Properties of an alternative off-shell formulation of 4D supergravity
aa r X i v : . [ h e p - t h ] J a n Properties of an alternative off-shell formulation of 4Dsupergravity
Friedemann Brandt
Institut f¨ur Theoretische Physik, Leibniz Universit¨at Hannover, Appelstraße 2,30167 Hannover, Germany
Abstract
This article elaborates on an off-shell formulation of D=4, N=1 super-gravity whose auxiliary fields comprise an antisymmetric tensor field withoutgauge degrees of freedom. In particular, the relation to new minimal super-gravity, a supercovariant tensor calculus and the construction of invariantactions including matter fields are discussed.
In the basic formulation [1, 2] of pure D=4, N=1 supergravity the commutatoralgebra of local symmetry transformations closes only on-shell. This complicatesvarious computations, such as the construction of couplings of the supergravitymultiplet (i.e. the supersymmetry multiplet with the vierbein and the gravitino) tomatter multiplets, of locally supersymmetric invariants with higher derivatives andof Faddeev-Popov terms. Fortunately there are off-shell formulations of the theorywith auxiliary fields that close the algebra of local symmetry transformations off-shell.The best-known off-shell formulations of pure D=4, N=1 supergravity are the so-called old minimal formulation [3, 4] and the so-called new minimal formulation[5]. The auxiliary fields of the new minimal supergravity multiplet are a real 2-form gauge potential and a real vector field which is the gauge field of local R-transformations (“R-gauge field”). This auxiliary field content of new minimalsupergravity hinders the algebraic elimination of the auxiliary fields because theequations of motion for these fields only contain the field strengths of the 2-formgauge potential and of the R-gauge field, i.e. derivatives of the auxiliary fields.The subject of this article is an off-shell formulation [6, 7] of D=4, N=1 supergravitywhich overcomes this obstacle of new minimal supergravity. This formulation is aconsistent deformation of new minimal supergravity coupled to an abelian gaugemultiplet wherein the 2-form gauge potential mutates into an ordinary auxiliary fieldwithout gauge degrees of freedom and the R-gauge field mutates from an auxiliaryfield into a physical gauge field. The physical fields of the supergravity multipletof this formulation are the vierbein, the R-gauge field, the gravitino and a spin-1/2 field, the auxiliary fields are a real antisymmetric two-component tensor fieldwithout gauge degrees of freedom and a real scalar field. This supergravity multiplet,according to the usual counting, has off-shell 16 bosonic degrees of freedom (6 fromthe vierbein, 3 from the R-gauge field, 6 from the auxiliary antisymmetric tensor1eld, 1 from the auxiliary scalar field) and 16 fermionic degrees of freedom (12 fromthe gravitino, 4 from the spin-1/2 field), and on-shell 4 bosonic degrees of freedom(from the vierbein and the R-gauge field) and 4 fermionic degrees of freedom (fromthe gravitino and the spin-1/2 field). Elimination of the auxiliary fields providesa supergravity model as in [8] with gauged R-symmetry and spontaneously brokensupersymmetry. A similar model has been found recently in [9].In [6, 7] it was noted already that the coupling of the supergravity multiplet to othersupersymmetry multiplets (“matter multiplets”) is analogous to the coupling of thenew minimal supergravity multiplet to these multiplets. However, details were notgiven. The present paper is committed to providing these details and elaborating onrelated features of the theory. In section 2 the model of [6, 7] is revisited. In section3 a supercovariant tensor calculus is presented and the symmetry transformationsof the component fields of matter multiplets are given explicitly as nilpotent Becchi-Rouet-Stora-Tyutin (BRST) transformations. In section 4 locally supersymmetricactions for the matter multiplets are constructed and the elimination of the auxiliaryfields is discussed. Section 5 contains a brief discussion of the results.The conventions used here are the same as in [10] and differ from those of [11] basi-cally only in the choice of the Minkowski metric which is η ab ∼ (1 , − , − , − Our starting point is the deformation [6, 7] of new minimal supergravity coupled toan abelian gauge multiplet. The fields of that supergravity model are the compo-nents fields of the new minimal supergravity multiplet