ON THE COHOMOLOGICAL DERIVATION OF D = 11, N = 1 SUGRA

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1 ON THE COHOMOLOGICAL DERIVATION OF D =, N = SUGRA E. M. CIOROIANU, E. DIACONU, S. C. SÃRARU Faculty of Physics, University of Craiova 3 A. I. Cuza Str., Craiova 585, Romania scsararu@central.ucv.ro Received August 3, 8 The couplings that can be introduced between a massless Rarita-Schwinger field, a Pauli-Fierz field and an Abelian three-form gauge field in eleven spacetime dimensions are analyzed in the context of the deformation of the solution of the master equation. We prove, by means of specific cohomological techniques, relying on local BRST cohomology, that the only consistent interactions are described by N =, D = SUGRA.. INTRODUCTION It is well known that the field spectrum of D =, N = SUGRA consists in a massless spin- field, a massless spin-3/ field and a 3-form gauge field. In the free limit the action of simple SUGRA in D = reduces to the sum between Pauli-Fierz, Rarita-Schwinger, and a standard abelian 3-form actions L PF RS 3F ν νρ ν νρ S [ h,ψ, A ] = S [ h ] S [ ψ ] S [ A ]. () In order to determine the consistent interactions that can be added to action () we must study, beside the self-interactions, which are known from the literature, also the cross-couplings. In this paper, we solve the latter problem, which is done in two steps: firstly, we determine the interaction vertices containing only two of the three types of fields, and then the vertices including all the three kinds. We investigate all these problems in the framework of the deformation theory [] based on local BRST cohomology.. CONSTRUCTION OF CONSISTENT INTERACTIONS In section we briefly address the problem of constructing consistent interactions. It has been shown in [] that if an interacting gauge theory can be consistently constructed, then the solution S to the master equation associated with the free theory, (S, S) =, can be deformed into a solution S Rom. Journ. Phys., Vol. 53, Nos. 9, P , Bucharest, 8

2 6 E. M. Cioroianu, E. Diaconu, S. C. Sãraru 3 D S S = Sλ S λ S = Sλ d xaλ d xb () of the master equation for the deformed theory ( S, S) =, (3) such that both the ghost and antifield spectra of the initial theory are preserved. Equation (3) splits, according to the various orders in the coupling constant (deformation parameter) λ, into a tower of equations: (S, S) = (4) (S, S) = (5) (S, S) (S, S ) = (6) (S 3, S) (S, S ) = (7) Equation (4) is fulfilled by hypothesis. The next equation requires that the firstorder deformation of the solution to the master equation, S, is a co-cycle of the free BRST differential s, ss =. However, only cohomologically nontrivial solutions to (5) should be taken into account, since the BRST-exact ones can be eliminated by some (in general nonlinear) field redefinitions. The analysis is performed under the hypotheses of smoothness of the interactions in the coupling constant, locality, Poincar e invariance, Lorentz covariance, and the presence of at most two derivatives in the Lagrangian of the interacting theory (the same number of derivatives like in the free Lagrangian). D 3. INTERACTIONS BETWEEN AN ABELIAN THREE-FORM GAUGE FIELD AND A PAULI-FIERZ GRAVITON In this section, we perform the first step of our approach, namely the analysis of the consistent couplings that can be introduced between a spin two field and an Abelian three-form gauge field. Our starting point is the Lagrangian action represented by the sum between Pauli-Fierz and an abelian three-form actions in eleven space-time dimensions h,a PF 3F [ ν, νρ] = [ ν ] [ νρ] = = dx ( h ) ( hνρ) ( hρ ν νρ ) ( hνρ) (8) S h A S h S A ( h)( hν ) ( h )( h) F Fνρλ ν νρλ. 4!

