa A ψ I = m A ψ I, I =, 4. 57

8 4848 E.-M. Cioroianu, E. Diaconu & S. C. Săraru Recalling the results from the previous subsection on the cohomology Hγ, it follows that the elements with pure ghost number four of a basis in the space of polynomials in the ghosts C and ξ can be chosen as { ξc, ξγ ξ ξγ ξ, ξγ ν ξ ξγ ν ξ, ξγ νρλσ ξ ξγ νρλσ ξ }. 58 The solution to 56 is obtained like in 4, by gluing the general representative of H inv δ d, namely C, to 58 a A ψ 4 = v C ξγ ξ ξγ ξ + v 2 C ξγ ν ξ ξγ ν ξ + v 3 C ξγ νρλσ ξ ξγ νρλσ ξ, 59 where v i i=,2,3 are some arbitrary constants the element ξc cannot be coupled to C to form a Lorentz invariant since ξ is a Majorana spinor, so it is not eligible to enter 59. Substituting 59 back in 57 for I = 4 and using definitions 2 28, we obtain a A ψ 3 = 4C α[ v ξγ ξ ξγ ψ α + v 2 ξγ ν ξ ξγ ν ψ α + v 3 ξγ νρλσ ξ ξγ νρλσ ψ α ]. 60 By applying the Koszul Tate differential on 60, we find δa A ψ 3 = γ { 8C [ v ξγ ψ α ξγ ψ β + v 2 ξγ ν ψ α ξγ ν ψ β + v 3 ξγ νρλσ ψ α ξγ νρλσ ψ β ] 4C [ v ξγ ξ ψ α γ ψ β + v 2 ξγ ν ξψ α γ ν ψ β + v 3 ξγ νρλσ ξ ψ α γ νρλσ ψ β ]} 4C [ v ξγ ξ ξγ [α ψ β] + v2 ξγ ν ξ ξγ ν [α ψ β] + v 3 ξγ νρλσ ξ ξγ νρλσ [α ψ β] ] + m A ψ 2. 6 Comparing 6 with 53 for I = 3 it follows that a A ψ 3 provides a consistent a A ψ 2 if the quantity π = 4C [ v ξγ ξ ξγ [α ψ β] + v2 ξγ ν ξ ξγ ν [α ψ β] + v 3 ξγ νρλσ ξ ξγ νρλσ [α ψ β] ], 62 can be written in a γ-exact modulo d form π = γw + θ. 63 Assume that 63 holds. Taking its Euler Lagrange EL derivatives with respect to C we get δ L π δ L δc = γ w δc. 64

9 On the Uniqueness of D = Interactions 4849 On the other hand, from 62 by direct computation we infer δ L π δc = 4[ v ξγ ξ ξγ [α ψ β] + v2 ξγ ν ξ ξγ ν [α ψ β] + v 3 ξγ νρλσ ξ ξγ νρλσ [α ψ β] ]. 65 Thus, Eq. 64 restricts δ L π/δc to be a trivial element of Hγ, while 65 emphasizes that δ L π/δc is a nontrivial element from Hγ because each term from the right-hand side of 65 is so, such that the only possibility is that δ L π/δc must vanish δ L π = δc This further implies, by means of 65, that we must set zero all the constants that parametrize a A ψ 4 so in the end we have that v = v 2 = v 3 = 0, 67 a A ψ 4 = a A ψ 3 = As a consequence, decomposition 55 can stop earliest at antighost number three, a A ψ = a A ψ 0 + a A ψ + a A ψ 2 + a A ψ 3, where a A ψ 3 satisfies the equation γa A ψ 3 = 0. According to 4, 46 and recalling the assumption that a A ψ 3 mixes the BRST generators of the three-form with those from the Rarita Schwinger sector, it results that the solution to this equation reads as a A ψ 3 = C e ξ, where e ξ denote the vector-like elements of pure ghost number three of a basis in the space of polynomials in the ghost ξ. Since pghξ =, it follows that e ξ necessarily contains three spinors of the type ξ and therefore we can set a A ψ 3 = 0 because one cannot construct a Lorentz -dimensional vector out of three spinors. Thus, we can write such that Eq. 53 becomes equivalent to δa A ψ I a A ψ = a A ψ 0 + a A ψ + a A ψ 2, 69 γa A ψ 2 = 0, 70 + γa A ψ I = m A ψ I, I =, 2. 7 Because the elements of pure ghost number two of a basis in the space of polynomials in the ghost ξ read as { ξγ ξ, ξγ ν ξ, ξγ νρλσ ξ } 72 the ghost for ghost for ghost C is not eligible as pghc = 3 and the representatives of H2 inv δ d are given by 47, we observe that the only combination that might generate cross-interactions remains a A ψ 2 = k 2 C ν ξγν ξ, 73