which are the vierbein e µa ,the gravitino ψ µ , a real 2-form gauge potential with components t µν = − t νµ anda real R-gauge field A ( r ) µ , and the component fields of the abelian gauge multipletwhich are a real gauge field A µ , a complex spinor field λ and an auxiliary real scalarfield D . The Lagrangian L SG for these fields derived in [6, 7] reads: L SG = M P l L + g − L + g L (2.1) L /e = R − ε µνρσ ( ψ µ σ ν ∇ ρ ¯ ψ σ + c.c. ) − H µ H µ − ε µνρσ A ( r ) µ ∂ ν t ρσ (2.2) L /e = − ( F µν + g t µν )( F µν + g t µν ) + D − g + λσ µ ¯ λ H µ − [ iλσ µ ∇ µ ¯ λ + ig ψ µ σ µ ¯ λ + ε µνρσ ( F µν + g t µν ) ψ ρ σ σ ¯ λ − ψ µ σ µν ψ ν ¯ λ ¯ λ + c.c. ] (2.3) L /e = D + λσ µ ¯ ψ µ + ψ µ σ µ ¯ λ + ε µνρσ ( A µ ∂ ν t ρσ + g t µν t ρσ ) (2.4)where e = det( e µa ) (2.5)2 µνρσ = e µa e ν b e ρc e σd ε abcd (2.6) R = 2 e νa e µb ( ∂ [ µ ω ν ] ab − ω [ µca ω ν ] cb ) (2.7) H µ = ε µνρσ ( ∂ ν t ρσ + iψ ν σ ρ ¯ ψ σ ) (2.8) F µν = 2( ∂ [ µ A ν ] + iλσ [ µ ¯ ψ ν ] + iψ [ µ σ ν ] ¯ λ ) (2.9) ∇ µ ψ ν = ∂ µ ψ ν − ω µab ψ ν σ ab − iA ( r ) µ ψ ν (2.10) ∇ µ λ = ∂ µ λ − ω µab λσ ab − iA ( r ) µ λ (2.11) ω µab = e νa e ρb ( ω [ µν ] ρ − ω [ νρ ] µ + ω [ ρµ ] ν ) (2.12) ω [ µν ] ρ = e ρa ∂ [ µ e ν ] a − iψ µ σ ρ ¯ ψ ν + iψ ν σ ρ ¯ ψ µ . (2.13) M P l , g , g und g are real coupling constants. e µa denotes the inverse vierbeinfulfilling e µa e ν a = δ µν , e µb e µa = δ ab . The three contributions L , L and L to the Lagrangian (2.1) are separately in-variant up to a total divergence, respectively, under general coordinate transfor-mations, local supersymmetry transformations, local Lorentz transformations, localR-transformations and local gauge transformations of t µν and A µ . The local super-symmetry transformations and the gauge transformations of t µν and A µ are, writtenas part ˆ s of the BRST transformations of these fields:ˆ se µa = 2 i ( ξσ a ¯ ψ µ − ψ µ σ a ¯ ξ ) (2.14)ˆ sψ µ = ∂ µ ξ − ω µab ξσ ab − iA ( r ) µ ξ − iξH µ − iξσ µν H ν (2.15)ˆ st µν = 2 ∂ [ ν Q µ ] − i ( ξσ [ µ ¯ ψ ν ] + ψ [ µ σ ν ] ¯ ξ ) (2.16)ˆ sA ( r ) µ = ξσ µ ¯ S + Sσ µ ¯ ξ (2.17)ˆ sA µ = ∂ µ C − iξσ µ ¯ λ + iλσ µ ¯ ξ + g Q µ (2.18)ˆ sλ = ξ ( g − iD ) − ξσ µν ( F µν + g t µν ) (2.19)ˆ sD = ξσ µ (cid:2) ∇ µ ¯ λ − ¯ ψ µ ( iD + g ) − ¯ σ νρ ¯ ψ µ ( F νρ + g t νρ ) + i ¯ λH µ (cid:3) + c.c. (2.20)where S in (2.17) is the spin- portion of the supercovariant gravitino field strength, S = 2( ∇ µ ψ ν ) σ µν + i ψ µ H µ , ¯ S = − σ µν ∇ µ ¯ ψ ν − i ¯ ψ µ H µ . (2.21) ξ α are ghosts of local supersymmetry transformations, Q µ are ghosts of reduciblegauge transformations of t µν and C is a ghost of local gauge transformations of A µ .The Lagrangian (2.1) and the symmetry transformations given above are deforma-tions of the Lagrangian and symmetry transformations of new minimal supergravitycoupled to A µ , λ and D with deformation parameters g and g . Only the symmetrytransformations of A µ , λ and D are deformed by the g -dependent terms in (2.18),(2.19) and (2.20).Now, a first observation is that the dependence on g of the symmetry transforma-tions can be completely removed by the following rescalings of fields: A ′ µ = g − A µ , λ ′ = g − λ, D ′ = g − D, C ′ = g − C. (2.22)3 second observation is that for g = 0 the Lagrangian (2.1), up to a total divergence,and the symmetry transformations depend on t µν and A µ only via the combination b µν = t µν + ∂ µ A ′ ν − ∂ ν A ′ µ (2.23)because A ′ µ does not contribute to H µ when H µ is written in terms of b µν and theterms in L depending on t µν are equal to g eε µνρσ b µν b ρσ up to a total divergence.Redefining also the deformation parameters as g ′ = ( g /g ) , g ′ = g g (2.24)the Lagrangian (2.1) can for g = 0 be written, up to a total divergence, as L SG = M P l L + g ′ L ′ + g ′ L ′ (2.25) L /e = R − ε µνρσ ( ψ µ σ ν ∇ ρ ¯ ψ σ + c.c. ) − H µ H µ − ε µνρσ A ( r ) µ ∂ ν b ρσ (2.26) L ′ /e = − b µν b µν + D ′ − + λ ′ σ µ ¯ λ ′ H µ + 2( λ ′ σ [ µ ¯ ψ ν ] )( ψ µ σ ν ¯ λ ′ ) − ( iλ ′ σ µ ∇ µ ¯ λ ′ + iψ µ σ µ ¯ λ ′ + 2 ib µν ψ µ σ ν ¯ λ ′ + ε µνρσ b µν ψ ρ σ σ ¯ λ ′ + ψ µ σ µν ψ ν ¯ λ ′ ¯ λ ′ − ψ µ ψ µ ¯ λ ′ ¯ λ ′ + c.c. ) (2.27) L ′ /e = D ′ + λ ′ σ µ ¯ ψ µ + ψ µ σ µ ¯ λ ′ + ε µνρσ b µν b ρσ (2.28)with H µ now defined in terms of b µν according to H µ = ε µνρσ ( ∂ ν b ρσ + iψ ν σ ρ ¯ ψ σ ) . (2.29)The fields in the Lagrangian (2.25) are the vierbein e µa , the gravitino ψ µ , the R-gauge field A ( r ) µ , the spin-1/2 field λ ′ and the auxiliary fields b µν and D ′ . Thecomplete BRST transformations