Counterpart of the Weyl tensor for Rarita-Schwinger type fields
aa r X i v : . [ h e p - t h ] A p r ITP-UH-24/16
Counterpart of the Weyl tensor for Rarita-Schwinger typefields
Friedemann Brandt
Institut f¨ur Theoretische Physik, Leibniz Universit¨at Hannover, Appelstraße 2,30167 Hannover, Germany
Abstract
In dimensions larger than 3 a modified field strength for Rarita-Schwingertype fields is constructed whose components are not constrained by the fieldequations. In supergravity theories the result provides a modified (superco-variant) gravitino field strength related by supersymmetry to the (superco-variantized) Weyl tensor. In various cases, such as for free Rarita-Schwingertype gauge fields and for gravitino fields in several supergravity theories, themodified field strength coincides on-shell with the usual field strength. A cor-responding result for first order derivatives of Dirac type spinor fields is alsopresented.
This note relates to Rarita-Schwinger type fields in D dimensions with D >
3. Thecomponents of these fields are denoted by ψ mα where m = 1 , . . . , D is a vectorindex and α = 1 , . . . , ⌊ D/ ⌋ is a spinor index, ⌊ D/ ⌋ being the largest integer ≤ D/
2. The fields are subject to field equations (equations of motion) which reador imply ∂ m ψ n Γ mnr + . . . ≈ ψ m ), ≈ denotesequality on-shell (i.e. equality whenever the field equations hold), spinor indiceshave been suppressed (as will be often done below) and Γ mnr = Γ [ m Γ n Γ r ] is thetotally antisymmetrized product of three gamma-matrices. The field equationsthus constrain the usual field strength whose components are T mnα = ∂ m ψ nα − ∂ n ψ mα + . . . (1)The main purpose of this note is a suitable definition of a modified field strengthfor Rarita-Schwinger type fields whose components are not constrained by the fieldequations. We denote the components of the modified field strength by W mnα anddefine it in terms of the usual field strength according to W mn = D − D − T mn − D − D − D − T r [ m Γ n ] r − D − D − T rs Γ rsmn , (2)where Γ mn = Γ [ m Γ n ] and Γ mnrs = Γ [ m Γ n Γ r Γ s ] . We shall now comment on thisdefinition. We use conventions as in [1]. δ ξ ψ mα = ( D m ξ ) α + . . . , (3)where ξ is an arbitrary spinor field with components ξ α (‘gauge parameters’), and( D m ξ ) α are the components of a covariant derivative of ξ defined by means of a(supercovariant) spin connection ω mnr , D m ξ = ∂ m ξ − ω mnr ξ Γ nr . (4)Accordingly, the (supercovariant) gravitino field strength T mn = D m ψ n − D n ψ m + . . . transforms under supersymmetry according to δ ξ T mn = [ D m , D n ] ξ + . . . = − R mnrs ξ Γ rs + . . . , (5)where R mnrs = ∂ m ω nrs − ∂ n ω mrs + . . . denote the components of the (supercovari-antized) Riemann tensor. This implies that the supersymmetry transformation ofthe modified gravitino field strength defined according to (2) in terms of the (super-covariant) gravitino field strength is δ ξ W mn = − C mnrs ξ Γ rs + . . . , (6)where C mnrs are the components of the (supercovariantized) Weyl tensor C mnrs = R mnrs + D − ( δ r [ m R n ] s − δ s [ m R n ] r ) − D − D − R δ r [ m δ sn ] , (7)where R mn = R mrrn and R = R mm denote the (supercovariantized) Ricci tensorand the (supercovariantized) Riemann curvature scalar, respectively. The modifiedgravitino field strength is thus related by supersymmetry to the Weyl tensor butnot to the Ricci tensor. Notice also that, like the Weyl tensor, the modified fieldstrength (2) vanishes for D = 3.In order to further discuss the modified field strength (2) we write it as W mnα = T rsβ P rsmnβα , (8)where P rsmnβ α are the entries of the matrix P rsmn = D − D − δ r [ m δ sn ] + D − D − D − ( δ r [ m Γ n ] s − δ s [ m Γ n ] r ) − D − D − Γ rsmn , (9)where is the 2 ⌊ D/ ⌋ × ⌊ D/ ⌋ unit matrix. This implies P mnrs = D − D − δ r [ m δ sn ] − D − D − D − ( δ r [ m Γ n ] s − δ s [ m Γ n ] r ) − D − D − Γ mnrs . (10)These matrices fulfill the identities P rsmn Γ mnk = 0 , Γ mnk P mnrs = 0 , (11)2 rskℓ P kℓmn = P rsmn , (12) P rsmn ⊤ = CP mnrs C − , (13)where in (13) P rsmn ⊤ denotes the transpose of P rsmn , and C and C − denote acharge conjugation matrix and its inverse which relate the gamma matrices to thetranspose gamma matrices according to Γ m ⊤ = − ηC Γ m C − with η ∈ { +1 , − } andare used to raise and lower spinor indices according to W mnα = C − αβ W mnβ , W mnα = C αβ W mnβ . (14)The first equation (11) implies that the modified field strength identically fulfills W mn Γ mnr = 0, i.e. indeed the components of the modified field strength are notconstrained by the field equations. (12) implies that the matrices P rsmn define aprojection operation