10 4850 E.-M. Cioroianu, E. Diaconu & S. C. Săraru where k is an arbitrary constant. Replacing 73 in 7 for I = 2 we determine a A ψ under the form a A ψ = 3 ka νρ ξγν ψ ρ + ā A ψ, 74 where ā A ψ is the general solution to the homogeneous equation It is expressed by γā A ψ = ā A ψ = ψ M + A νρ N νρ ξ, 76 with N νρ the components of a real, fermionic, gauge-invariant, completely antisymmetric spinor tensor and M some bosonic, gauge-invariant, matrices, which in addition must explicitly depend on the three-form field strength F νρλ in order to provide cross-interactions. By applying δ on 74 with ā A ψ of the form 76, we obtain where d 0 = δa A ψ = γd 0 + e 0 + m, 77 4 k ψ γ νρ ψ λ + 3! N νρψ λ F νρλ, 78 e 0 = 4 k ξγ ν ρ ψ λ F νρλ + i ψ λ γ ρλ ρ M ξ + i ψ λ γ ρλ M ρ ξ + 3! λn νρ ξf νρλ. 79 The condition that 77 is expressed as in 7 for I = restricts e 0 expressed by 79 to be γ-exact modulo d e 0 = γp + n. 80 Recalling the requirement that the quantities N νρ are spinor-like and gaugeinvariant, we deduce that the most general representation of these elements is N νρ = [α ψβ] Nνρ, 8 where Nνρ are also gauge-invariant. As c 0 from 79 involves terms with different numbers of derivatives, it is useful to decompose the functions M and Nνρ according to the number of space time derivatives N M = M + M + 3 M +, 82 νρ = 0 Nνρ + N νρ +, 83 where k M and k Nνρ contain precisely k derivatives 82 cannot contain a derivative-free term because, as we have emphasized before, M depends at least

11 On the Uniqueness of D = Interactions 485 linearly on F νρλ. Inserting 82 and 83 in 79 and projecting 80 on the various numbers of derivatives, we find the equivalent tower of equations 4 k ξγ ν ρ ψ λ F νρλ + i ψ λ γ ρλ ρ M i ψ λ γ ρλ ρ M ξ + i ψ ρλ λ γ M ρ ξ + 3! λ 0 [α ψβ] N ξ + i ψ ρλ λ γ M ρ ξ = γ νρ ξf νρλ = γ i ψ λ γ ρλ k ρ M ξ + i ψ k ρλ λ γ M ρ ξ + 3! λ k 2 [α ψβ] N νρ ξf νρλ = γ k p + n p + n p + k, 84, 85 n, k Equations 86 would lead to interaction vertices with more than two space time derivatives, so, in agreement with our hypothesis on the conservation of the number of derivatives on each field with respect to the free theory, they must be discarded k k M = 0, k 3, 87 Nνρ = 0, k, 88 which ensures k p = 0 for k 3 in 80. As the matrices M are linear in the three-form field strength, they can be generally represented in the form M = k F γ γ γ + k 2 F γδ γ γδ, 89 with k and k 2 some arbitrary constants. Based on 89, the left-hand side of 84 becomes 4 k ξγ ν ρ ψ λ F νρλ + i ψ λ γ ρλ ρ M ξ + i ψ ρλ λ γ M γψ ρ = γ [ 38 k + 8k 2 i ψ α γ νρλ ψ α 2 ] k + 5k 2 i ψ α γ νρλ ψ F β νρλ + 2 k 2 3!ik 7 4!ik 2 ξγ ν ρ ψ λ F νρλ + k + 8k 2 [ 3 ] 4 i α ψα γ νρλ ξ + 3i ψα γ ανρλ ξ + 3i ψ γ ανρλ γψ α n F νρλ +. 90