of these fields are: se µa = C ν ∂ ν e µa + ( ∂ µ C ν ) e νa + C ba e µb + 2 i ( ξσ a ¯ ψ µ − ψ µ σ a ¯ ξ ) (2.30) sψ µ = C ν ∂ ν ψ µ + ( ∂ µ C ν ) ψ ν + C ab ψ µ σ ab + iC ( r ) ψ µ + ∂ µ ξ − ω µab ξσ ab − iA ( r ) µ ξ − iξH µ − iξσ µν H ν (2.31) sA ( r ) µ = C ν ∂ ν A ( r ) µ + ( ∂ µ C ν ) A ( r ) ν + ∂ µ C ( r ) + ξσ µ ¯ S + Sσ µ ¯ ξ (2.32) sλ ′ = C µ ∂ µ λ ′ + C ab λ ′ σ ab + iC ( r ) λ ′ + ξ ( − iD ′ ) − ξσ µν b µν (2.33) sb µν = C ρ ∂ ρ b µν + 2 b ρ [ ν ∂ µ ] C ρ − i (cid:0) ∂ [ µ [ ξσ ν ] ¯ λ ′ ] + ξσ [ µ ¯ ψ ν ] − c.c. (cid:1) (2.34) sD ′ = C µ ∂ µ D ′ + (cid:0) ξσ µ [ ∇ µ ¯ λ ′ − ¯ ψ µ ( + iD ′ ) − ¯ σ νρ ¯ ψ µ b νρ + i ¯ λ ′ H µ ] + c.c. (cid:1) (2.35)where S and ¯ S are as in (2.21) with H µ as in (2.29). C µ are ghosts of generalcoordinate transformations, C ab = − C ba are ghosts of local Lorentz transformationsand C ( r ) is a ghost of local R-transformations. The ghosts C µ , C ab and C ( r ) arereal and Graßmann odd, the supersymmetry ghosts ξ α are complex and Graßmanneven, with ¯ ξ ˙ α denoting the complex conjugate of ξ α . The BRST transformations ofthe ghosts are sC µ = C ν ∂ ν C µ + 2 iξσ µ ¯ ξ (2.36) sξ = C µ ∂ µ ξ + C ab ξσ ab + iC ( r ) ξ − iξσ µ ¯ ξ ψ µ (2.37)4 C ab = C µ ∂ µ C ab + C ca C cb − iξσ µ ¯ ξ ω µab + 2 iε abcd ξσ c ¯ ξ H d (2.38) sC ( r ) = C µ ∂ µ C ( r ) − iξσ µ ¯ ξ A ( r ) µ . (2.39)In the BRST transformations (2.30) through (2.39) all spinor fields have upper spinorindices. These transformations are strictly nilpotent off-shell ( s = 0), i.e. the alge-bra of the corresponding local symmetry transformations closes off-shell. The gaugefield A µ and the ghosts Q µ and C have completely disappeared from the theory,along with the corresponding gauge symmetries. We shall use this formulation inthe following analysis. Of course one can return to the formulation with Lagrangian(2.1) by undoing the field redefinitions (2.22) and (2.23). The Lagrangian (2.25)is quite similar to the one given in equations (4.16) and (4.17) of [9]. Apart fromdifferent conventions, (2.25) seems to differ from equations (4.16) and (4.17) of [9]basically only in the use of b µν instead of its supercovariant counterpart B ab givenbelow in equation (3.4), and in the term λ ′ σ µ ¯ λ ′ H µ in (2.27) which seems not tohave a counterpart in [9] but is needed in order that (2.27) is invariant off-shell upto a total divergence under the BRST transformations (2.30) through (2.35). In order to couple matter multiplets to the supergravity multiplet and to constructsupersymmetric actions for these multiplets we use a supercovariant tensor calculus.The calculus comprises supercovariant derivatives D a , spinorial anti-derivations D α ,¯ D ˙ α and generators δ I of a structure group which are realized on supercovarianttensors (see below) and fulfill the graded commutator algebra[ D A , D B } = − T AB C D C − F ABI δ I , [ δ I , D A ] = − g IAB D B , [ δ I , δ J ] = f IJ K δ K (3.1)where the index A of D A runs over Lorentz vector indices a and spinor indices α, ˙ α .[ D A , D B } denotes the commutator [ D A , D B ] if A or B is a Lorentz vector indexand the anticommutator {D A , D B } if both A and B are spinor indices. The f IJ K denote structure constants of the Lie algebra G of the structure group which isthe direct sum of the Lorentz algebra and a further reductive Lie algebra which atleast comprises the generator δ ( r ) of R-transformations and may comprise furthergenerators δ i of a Yang-Mills gauge group with or without abelian factors. Denotingthe generators of the Lorentz algebra by l ab = − l ba , we have {D A } = {D a , D α , ¯ D ˙ α } , { δ I } = { l ab , δ ( r ) , δ i } . The sum over indices of G is defined with a factor 1/2 for the Lorentz generators,such as F ABI δ I = F ABab l ab + F AB ( r ) δ ( r ) + F ABi δ i and the sum over indices A, B, . . . is defined with upper first spinor index, such as T ABC D C = T ABc D c + T ABγ D γ + T AB ˙ γ ¯ D ˙ γ . IAB are the entries of a matrix g I which represents δ I on the D A . The only nonva-nishing g IAB occur for δ I ∈ { l ab , δ ( r ) } with[ l ab , D c ] = η bc D a − η ac D b , [ l ab , D α ] = − σ abαβ D β , [ l ab , ¯ D ˙ α ] = ¯ σ ab ˙ β ˙ α ¯ D ˙ β (3.2)[ δ ( r ) , D a ] = 0 , [ δ ( r ) , D α ] = − i D α , [ δ ( r ) , ¯ D ˙ α ] = i ¯ D ˙ α . (3.3)The matter multiplets treated here are chiral multiplets [12], super-Yang-Mills mul-tiplets (in WZ gauge) [13, 14, 15] and linear multiplets [16]. The component