3 4 Cohomological derivation of D =, N = SUGRA 6 The theory (8) is invariant under the gauge transformations δ h = ε, δ A = ε. (9) ε ν ( ν) ε νρ [ νρ] The gauge algebra of the free theory (8) is Abelian and second-stage reducible. In order to construct the BRST symmetry for (8) we introduce the field, ghost, and antifield spectra ( hν Aνρ ) Δ ( h A ) Δ ν νρ Φ =,, Φ =, () ( Cν ) Δ ( C ) Δ ν η = η,, η = η,, () ( C ) Δ ( C ) Δ η =, η =, () ( C) Δ ( C ) Δ η 3 =, η =. (3) 3 In this case the solution to the master equation reads as h,a Sh,A = S [ h A ] d x( h ν ( ) A νρ ν, νρ η ν [ Cνρ] (4) C ν C C C. [ ν] ) By direct computation we obtain the first-order deformation in the interacting sector like h,a ha A h S = S S S = [ ] ( ) ( ν ( ν ) ( ) ν ) ν ρ ( ) ρ ρ ν ν = d x k C C η C C η C h νc C h ρcν ρcν C [ hν] ρ ν [ ρ] λ νρ 3 νρ [ λ] 3 λ νρ ρ ) νρ λ λ ( ) λ [ ] Cρλ ν h F ( Aνρσhλ ) F νρλ λ νρλ σ ( h Fνρσhλ q A F F Sh σ η η C η A η A A η C h 3 4 4! ε, 3 where k and q are arbitrary real constants and S h reads as h ν ρ ν ν [ ] ν ρ ν ν ρ HE In (6) we used the notations HE [ ] ( ) } (5) S = d x η η η h η h η h Λ h. (6) and Λ for the cubic vertex of the Einstein-Hilbert Lagrangian and respectively the cosmological constant. The

4 6 E. M. Cioroianu, E. Diaconu, S. C. Sãraru 5 consistency of the first-order deformation requires that the real constant k satisfies the equation k( k ) =, (7) with the non-trivial solution k =, (8) and q arbitrary constant. The previous results can be summarized in the following theorem. Theorem. Under the working assumptions, specified in section (), the only consistent deformation of (8) involving a spin- field and an abelian 3-form gauge field reads as h,a PF 3F ν νρ ν νρ S [ h, A ] = S [ h ] S [ A ] λ dx HE h F Fνρλ Λ νρλ h 4 4! F Fνρσhλ F Aνρσhλ νρλ σ νρλ σ 3! 4 ε λ, q A F F O and it is invariant under the gauge transformations ( ) ( ) ρ ρ ρ ( ) ( ) ν ( ν) ρ( ν) ( ν) ρ ρ ν (9) δ h = λ h h h O λ, () δ, A λ [ ] A Aλ σ ε νρ = ε νρ λ λ νρ [ ] [ ] νδρ σλ λ ( ε [ ) h λ ν ρ] λ ε [ νh ] O( λ ), which remain second-order reducible. The results summarized here are presented in detail (including the computation of the second order deformation for the solution to the classical master equation) in []. In this way we obtain that the first two orders of the interacting Lagrangian resulting from our setting originate in the development of the full interacting Lagrangian (in eleven spacetime dimensions) = g h A ( Rλ Λ ), λ where the cross-coupling part reads as ρλ ha gf Fνρλ q νρλ A F 3 4 F 7 8 = λ, 4! with g= det g ν, Λ the cosmological constant, λ the coupling constant, and q an arbitrary, real constant. Consequently, we showed [] the uniqueness of ()

5 6 Cohomological derivation of D =, N = SUGRA 63 interactions described by. The above interacting Lagrangian for Λ = is a part of D =, N = SUGRA Lagrangian. 4. CONSISTENT COUPLINGS BETWEEN AN ABELIAN THREE-FORM GAUGE FIELD AND A MASSLESS SPIN-3/ FIELD We start with a free theory whose Lagrangian action is written as the sum between an Abelian three-form and a non-massive Rarita-Schwinger actions in eleven space-time dimensions A,ψ 3F RS νρ,ψ = [ νρ] [ ψ ] = d x F Fνρλ i νρ νρλ ψ ν ρ 4! S A S A S = γ ψ. In the above the fermionic fields ψ are considered to be real (Majorana) spinors ( ( ψ=ψ c ). We work with the metric σ ν = diag(,,, ), the gamma matrices (γ ) are all purely imaginary. We take the matrix γ to be antisymmetric and Hermitian, γ i ( i =, ) are symmetric and anti-hermitian. The theory described by action () possesses an Abelian, off-shell, second-order reducible generating set of gauge transformations ε νρ [ νρ] ε () δ A = ε, δ ψ = ε. (3) In order to construct the BRST symmetry for () we introduce the field, ghost, and antifield spectra ( Aνρ ) Δ ( A ) Δ νρ Φ =,ψ, Φ =,ψ, (4) ( Cν ) Δ ( C ) Δ ν η =,ξ, η =,ξ, (5) ( C ) Δ ( C ) Δ η =, η =, (6) ( C) Δ ( C ) Δ η 3 =, η =. (7) 3 The solution to the master equation reads as A SA,ψ,ψ = S [ A ] d x ( A νρ νρ,ψ ψ ξ [ Cνρ] (8) C ν C C C. [ ν] By direct computation we obtain the first-order deformation )