12 4852 E.-M. Cioroianu, E. Diaconu & S. C. Săraru Asking now that the right-hand side of 90 satisfies 84, we find the restrictions which further produce M = i k k = 9 i k, k 2 = 3 4! i k, 9 9 F γ γ γ 3 4! F γδ γ γδ, 92 p = 2 4! k ψ α γ νρλ ψ β F νρλ. 93 Next, we approach Eq. 85. Due to the fact that each M is a gauge-invariant, matrix with two space time derivatives, it contains precisely two threeform field strengths since it cannot depend on [α ψβ], which is a spinor. As the elements 0 Nνρ are derivative-free and gauge-invariant, they can only be constant. Based on the last two observations, we observe that each of the first two terms from the left-hand side of Eq. 85 comprises two three-form field strengths, while the last term is only linear in F νρλ, such that 85 splits into two separate equations i ψ λ γ ρλ ρ M ξ + i ψ ρλ λ γ M ρ ξ = γ p + n, 94 3! λ 0 [α ψβ] Nνρ ξf νρλ = γ p 2 + n The left-hand side of 94 is γ-exact modulo d if the following conditions are simultaneously satisfied: T ρλ λρ γ 0 γ M = γ 0 γ M, 96 [ρ M] = In order to investigate the former condition, we represent M in terms of a basis in the space of γ-matrices M = M + α γ γδ γδε M γ α + M γ + M γ γ + M γ γδ + M γ γδε, α where each of the coefficients M, M, M, M, M and M is a function with precisely two three-form field strengths. This dependence implies the vanishing of all coefficients with an odd number of indices γ γδ γδε 98 M = 0, γδ M = 0, M = 0. 99

13 On the Uniqueness of D = Interactions 4853 Inserting 99 in 98 we find by direct computation the relation ρλ γ M = α M + + δ α [ γ ρλ] + γ ρλ α γ M γδε M δ [α δλ βδ ρ γ] + δ[ [α δλ βγ ρ] γ] + δ[ [α γρλ] βγ] + γρλ γ δ [α δλ βδ ρ γγ δε] + δ [ [α δλ βγ ρ] γδε] + δ[ [α γρλ] βγδε] + γρλ γδε λρ γδε. 00 Looking at 00, we remark that the terms 3! M and M γ ρλ γδε appear- ρλ ing in γ M break condition 96, so we must set γ M = 0, The last result replaced in 00 yields It is clear that α ρλ γ M = α M γδε M = 0. 0 δ [ α γρλ] + γ ρλ α. 02 M γ ρλ α from 02 cannot fulfill 96, so we must take α M = 0, 03 which, together with 99, 0 and 03, lead to the result M = 0, 04 so we can only have p = 0 in 94. Now, we investigate Eq. 95. Direct computation provides 3! λ 0 [α ψβ] Nνρ ξf νρλ = γ 3! ψ 0 [α β] Nνρ ψ λ F νρλ 3! ψ 0 [α β] Nνρ ξ λ F νρλ + s. 05 Assume that the second term from the right-hand side of 05 would give a γ-exact modulo d quantity. Comparing 05 to 85, we find that p 2 = 3! ψ 0 [α β] Nνρ ψ λ F νρλ It is simple to see that p 2 which contributes to a A ψ 0 produces field equations for the Rarita Schwinger field with two space time derivatives, which disagrees with

14 4854 E.-M. Cioroianu, E. Diaconu & S. C. Săraru requirement i from the beginning of this section related to the derivative order assumption. Thus, we must set 0 Nνρ = 0, 07 which yields p 2 = 0. Inserting 87, 92 and 04 into 82 and respectively 88 and 07 into 83, and then substituting the resulting expressions of 82 and 83 in 76, we obtain the general form of the solution ā A ψ, such that 74 takes the final form a A ψ = 3 ka νρ ξγν ψ ρ + i kψ 9 F νρλγ νρλ 3 4! F νρλσ γ νρλσ Accordingly, we find that a A ψ 0 as solution to Eq. 7 for I = reads as ξ. 08 a A ψ 0 = 4 k ψ γ νρ ψ λ F νρλ 2 4! k ψ α γ νρλ ψ β F νρλ. 09 Replacing now 73 and into 69, we find that the interacting part of the first-order deformation of the solution to the master equation becomes S A ψ = d x a A ψ 2 + a A ψ + a A ψ 0 k d x 2 C ν ξγν ξ 3A νρ ξγν ψ ρ + i 9 ψ F νρλ γ νρλ ξ i 3 4! ψ F νρλσ γ νρλσ ξ 4 ψ γ νρ ψ λ F νρλ 2 4! ψ α γ νρλ ψ β F νρλ. 0 In what follows we employ the notation S A ψ = S A ψ + d x a A, with a A given by 54, so the complete expression of the first-order deformation of the solution to the master equation for the model under consideration is see 49 S A,ψ = S A ψ + S ψ, 2 where S ψ is the component corresponding to the Rarita Schwinger sector = d x a ψ, 3 S ψ and the integrand a ψ can be read from 50 and 5.