fieldsof the chiral multiplets are denoted by φ m , χ mα , F m and their complex conjugates¯ φ ¯ m , ¯ χ ¯ m ˙ α , ¯ F ¯ m where φ m , F m are complex scalar fields and χ m are complex spinorfields. The component fields of the super-Yang-Mills multiplets are denoted by A iµ , λ iα , ¯ λ i ˙ α , D i where A iµ are real gauge fields, D i are real scalar fields, λ i are complexspinor fields and ¯ λ i is the complex conjugate of λ i . The component fields of thelinear multiplets are denoted by ϕ M , A Mµν , ψ Mα , ¯ ψ M ˙ α where ϕ M are real scalar fields, A Mµν = − A Mνµ are real components of 2-form gauge potentials, ψ M are complex spinorfields and ¯ ψ M is the complex conjugate of ψ M .The supercovariant tensors which the supercovariant algebra (3.1) is realized on are φ m , χ m , F m , ¯ φ ¯ m , ¯ χ ¯ m , ¯ F ¯ m , λ i , ¯ λ i , D i , ϕ M , ψ M , ¯ ψ M , λ ′ , ¯ λ ′ , B ab , D ′ , H a = e µa H µ , T abα , ¯ T ab ˙ α , F abI , L Ma and supercovariant derivatives of these tensors, with H µ as in(2.29) and B ab , T abα , ¯ T ab ˙ α , F abI and L Ma given by: B ab = e µa e νb ( b µν + 2 iλ ′ σ [ µ ¯ ψ ν ] + 2 iψ [ µ σ ν ] ¯ λ ′ ) (3.4) T abα = 2 e µa e νb ( ∇ [ µ ψ ν ] − iH [ µ ψ ν ] + iH ρ ψ [ µ σ ν ] ρ ) α (3.5)¯ T ab ˙ α = 2 e µa e νb ( ∇ [ µ ¯ ψ ν ] + iH [ µ ¯ ψ ν ] + iH ρ ¯ σ ρ [ µ ¯ ψ ν ] ) ˙ α (3.6) F abcd = 2 e µa e νb [ ∂ [ µ ω ν ] cd − ω [ µec ω ν ] ed + iψ [ µ ( σ ν ] ¯ T cd − σ [ c ¯ T d ] ν ] )+ i ( T cd σ [ µ + 2 T [ c [ µ σ d ] ) ¯ ψ ν ] − iψ [ µ σ e ¯ ψ ν ] ε cdef H f ] (3.7) F ab ( r ) = 2 e µa e νb ( ∂ [ µ A ( r ) ν ] − ψ [ µ σ ν ] ¯ S + Sσ [ µ ¯ ψ ν ] ) (3.8) F abi = 2 e µa e νb ( ∂ [ µ A iν ] + f jki A jµ A kν + iψ [ µ σ ν ] ¯ λ i + iλ i σ [ µ ¯ ψ ν ] ) (3.9) L Ma = e µa ε µνρσ ( ∂ ν A Mρσ − ψ ν σ ρσ ψ M + ¯ ψ M ¯ σ ρσ ¯ ψ ν − iϕ M ψ ν σ ρ ¯ ψ σ ) + 2 ϕ M H a = e µa ε µνρσ ( ∂ ν A Mρσ + ϕ M ∂ ν b ρσ − ψ ν σ ρσ ψ M + ¯ ψ M ¯ σ ρσ ¯ ψ ν ) . (3.10) δ ( r ) is represented on supercovariant tensors according to table 1 where r m are realconstants (“R-charges” of the φ m ). For the respective complex conjugate superco-variant tensors we have δ ( r ) ¯Φ = δ ( r ) Φ where ¯Φ denotes the complex conjugate of Φ,and δ ( r ) Φ denotes the complex conjugate of δ ( r ) Φ. Real supercovariant tensors, suchas D i , ϕ M , B ab , D ′ , F abI and L Ma , have vanishing R -transformation.Φ φ m χ m F m λ i ψ M λ ′ T abα δ ( r ) Φ ir m φ m i ( r m − χ m i ( r m − F m iλ i − iψ M iλ ′ iT abα Table 16 α is realized on supercovariant tensors according to table 2 with (in the last row) T αaβ , F α ˙ αcd and F aαcd as in equations (3.13), (3.15) and (3.16), respectively.Φ D α Φ φ m χ mα χ mβ ε βα F m F m φ m χ m ˙ α − i D α ˙ α ¯ φ m ¯ F m (2 i D α ˙ α + H α ˙ α ) ¯ χ m ˙ α − λ iα δ i − iS α δ ( r ) ) ¯ φ m λ iβ − iδ βα D i − σ abαβ F abi ¯ λ i ˙ α D i ( D α ˙ α + i H α ˙ α )¯ λ i ˙ α F abi − i D [ a ¯ λ i ˙ α σ b ] α ˙ α + H [ a σ b ] α ˙ α ¯ λ i ˙ α − iε abcd H c σ dα ˙ α ¯ λ i ˙ α ϕ M ψ Mα ψ Mβ ψ M ˙ α − i D α ˙ α ϕ M − L Mα ˙ α L Ma iσ abαβ D b ψ Mβ + ψ Mα H a λ ′ β δ βα ( − iD ′ ) − σ abαβ B ab ¯ λ ′ ˙ α D ′ ( D α ˙ α + i H α ˙ α )¯ λ ′ ˙ α B ab − i D [ a ¯ λ ′ ˙ α σ b ] α ˙ α + H [ a σ b ] α ˙ α ¯ λ ′ ˙ α − iε abcd H c σ dα ˙ α ¯ λ ′ ˙ α H a i ε abcd σ bα ˙ α ¯ T cd ˙ α F ab ( r ) D [ a ¯ S ˙ α σ b ] α ˙ α + iH [ a σ b ] α ˙ α ¯ S ˙ α + ε abcd H c σ dα ˙ α ¯ S ˙ α T abβ − iδ βα ( F ab ( r ) + 2 D [ a H b ] ) − F abcd σ cdαβ − i D a H c σ bcαβ + i D b H c σ acαβ − H c H [ a σ bc ] αβ ¯ T ab ˙ α F abcd − D [ a F b ] αcd + 2 T α [ aβ F b ] βcd + ¯ T ab ˙ α F α ˙ αcd Table 2 S α and ¯ S ˙ α are as in (2.21) and are the spin-1/2 parts of T abα and ¯ T ab ˙ α : S α = T abβ σ abβα , ¯ S ˙ α = − ¯ σ ab ˙ α ˙ β ¯ T ab ˙ β . (3.11)¯ D ˙ α is obtained from D α by complex conjugation, using¯ D ˙ α ¯ T = ( − ) |T | D α T where |T | denotes the Graßmann parity of T .The nonvanishing T ABC and F ABI in (3.1) are T abα , T ab ˙ α = − ¯ T ab ˙ α and F abI given inequations (3.5) through (3.9), and the T ABC and F ABI given by T α ˙ βc = T ˙ βαc = 2 iσ cα ˙ β (3.12) T αbγ = − T bαγ = − iδ γα H b + iH c σ cbαγ (3.13) T ˙ αb ˙ γ = − T b ˙ α ˙ γ = iδ ˙ γ ˙ α H b + iH c ¯ σ cb ˙ γ ˙ α (3.14)7 α ˙ βcd = F ˙ βαcd = 2 iε abcd σ aα ˙ β H b (3.15) F αbcd = − F bαcd = − i ¯ T cd ˙ α σ b α ˙ α + 2 iσ [ cα ˙ α ¯ T d ] b ˙ α (3.16) F ˙ αbcd = − F b ˙ αcd = iT cdα σ b α ˙ α − iσ [ cα ˙ α T d ] bα (3.17) F αb ( r ) = − F bα ( r ) = σ bα ˙ α ¯ S ˙ α (3.18) F ˙ αb ( r ) = − F b ˙ α ( r ) = σ bα ˙ α S α (3.19) F αbi = − F