6 64 E. M. Cioroianu, E. Diaconu, S. C. Sãraru 7 A,ψ Aψ A ψ ν S = S S S = ( = d xk C 3A νρ ξγνξ ξγνψ ρ ψ F νρλ Fνρλσ νρλγ ξ ψ γνρλσξ 9 3 4! F α νρλ βfνρλ ψ 4 γνρψλ ψ γαβνρλψ 4! m 9i ν q ψ γ ξ ψγ ψ ν ε A F F , (9) where k, m and q are arbitrary constants. The consistency of the first-order deformation (the existence of the second-order deformation) requires k = and m=, (3) and q remains an arbitrary constant. The previous results can be summarized in the following theorem. Theorem. Under the working hypotheses, the only consistent deformation of () involving an Abelian three-form gauge field and a massless spin- 3/ field reads as A,ψ 3F RS S [ Aνρ ] S [ Aνρ] S [ ] q A F F ,ψ = ψ λε, (3) and it is invariant under the original gauge transformations. The results synthesized in this part are developped in [3]. 5. CROSS-COUPLINGS BETWEEN A PAULI-FIERZ GRAVITON AND A MASSLESS RARITA-SCHWINGER FIELD The Lagrangian action for the free theory is represented by the sum between Pauli-Fierz and a non-massive Rarita-Schwinger actions in eleven space-time dimensions h,ψ PF RS ν ν S h,ψ = S [ h ] S [ ψ ] = νρ ρ ν ( νρ) ( ) ( ) ( νρ) = d x h h h h (3) i ( h)( hν ) ( h )( h νρ ν ) ψ ν ρ γ ψ. Action (3) possesses an irreducible and Abelian generating set of gauge transformations

7 8 Cohomological derivation of D =, N = SUGRA 65 δ h =, δ ψ = ε, (33) ε ν ( ν) ε with bosonic and ε fermionic gauge parameters. In order to construct the BRST symmetry for (3) we introduce the field, ghost, and antifield spectra ( hν ) Δ ( h ) ( ) Δ ( ) Δ ν Φ =,ψ, Φ =,ψ, (34) Δ η = η,ξ, η = η,ξ, (35) For this model, the solution to the master equation reads as ( ( ) ) h h S,ψ =,ψ S h d x h ν ν,ψ η ν ψ ξ. (36) By direct computation we obtain the first-order deformation h,ψ hψ ψ h = = d x k i ρ ν ( νρ λ ) [ h ] h νρ ψ λ ν ρ ψ ν ρ 4 S S S S = γ ψ σ γ ψ γ ψ ( ) ψ [ ] ( νρ λ νρ λ ν ψγ ψρ hνλ γ ψ νhρ λ ih ξγψ ν 4 ψ γαβ ( ψ [ ] [ h ] ) 4 ν ( ) ν αηβ ξ α β ψ νψ η ψ ψ [ ην] ν ( ) h ψ ξ ν ) ξ ( ξ) η i ν η ξγξξ γ ξ [ ην] 8 im 9m ν ψ Sh γ ξ ψ ν γ ψ, with k and m some arbitrary constants. The consistency of the first-order deformation imposes (37) k = and m=. (38) The previous results can be summarized in the following theorem. Theorem 3. Under the working assumptions, the only consistent deformation of (3) involving a spin- field and a massless spin-3/ field reads as H E ( ) ( ) h S,ψ PF RS [ hν,ψ ] = S [ hν ] S [ ψ ] λ d x Λ h O λ, (39) and it is invariant under the gauge transformations ρ ρ ρ ( ) ( ) δ h = λ h h h O λ, (4) ε ν ( ν) ρ( ν) ( ν) ρ ρ ν δψ ε =δψ ε = ε. (4) The results shortly addressed here are presented in detail in [4].