15 On the Uniqueness of D = Interactions Second-order deformation In this section we investigate the consistency of the first-order deformation, described by Eq. 33. Along the same line as before, we can write the second-order deformation like the sum between the Rarita Schwinger contribution and the interacting part The piece S ψ 2 where S A,ψ A ψ 2 = S 2 + S ψ 2. 4 is subject to the equation 2 S, S ψ + s S ψ 2 = 0, 5 S, S ψ = In formula 6 we used the notation S A ψ, S A ψ that contain only BRST generators from the Rarita Schwinger spectrum. The component where and ψ S ψ, Sψ + S A ψ, S A ψ. 6 ψ for those pieces from S A ψ, S A ψ S, S A ψ = 2 A ψ S 2 results as solution to the equation 2 S, S A ψ A ψ + s S 2 = 0, 7 S ψ, S A ψ A ψ S A ψ, S A ψ = S A ψ, S A ψ A ψ + S A ψ, S A ψ 8 ψ. If we denote by ψ S A ψ, S A ψ and b ψ the nonintegrated densities of the functionals S, S ψ and respectively S ψ 2, then the local form of 5 becomes with ψ = 2s b ψ + ñ, 9 gh ψ =, gh b ψ = 0, ghñ =, 20 for some local currents n. Direct computation shows that ψ decomposes as ψ = ψ + ψ 0, agh ψ =, agh ψ 0 = 0, 2 where [ ψ = τ + γ i k 2 ψ[ 3 γ νρλ]ξ ] 2 ψ σ γ νρλσ ξ ψ γ νρ ψ λ i k 2 ψ[ 3 γ νρλ]ξ 2 ψ σ γ νρλσ ξ ξγ ν [ρ ψ λ], 22 ψ 0 = τ im2 ξγ ψ + i k 2 ψ [ γ νρ ψ λ] + 2 ψ α γ νρλ ψ β ξγ νρ ψ λ. 23

16 4856 E.-M. Cioroianu, E. Diaconu & S. C. Săraru Because S, S ψ contains terms of maximum antighost number equal to one, we can assume without loss of generality that b ψ stops at antighost number two bψ = ñ = 2 I=0 2 I=0 bψ I, agh b ψ I = I, I = 0, 2, 24 ñ I, aghñ I = I, I = 0, By projecting Eq. 9 on the various decreasing values of the antighost number, we then infer the equivalent tower of equations 0 = 2γ b ψ 2 + ñ 2, 26 ψ I = 2 δ b ψ I+ + γ b ψ I + ñ I, I = 0,. 27 Equation 26 can always be replaced with γ b ψ 2 = Thus, b ψ 2 belongs to the Rarita Schwinger sector of cohomology of γ, Hγ. By means of definitions we get that Hγ in the Rarita Schwinger sector is generated by the objects ψ, ξ, [ ψ ν], by their space time derivatives up to a finite order, and also by the undifferentiated ghosts ξ the space time derivatives of ξ are γ-exact according to the second relation in 26. As a consequence, we can write bψ 2 = β ψ [ ] 2 [ ψ ν], [ψ ], [ξ ] e 2 ξ, where e 2 ξ are the elements of pure ghost number two of a basis in the space of polynomials in the ghosts ξ, 72. We observe that ψ from 22 can be written as in 27 for I = if and only if χ = i k 2 ψ[ 3 γ νρλ]ξ 2 ψ σ γ νρλσ ξ ξγ ν [ρ ψ λ] 29 reads as where χ = 2δ b ψ 2 + γ ρ + l, 30 ρ = i k 2 ψ[ 3 γ νρλ]ξ 2 ψ σ γ νρλσ ξ ψ γ νρ ψ λ 2 b ψ. 3 Assume that 30 holds. Then, by taking its left Euler Lagrange EL derivatives with respect to ψ and using the commutation between γ and each EL derivative δ L /δψ, we infer the relations δ L χ + 2δ b ψ 2 δl ρ = γ. 32 δψ δψ