bαi = − iσ bα ˙ α ¯ λ i ˙ α (3.20) F ˙ αbi = − F b ˙ αi = iσ bα ˙ α λ iα . (3.21)The supercovariant derivative D a is defined on supercovariant tensors T accordingto D a T = e µa ( ∂ µ − ω µab l ab − A ( r ) µ δ ( r ) − A iµ δ i − ψ αµ D α + ¯ ψ ˙ αµ ¯ D ˙ α ) T . (3.22) D a H a and D a L Ma fulfill the identities D a H a = 0 , D a L Ma = 2 H a D a ϕ M − iSψ M + i ¯ ψ M ¯ S. (3.23) D [ a B cd ] and H a are related by D [ a B cd ] = ε abcd H d + iT [ ab σ c ] ¯ λ ′ − iλ ′ σ [ a ¯ T bc ] . (3.24)For later purpose we remark that − iS plays the role of the gaugino of R-transformations,cf. equations (3.19) and (3.21), and that one has D α S β = δ βα ( − F abba + H a H a ) − iF ab ( r ) σ abαβ (3.25)which shows that − F abba + H a H a plays the role of the D -field of R-transformations.The BRST transformation s T of a supercovariant tensor T is s T = ( C µ ∂ µ + ξ α D α + ¯ ξ ˙ α ¯ D ˙ α + C I δ I ) T = ( ˆ ξ A D A + ˆ C I δ I ) T (3.26)wherein C i are ghosts of Yang-Mills gauge transformations and ˆ ξ A and ˆ C I are “co-variant ghosts” given byˆ ξ a = C µ e µa , ˆ ξ α = ξ α + C µ ψ αµ , ˆ ξ ˙ α = ¯ ξ ˙ α − C µ ¯ ψ ˙ αµ (3.27)ˆ C ab = C ab + C µ ω µab , ˆ C ( r ) = C ( r ) + C µ A ( r ) µ , ˆ C i = C i + C µ A iµ . (3.28)The BRST transformations of A iµ and A Mµν are sA iµ = C ν ∂ ν A iµ + ( ∂ µ C ν ) A iν + ∂ µ C i + f jki A jµ C k − iξσ µ ¯ λ i + iλ i σ µ ¯ ξ (3.29) sA Mµν = C ρ ∂ ρ A Mµν + ( ∂ µ C ρ ) A Mρν + ( ∂ ν C ρ ) A Mµρ + ∂ ν Q Mµ − ∂ µ Q Mν +2( ξσ µν ψ M − ¯ ψ M ¯ σ µν ¯ ξ ) + 4 i ( ψ [ µ σ ν ] ¯ ξ + ξσ [ µ ¯ ψ ν ] ) ϕ M (3.30)where Q Mµ are real Graßmann odd ghosts of reducible gauge transformations of A Mµν .The BRST transformations of the ghosts C i and Q Mµ are sC i = C µ ∂ µ C i + f jki C k C j − iξσ µ ¯ ξ A iµ (3.31)8 Q Mµ = ∂ µ Q M + ( ∂ µ C ν ) Q Mν − iξσ ν ¯ ξ A Mµν + 2 iξσ µ ¯ ξ ϕ M (3.32)where Q M are purely imaginary Graßmann even “ghosts for ghosts” with ghostnumber 2 whose BRST transformations are sQ M = C µ ∂ µ Q M − iξσ µ ¯ ξ Q Mµ . (3.33)Covariant ghosts for ghosts ˆ Q M are defined analogously to (3.27) and (3.28) accord-ing to ˆ Q M = Q M + C µ Q Mµ + C µ C ν A Mµν . (3.34)The BRST transformations of the covariant ghosts and ghosts for ghosts are s ˆ ξ A = − ( − ) | B | ˆ ξ B ˆ ξ C T CBA + ˆ C I g IBA ˆ ξ B (3.35) s ˆ C I = − ( − ) | A | ˆ ξ A ˆ ξ B F BAI + f KJ I ˆ C J ˆ C K (3.36) s ˆ Q M = ˆ ξ a ˆ ξ b ˆ ξ c ε abcd ( L Md − H d ϕ M ) − i ˆ ξ α ˆ ξ α ˙ α ˆ ξ ˙ α ϕ M + ˆ ξ a ˆ ξ b ( ˆ ξ α σ abαβ ψ Mβ − ¯ ψ M ˙ α ¯ σ ab ˙ α ˙ β ˆ ξ ˙ β ) (3.37)where | a | = 0 and | α | = | ˙ α | = 1.The BRST transformations given above are strictly nilpotent off-shell. As is reca-pitulated in appendix A, the off-shell nilpotency of s ( s = 0) on all fields (includingthe ghosts) except on A Mµν , Q Mµ and Q M , and the construction of T abα , ¯ T ab ˙ α and F abI according to equations (3.5) through (3.9) can be deduced elegantly from the super-covariant algebra (3.1) and the corresponding Bianchi identities. The nilpotency of s on A Mµν , Q Mµ and Q M and the construction of L Ma according to equation (3.10) canbe checked separately. Furthermore the identities (3.23) can be checked explicitly.
The supercovariant algebra (3.1) and the way it is realized on the matter multipletsare exactly the same as in new minimal supergravity. In particular the additionalfields λ ′ and D ′ do not occur in the supersymmetry transformations of any compo-nent field of the matter multiplets, and the field b µν contributes to these supersym-metry transformations only via H µ given in (2.29), precisely as the 2-form gaugepotential t µν in new minimal supergravity. In addition the supercovariant algebra(3.1) is realized off-shell also on λ ′ , D ′ and on the supercovariant tensor B ab given in(3.4). For these reasons one can adopt methods and results derived in [17, 18, 19, 10]for new minimal supergravity to construct locally supersymmetric actions involvingthe matter multiplets and the fields λ ′ and D ′ in the present theory. In other words, one can check explicitly that s squares to zero on A Mµν , Q Mµ and Q M and thatthe BRST transformation of A Mµν given in (3.30) and the BRST transformations of ϕ M and ψ M arising from (3.26) imply that L Ma defined according to equation (3.10) transforms according toequation (3.26) with D α L Ma as in table 2. The BRST transformations of A Mµν , ϕ M and ψ M andthe construction of L Ma are compatible with equations (2.4) of [17] and (3.2) and (3.5) of [18].