8 66 E. M. Cioroianu, E. Diaconu, S. C. Sãraru 9 6. ALL POSSIBLE INTERACTIONS AMONG A GRAVITON, A MASSLESS SPIN-3/, FIELD AND A THREE-FORM GAUGE FIELD The starting point of the last problem is represented by the free model with the Lagrangian action given by the sum between Pauli-Fierz, an Abelian threeform and a non-massive Rarita-Schwinger actions in D = ha,,ψ PF 3F RS [ ν, νρ,ψ ] = [ ν ] [ νρ ] [ ψ ] = d x h hνρ hρ νh h hν νρ νρ ν i ( h )( h) F Fνρλ νρ νρλ ψ ν ρ 4 γ ψ.! S h A S h S A S = ( ) ( ) ( ) ( ) ( )( ) (4) The theory described by action (4) possesses an Abelian, off-shell, secondorder reducible generating set of gauge transformations (9), (3). The field, ghost, and antifield spectra have been introduced in () (3) and (4) (7). The solution to the master equation takes the form ha ShA,,ψ,,ψ = S h A d x( h ν ν, νρ,ψ ( η ν) ψ ξ (43) A νρ C C ν C C C. [ νρ] [ ν] For this situation, the first-order deformation of the solution to the classical master equation can be written as ha,,ψ ha hψ Aψ ψ h A S = S S S S S S. (44) The consistency of the first-order deformation requires that the constants k, k, k, q, m and Λ should satisfy the following algebraic equations k k =, 8m kλ=, 3 (45) k ( k ) =, k kk =, 3 (46) k mk =, k q ( ) 3 =, (47) 3 ( ) ( ) k k =, k k k =. (48) There are two types of nontrivial solutions, namely k = or k =, k = k = m=, Λ, q= arbitrary, (49) 4k, k = k =, k i, =±, q, =, m= =Λ. (5) 8 4 ( ) )

9 Cohomological derivation of D =, N = SUGRA 67 The former type is less interesting from the point of view of interactions since it maximally allows the graviton to be coupled to the 3-form (if k = ). For this reason in the sequel we will extensively focus on the latter solution, (5), which forbids both the presence of the cosmological term for the spin- field and the appearance of gravitini mass constant. The previous results can be summarized in the following theorem. Theorem 4. Under the working assumptions, the only consistent deformation of (4) involving a spin- field, an Abelian three-form gauge field and a massless spin-3/ field reads as ha,,ψ PF 3F RS ν νρ ν νρ S [ h, A,ψ ] = S [ h ] S [ A ] S [ ψ ] λ dx HE F Fνρλ νρλ h 4 4! F Fνρσhλ F 3 4 ( Aνρσhλ νρλ σ νρλ σ )! ε q A F F α k i Fνρλ βfνρλ ψ 4 γνρψ λ ψ γαβνρλψ 4! i ρ ν 4 ( νρ λ ) [ h ] h νρ ψ γψ σ γψ λ ν ρ ψ γ ψ ν ρ νρ λ ( ) h νρ λ ψγ ψρ νλ ψ [ νhρ] λ O γ ψ ( λ ) and it is invariant under the gauge transformations, h ( ) h ρ ρ ( ) ( h ρ ε ν ν ρ ν ν) ρ ρhν ( ν) 8 (5) δ = λ i εγ ψ..., (5) ( ) δ ψ = ελ ε ψ ψ γ ψ (53), hν α ν αβ ε ν α [ ν] [ α β] 8 ik i γαβε [ h νρλ ] F Fνρλσ α β γ ε νρλ γνρλσε..., δ = ε λ δ, A δ [ ] A A δ σ ε αβγ α βγ δ αβγ [ ] [ σ δ] αβ γ δ ( ) ε ε iξ γ ψ..., which remain second-order reducible. [ h δ αβ γ] δ [ α βhγ] δ k [ αβ γ] (54) The effective expression of the second-order deformation and the interpretation of the interacting theory are exposed in [5].

10 68 E. M. Cioroianu, E. Diaconu, S. C. Sãraru Acknowledgements. 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-4-54 and by the grant AT4/5 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. G. Barnich, M. Henneaux, Phys. Lett. B 3, 3 (993).. E. M. Cioroianu, E. Diaconu, S. C. Sararu, On the uniqueness of D = interactions among a graviton, a massless gravitino and a three-form. I: Pauli-Fierz and three-form, Accepted by IJMPA. 3. E. M. Cioroianu, E. Diaconu, S. C. Sararu, On the uniqueness of D = interactions among a graviton, a massless gravitino and a three-form. II: Three-form and gravitini, Accepted by IJMPA. 4. E. M. Cioroianu, E. Diaconu, S. C. Sararu, On the uniqueness of D = interactions among a graviton, a massless gravitino and a three-form. III: Graviton and gravitini, Accepted by IJMPA. 5. E. M. Cioroianu, E. Diaconu, S. C. Sararu, On the uniqueness of D = interactions among a graviton, a massless gravitino and a three-form. IV: Putting things together, Accepted by IJMPA.