17 On the Uniqueness of D = Interactions 4857 As b ψ 2 is γ-invariant, then δ b ψ 2 will also be γ-invariant. Recalling the previous results on the cohomology of γ in the Rarita Schwinger sector, we find that δ b ψ 2 = e2 ξψ v, with v fermionic, γ-invariant functions of antighost number zero and e 2 ξ the elements of pure ghost number two of a basis in the space of polynomials in the ghosts ξ. By using 29 and the last expression of δ b ψ 2, direct computation provides the equation δ L χ + 2δ b ψ 2 = i k 2 [ 23 3 γ νρλξ ξγ [ν ρ ψ λ] + 2 γνρλσ ξ ξγνρ ] [λ ψ σ] δψ + 2e 2 ξv. 33 On the one hand, Eq. 32 shows that δ L χ + 2δ b ψ 2 /δψ is trivial in Hγ. On the other hand, relation 33 emphasizes that δ L χ + 2δ b ψ 2 /δψ is a nontrivial element from Hγ because each term on the right-hand side of 33 is nontrivial in Hγ. Then, δ L χ + 2δ b ψ 2 /δψ must be set zero δ L χ + 2δ b ψ 2 = 0, 34 which yields a δψ By acting with δ on 35 we deduce χ + 2δ b ψ 2 = l. 35 δ χ = j. 36 From 29, by direct computation we find δ χ = k 2 [ 2 3 α ψβ γ 3 γ νρλξ ξγ [ν ρ ψ λ] ] 2 γνρλσ ξ ξγ νρ [λ ψ σ]. 37 Comparing 36 with 37 and recalling the Noether identities corresponding to the Rarita Schwinger action, we obtain that the right-hand of 37 reduces to a total derivative iff 2 3 γ νρλξ ξγ [ν ρ ψ λ] 2 γνρλσ ξ ξγ νρ [λ ψ σ] = p. 38 Simple computation exhibits that the left-hand side of 38 cannot be written like a total derivative, so neither relation 36 nor Eq. 30 hold. As a consequence, χ must vanish and hence we must set k = a In fact, the general solution to Eq. 35 takes the form χ+2δ b ψ 2 = u+ l, where u is a function of antighost number one depending on all the BRST generators from the Rarita Schwinger sector but the antifields ψ. As the antifields ψ are the only Rarita Schwinger antifields of antighost number one, the condition aghu = automatically produces u = 0.

18 4858 E.-M. Cioroianu, E. Diaconu & S. C. Săraru Inserting 39 in 22 23, we obtain that ψ = τ, 40 ψ 0 = τ im2 ξγ ψ. 4 From 40 it results that we can safely take b ψ 2 = 0 and b ψ = 0, which replaced in 4 lead to the necessary condition that ψ 0 must be a trivial element from the local cohomology of γ, i.e. ψ 0 = 2γ b ψ 0 + ñ 0. In order to solve this equation with respect to b ψ 0, we will project it on the number of derivatives. Since γ b ψ 0 contains at least one space time derivative, the above equation projected on the number of derivatives equal to zero reduces to ψ 0 = 80im2 ξγ ψ = 0, which further implies so m = 0, 42 S ψ = Replacing 42 and 43 in 2, we obtain that the general form of the firstorder deformation for the free model under study that is consistent to the second order in the coupling constant reads as S A,ψ = S A q d x ε A 2 3 F 4 7 F Inserting 44 into 32 34, etc., we find that all the higher-order deformations can be taken to vanish S k = 0, k >, 45 so the full deformed solution to the master that is consistent to all orders in the coupling constant takes the simple form S = S + λs A, 46 where S is the solution to the master equation for the starting free model, 29. Relation 46 emphasizes that under the hypotheses mentioned at the beginning of this section, there are neither cross-couplings that can be added between an Abelian three-form gauge field and a massless gravitino nor self-interactions for the gravitino in D =. 4. Conclusion To conclude with, in this paper we have investigated the consistent interactions in space time dimensions that can be added to a free theory describing a massless gravitino and an Abelian three-form gauge field. Our treatment is based on the Lagrangian BRST deformation procedure, which relies on the construction of consistent deformations of the solution to the master equation with the help of standard cohomological techniques. We worked under the hypotheses of smoothness in the coupling constant, locality, Lorentz covariance, Poincaré invariance, and

19 On the Uniqueness of D = Interactions 4859 the preservation of the number of derivatives on each field. Our main result is that there are neither cross-couplings that can be added between an Abelian three-form gauge field and a massless gravitino nor self-interactions for the gravitino in D =. The only possible term that can be added to the deformed solution to the master equation is nothing but a generalized Chern Simons term for the three-form gauge field, 44, which brings contributions to the deformed Lagrangian, but does not modify the original, Abelian gauge transformations. Acknowledgments The authors wish to thank Constantin Bizdadea and Odile Saliu for useful discussions and comments. This work is partially supported by the European Commission FP6 program MRTN-CT and by the grant AT24/2005 with the Romanian National Council for Academic Scientific Research C.N.C.S.I.S. and the Romanian Ministry of Education and Research M.E.C.. References. E. M. Cioroianu, E. Diaconu and S. C. Săraru, Int. J. Mod. Phys. A 23, E. M. Cioroianu, E. Diaconu and S. C. Săraru, Int. J. Mod. Phys. A 23, E. M. Cioroianu, E. Diaconu and S. C. Săraru, Int. J. Mod. Phys. A 23, N. Boulanger and M. Esole, Class. Quantum Grav. 9,