9n particular the results derived in [19, 10] for invariant actions in new minimalsupergravity can be extended straightforwardly to the presence of linear multipletsand of the additional supercovariant tensors λ ′ , D ′ and B ab . One obtains thatLagrangians which are invariant off-shell up to a total divergence, respectively, underthe BRST transformations given in sections 2 and 3 are L /e = ( ¯ D − iψ µ σ µ ¯ D + 16 ψ µ σ µν ψ ν )[ A ( ¯ φ, ¯ λ, ¯ λ ′ , ¯ S, ¯ W ) + D B ( T )] + c.c. (4.1) L /e = µ i a ( D i a + λ i a σ µ ¯ ψ µ + ψ µ σ µ ¯ λ i a + ε µνρσ A i a µ ∂ ν b ρσ ) (4.2) L /e = κ i a M [ ε µνρσ A i a µ ∂ ν A Mρσ − ϕ M D i a + ( iλ i a ψ M − ϕ M λ i a σ µ ¯ ψ µ + c.c. )] (4.3)where in (4.2) and (4.3) µ i a and κ i a M denote real coupling constants and the sumover i a runs over the abelian factors of G including the R-transformation with theidentifications (cf. text around equation (3.25)) λ ( r ) α ≡ − iS α , ¯ λ ( r )˙ α ≡ i ¯ S ˙ α , D ( r ) ≡ − F abba + H a H a . (4.4)In (4.1) we used the notation¯ D = ¯ D ˙ α ¯ D ˙ α , D = D α D α , ¯ W ˙ α ˙ β ˙ γ = − ¯ T ab ( ˙ α ¯ σ ab ˙ β ˙ γ ) (4.5)and A ( ¯ φ, ¯ λ, ¯ λ ′ , ¯ S, ¯ W ) denotes any function of the supercovariant tensors ¯ φ m , ¯ λ i ˙ α , ¯ λ ′ ˙ α ,¯ S ˙ α and ¯ W ˙ α ˙ β ˙ γ (but not of supercovariant derivatives thereof) which has R -charge − δ I , and B ( T ) is any funtion of superco-variant tensors which is invariant under all δ I , δ ( r ) A = − i A , ∀ I = ( r ) : δ I A = 0 , ∀ I : δ I B = 0 . (4.6) L is a generic Lagrangian which provides, amongst others, a standard locally su-persymmetric Yang-Mills portion arising from a contribution to A proportional to¯ λ i ¯ λ j g ij (with G -invariant metric g ij ), locally supersymmetric kinetic terms for thechiral multiplets arising from a contribution b ( φ, ¯ φ ) to B , superpotential terms forthe chiral multiplets arising from a contribution a ( ¯ φ ) to A and locally supersym-metric kinetic terms for the linear multiplets arising from a contribution c ( ϕ ) to B .In addition L provides Lagrangians with various higher derivative terms, such asfour-derivative terms with the square of the Weyl tensor arising from a contribution¯ W ¯ W to A and/or with quartic terms in the Yang-Mills field strengths arising from acontribution to B bilinear both in λ ’s and ¯ λ ’s (of course, L provides further higherderivative terms, in particular terms with more than four derivatives). Furthermore,contributions to A given by ¯ λ ′ ¯ λ ′ and i ¯ λ ′ ¯ λ ′ reproduce L ′ and L ′ given in (2.27)and (2.28), respectively. L and L are “exceptional Lagrangians” that cannot be written in the form of L .A contribution to L with i a = ( r ) contains a Fayet-Iliopoulos term eD i a and canthus contribute to supersymmetry breaking [20] and to the cosmological constant.The contribution to L with i a = ( r ) reproduces for µ ( r ) = − M P l the L -portionof the Lagrangian (2.25) as may be verified by explicitizing F abba , S and ¯ S . L cancontribute, amongst others, mass terms for component fields of linear multiplets.10n appendix B solutions of the so-called descent equations are given which correspondto L , L and L , respectively. It is easier to verify these solutions using reasoninggiven in [19] than to check the invariance of L , L and L directly.We now discuss the elimination of auxiliary fields for Lagrangians L = L + L + L with A and B of the form A = F ( ¯ φ, ¯ λ, ¯ λ ′ ) , B = G ( φ, ¯ φ, ϕ ) = G ( φ, ¯ φ, ϕ ) . (4.7)Such Lagrangians contain only terms with at most two derivatives and thus maybe termed “low energy Lagrangians”. The fields F m , ¯ F ¯ m , D i and D ′ occur in L undifferentiated and at most quadratically as can easily be checked. Hence, thesefields can be eliminated algebraically using their equations of motion. However,the direct algebraic elimination of the field b µν is hindered by terms in L that arequadratic in H µ . Such terms are present both in the contribution to L with i a = ( r )and (generically) in L because L contains, amongst others, G m ¯ D D φ m = − G m ¯ D F m = 8 iG m ( F abba − H a H a ) δ ( r ) φ m + . . . (4.8)where G m = ∂G ( φ, ¯ φ, ϕ ) ∂φ m . Now, one may remove the terms quadratic in H µ by a suitable redefinition of the R -gauge field A ( r ) µ . To show this we collect all terms in L containing A ( r ) A ( r ) , A ( r ) H or HH . A straightforward computation yields that these terms can be written as∆ L with ∆ L /e = ( µ ( r ) − G ( r ) ) H µ H µ + G ( r )( r ) A ( r ) µ A ( r ) µ + 2( µ ( r ) + G ( r ) ) A ( r ) µ H µ (4.9)where G ( r ) = 48 iG m δ ( r ) φ m , G ( r )( r ) = 32 G m ¯ n ( δ ( r ) φ m )( δ ( r ) ¯ φ ¯ n ) (4.10)with G m ¯ n = ∂ G ( φ, ¯ φ, ϕ ) ∂ ¯ φ ¯ n ∂φ m . We now make the following ansatz for a redefined R-gauge field: A ′ ( r ) µ = A ( r ) µ + mH µ . (4.11)Using (4.11) in (4.9) one obtains∆ L /e = u H µ H µ + G ( r )( r ) A ′ ( r ) µ A ′ ( r ) µ + 2( µ ( r ) + G ( r ) − mG ( r )( r ) ) A ′ ( r ) µ H µ (4.12)with u = m G ( r )( r ) − m ( µ ( r ) + G ( r ) ) + µ ( r ) − G ( r ) . (4.13) Here we used δ ( r ) G = G m δ ( r ) φ m + G ¯ m δ ( r ) ¯ φ ¯ m = 0 where G ¯ m = ∂G/∂ ¯ φ ¯ m .
11n order to remove all HH -terms from L by the redefinition (4.11) m has to bechosen such that u vanishes. Obviously this can be achieved if G ( r )( r ) vanishes: G ( r )( r ) = 0 : u = 0 ⇔ m = 3 µ ( r ) − G ( r ) µ ( r ) + G ( r ) ) . (4.14)However, G ( r )( r ) = 0 is a rather special case. For instance, G ( r )( r ) = 0 holds whenall φ m have vanishing R -charge which also implies G ( r ) = 0 and m = but forbidsa superpotential. When G ( r )( r ) does not vanish, u vanishes if (cid:18) m − µ ( r ) + G ( r ) G ( r )( r ) (cid:19) = G ( r )( r ) ( G ( r ) − µ ( r ) ) + ( µ ( r ) + G ( r ) ) ( G ( r )( r ) ) . (4.15)Now, m must be real in order that A ′ ( r ) µ is real. In order to solve (4.15) with real m the numerator on the right hand side of equation (4.15) must not be negative. Whether or not this numerator is non-negative depends on the R -charges of the φ m and is not further discussed here.When m is chosen such that u vanishes and A ′ ( r ) µ instead of A ( r ) µ is used, the La-grangian L does not contain terms which are quadratic in derivatives of b µν , andthen b µν may be eliminated algebraically using the equations of motion if F ( ¯ φ, ¯ λ, ¯ λ ′ )contains a term quadratic in ¯ λ ′ .We also remark that ( G ( r ) − µ ( r ) ) is a generally field dependent prefactor of theRiemann curvature scalar in L /e , cf. equation (4.8) (due to F abba = R + . . . ). Thisprefactor may be made field independent by a Weyl rescaling of the vierbein andcorresponding redefinitions of other fields that convert the Lagrangian from Brans-Dicke form into conventional Einstein form, cf. [18] for a detailed discussion of thesefield redefinitions in new minimal supergravity which analogously applies in our case.Furthermore we note that G m ¯ n is the “metric” in the kinetic terms eG m ¯ n ∂ µ φ m ∂ µ ¯ φ ¯ n of the φ m and ¯ φ ¯ m in L (after integration by parts). If this metric is positive definite, G ( r )( r ) is non-negative. The formulation of D=4, N=1 supergravity studied in this paper is similiar to newminimal supergravity. This by itself is not surprising as this formulation was ob-tained as a consistent deformation of new minimal supergravity. Nevertheless thedeformation has some unusual and surprising features.One of these features is that in this formulation, using the fields b µν , λ ′ and D ′ ,a Lagrangian, without or with matter fields included, in the simplest case differs G ( r ) and G ( r )( r ) are real. Indeed, G ( r )( r ) obviously is real because G is real. Furthermore δ ( r ) G = 0 implies (cf. footnote 2) G ( r ) = 24 iG m δ ( r ) φ m − iG ¯ m δ ( r ) ¯ φ ¯ m = 24 iG m δ ( r ) φ m + c.c. . Forthis reason m in (4.14) is real too. However, according to [18], G m ¯ n need not be positive definite in order that the kinetic termsfor the φ m and ¯ φ ¯ m are positive after converting the Lagrangian into Einstein form. L ′ given in (2.27). This extraportion is separately invariant up to a total divergence under local supersymmetrytransformations, and the remaining contributions to the Lagrangian are the sameas in new minimal supergravity. The reason is that the fields λ ′ and D ′ which arenot present in new minimal supergravity do not occur in the symmetry transfor-mations of other fields except in the transformations of λ ′ and D ′ themselves andin the modified supersymmetry transformation of b µν . Furthermore, even thoughthe symmetry transformations of b µν are modified as compared to new minimal su-pergravity, the symmetry transformations of H µ defined in (2.29) are not modified.Since the symmetry transformations of all other fields except λ ′ and D ′ depend on b µν at most via H µ the modification of the symmetry transformations of b µν thenhas no impact on the Lagrangian as compared to new minimal supergravity exceptfor the added extra portion.In particular this implies that one may simply add to any Lagrangian of new minimalsupergravity, without or with matter fields included, the extra portion proportionalto L ′ . The resultant theory again is an off-shell formulation of supergravity in whichnow b µν is a standard auxiliary field without gauge degrees of freedom (which may beeliminated algebraically, at least for reasonable low energy Lagrangians, cf. section4) and the R-gauge field is a physical field. Furthermore, if supersymmetry was un-broken before adding the extra portion, the inclusion of the extra portion inevitablyintroduces a cosmological constant, cf. equation (2.27), and breaks supersymmetryspontaneously (recall that sλ ′ = ξ + . . . , cf. equation (2.33), i.e. λ ′ then is agoldstino that may be eaten by the gravitino). Thus, the addition of the extra por-tion particularly provides an alternative mechanism for spontaneously breaking localsupersymmetry in new minimal supergravity, different from the familiar breakingmechanisms by Fayet-Iliopoulos terms or F -terms.On the other hand, in addition to or in place of L ′ given in (2.27) and optionally L ′ given in (2.28), one may include other terms in the Lagrangian depending on b µν , λ ′ and D ′ , such as terms arising from contributions to A in (4.7) which depend on both¯ λ ′ and some ¯ φ ¯ m . This may have a more subtle effect on the theory as compared tonew minimal supergravity and may be worth a further study. A BRST approach to off-shell supergravity
In this appendix we recapitulate briefly a BRST approach to off-shell supergravitytheories which was used also in [19, 10] and applies a general framework [21] oftreating theories with local and/or global symmetries. The approach is based on asupercovariant algebra (3.1) and corresponding Bianchi identities X ABC ◦ ( − ) | A | | C | ( D A T BC D + T ABE T EC D + F ABI g ICD ) = 0 (A.1) In addition one may include L ′ given in (2.28) but this does not make much difference. ABC ◦ ( − ) | A | | C | ( D A F BC I + T ABD F DC I ) = 0 (A.2)where X ◦ X ABC = X ABC + X BCA + X CAB denotes the cyclic sum.The supercovariant algebra (3.1) and the Bianchi identities (A.1), (A.2) are sup-posed to be realized off-shell on supercovariant tensor fields. The approach uses adifferential which is the sum of the BRST differential s and the exterior derivative d = dx µ ∂ µ and is denoted by ˜ s , ˜ s = s + d. (A.3)This differential acts on total forms and has total degree 1. A total form generallyis a sum of local p -forms with various form degrees p . The total degree is the sumof the ghost number and the form degree. Hence, a total form ˜ ω g with definite totaldegree g (“total g -form”) generally is a sum ˜ ω g = P p ω p,g − p of local p -forms ω p,g − p where ω p,g − p has ghost number g − p .The generators D A and δ I of the algebra (3.1) are related to total 1-forms constructedof the ghosts and corresponding 1-forms according to˜ ξ a = ˆ ξ a + dx µ e µa = ( C µ + dx µ ) e µa (A.4)˜ ξ α = ˆ ξ α + dx µ ψ µα = ξ α + ( C µ + dx µ ) ψ µα (A.5)˜ ξ ˙ α = ˆ ξ ˙ α − dx µ ¯ ψ µ ˙ α = ¯ ξ ˙ α − ( C µ + dx µ ) ¯ ψ µ ˙ α (A.6)˜ C I = ˆ C I + dx µ A Iµ = C I + ( C µ + dx µ ) A Iµ (A.7)where in the present case { A Iµ } = { ω µab , A ( r ) µ , A iµ } . ˜ s acts on supercovariant tensors T and on the total 1-forms ˜ ξ A and ˜ C I according to˜ s T = ( ˜ ξ A D A + ˜ C I δ I ) T (A.8)˜ s ˜ ξ A = − ( − ) | B | ˜ ξ B ˜ ξ C T CBA + ˜ C I g IBA ˜ ξ B (A.9)˜ s ˜ C I = − ( − ) | A | ˜ ξ A ˜ ξ B F BAI + f KJ I ˜ C J ˜ C K . (A.10)Using the algebra (3.1) and the Bianchi identities (A.1), (A.2) one easily checks that˜ s squares to zero on supercovariant tensors T and on the total 1-forms ˜ ξ A and ˜ C I .The ghost number 0 part of (A.8) provides the supercovariant derivatives D a ofsupercovariant tensors as it yields ∂ µ T = ( e µa D a + ψ µα D α − ¯ ψ µ ˙ α ¯ D ˙ α + A Iµ δ I ) T which can be solved for D a T and then gives equation (3.22) in the present case.The ghost number 1 part of (A.8) provides the BRST transformations (3.26) ofsupercovariant tensors.The ghost number 0 parts of (A.9) for A = α and A = ˙ α and of (A.10) provide thefield strengths (or curvatures) T abα , T ab ˙ α and F abI , respectively. For instance, theghost number 0 part of (A.9) yields for A = α∂ µ ψ να − ∂ ν ψ µα = e µa e ν b T abα + . . . T abα and then gives (3.5) in the present case. Analogouslyone obtains equations (3.6) through (3.9) from (A.9) and (A.10). The ghost number0 part of (A.9) for A = a provides analogously either T bca , or it determines ω µab when the constraint T bca = 0 is imposed. In the present case this gives ω µab as inequations (2.12) and (2.13).The ghost number 1 parts of (A.9) and of (A.10) provide the BRST transformationsof e µa , ψ µα and A Iµ . In the present case one obtains in particular equations (2.30),(2.31), (2.32) and (3.29). The ghost number 2 parts of (A.9) and of (A.10) thenprovide the BRST transformations of the ghosts C µ , ξ α and C I . In the presentcase this gives equations (3.35) and (3.36) and then, using in addition the BRSTtransformations of e µa , ψ µα and A Iµ , equations (2.36) through (2.39) and equation(3.31).As we remarked at the end of section 3, the BRST transformations of A Mµν , Q Mµ and Q M given in equations (3.30), (3.32) and (3.33) and the field strength L Ma given in equation (3.10) cannot be deduced from the algebra (3.1) in the samemanner. Nevertheless these equations can also be written in a compact form whichis analogous to (A.9) and of (A.10) and reads˜ s ˜ Q M = ˜ ξ a ˜ ξ b ˜ ξ c ε abcd ( L Md − H d ϕ M ) − i ˜ ξ a ˜ ξ α σ aα ˙ α ˜ ξ ˙ α ϕ M + ˜ ξ a ˜ ξ b ( ˜ ξ α σ abαβ ψ Mβ − ¯ ψ M ˙ α ¯ σ ab ˙ α ˙ β ˜ ξ ˙ β ) (A.11)where ˜ Q M = ˆ Q M + dx µ Q Mµ + dx µ dx ν A Mµν = Q M + ( C µ + dx µ ) Q Mµ + ( C µ + dx µ )( C ν + dx ν ) A Mµν . (A.12) B Solutions of the descent equations
In this appendix we provide solutions of the so-called descent equations that containthe Lagrangians L , L and L given in equations (4.1), (4.2) and (4.3), respectively.These descent equations read0 < p ≤ sω p, − pi + dω p − , − pi = 0 , sω , i = 0 (B.1)where ω p, − pi is a local p -form with ghost number 4 − p . The 4-form ω , i = L i d x contains the respective Lagrangian L i for i = 4 , ,