on the field strength as they define an idempotent operation.Hence, this operation projects to linear combinations of components of the usualfield strength which are not constrained by the field equations. Owing to (13) thecomponents of the modified field strength with lowered spinor index are related tothe usual field strength according to W mnα = P mnrsαβ T rsβ . (15)The second equation (11) thus implies that the modified field strength with loweredspinor index identically fulfills Γ mnr W mn = 0.Moreover, the modified field strength coincides with the usual field strength on-shellwhenever T mn Γ mnr vanishes on-shell: T mn Γ mnr ≈ ⇒ W mn ≈ T mn . (16)This is obtained from (2) by means of the following implications of T mn Γ mnr ≈ T mn Γ mnr ≈ · Γ r ⇒ T mn Γ mn ≈ · Γ r ⇒ T rm Γ m ≈ · Γ s ⇒ T m [ r Γ s ] m ≈ − T rs , (17) T mn Γ mn ≈ · Γ rs ⇒ T mn Γ mnrs − T m [ r Γ s ] m − T rs ≈ (17) ⇒ T mn Γ mnrs ≈ − T rs . (18)In particular, (16) applies to free Rarita-Schwinger type gauge fields satisfying T mn Γ mnr ≈ T mn = 2 ∂ [ m ψ n ] . Hence, (16) also applies to linearized super-gravity theories. Moreover (16) applies to several supergravity theories at the full(nonlinear) level, such as to N = 1 pure supergravity in D = 4 [4, 5] and to su-pergravity in D = 11 [6], where the equations of motion imply T mn Γ mnr ≈ T mn . Accordingly, in these casesthe modified gravitino field strength fulfills on-shell the same Bianchi identities asthe usual gravitino field strength, such as D [ m W nr ] ≈ N = 1, D = 4 puresupergravity for the supercovariant derivatives of W mn .If T mn Γ mnr ≈ X r for some X r , one obtains in place of equations (17) and (18) thefollowing implications: T mn Γ mnr ≈ X r ⇒ T m [ r Γ s ] m ≈ − T rs − D − X m Γ mrs + D − D − X [ r Γ s ] , (19) See also, e.g., equations (35) of [3]. mn Γ mnrs ≈ − T rs − D − X m Γ mrs + D − D − X [ r Γ s ] . (20)Using (19) and (20) in (2) gives in place of (16): T mn Γ mnr ≈ X r ⇒ W mn ≈ T mn + X mn , (21) X mn = D − D − X r Γ rmn − D − D − D − X [ m Γ n ] (22)which implies D [ m W nr ] ≈ D [ m T nr ] + D [ m X nr ] , relating the on-shell Bianchi identitiesfor the modified gravitino field strength to those for the usual gravitino field strength.I remark that X mn fulfills X mn Γ mnr = − X r (identically). Hence, ˆ W mn = W mn − X mn fulfills ˆ W mn ≈ T mn and ˆ W mn Γ mnr = X r , and may be used in place of W mn as amodified field strength that coincides on-shell with the usual field strength in the case T mn Γ mnr ≈ X r . Moreover, ˆ T mn = T mn + X mn fulfills ˆ T mn Γ mnr ≈ T mn Γ mnr ≈ X r ,i.e., alternatively one may redefine T mn to ˆ T mn and use W mn = ˆ T rs P rsmn which thencoincides on-shell with the redefined field strength, i.e., W mn ≈ ˆ T mn .(16) and the identities W mn Γ mnr = 0 also show that the matrices P rsmn remove pre-cisely those linear combinations of components from the usual field strength whichoccur in T mn Γ mnr . For instance, in N = 1 pure supergravity in D = 4 the modifiedgravitino field strength W mn contains precisely those linear combinations of compo-nents of T mn which, using van der Waerden notation with α = 1 , , ˙1 , ˙2, are denotedby W αβγ | and W ˙ α ˙ β ˙ γ | in the chapters XV and XVII of [7] but no linear combinationoccurring in T mn Γ mnr . The modified gravitino field strengths are thus particularlyuseful for the construction and classification of on-shell invariants, counterterms andconsistent deformations of supergravity theories.We end this note with the remark that there is an analog of the modified fieldstrength (2) for first order derivatives of Dirac type spinor fields ψ which are subjectto field equations which read or imply T m Γ m + . . . ≈ T m = ∂ m ψ + . . . . Thisanalog is defined according to W m = D − D T m − D T n Γ nm = T n P nm , P nm = D − D δ nm − D Γ nm . (23)The matrices P nm fulfill identities analogous to (11)-(13): P nm Γ m = 0 , Γ m P mn = 0 , P nr P rm = P nm , P nm ⊤ = CP mn C − . (24)In particular W m thus identically fulfills W m Γ m = 0. Furthermore, analogously to(16), T m Γ m ≈ W m ≈ T m because T m Γ m ≈ T n Γ nm ≈ − T m . Hence,the matrices P nm remove precisely those linear combinations of components from T m which occur in T m Γ m . For W m with lowered spinor index one has W m = P mn T n and Γ m W m = 0. Added note:
After this paper was published the author learned that D = 5 versionsof the modified gravitino field strength (2) and of the projection operator (9) werepublished already in [8]. 4 eferences [1] F. Brandt, “Supersymmetry algebra cohomology I: Definition and general struc-ture,” J. Math. Phys. (2010) 122302. [arXiv:0911.2118 [hep-th]].[2] P. van Nieuwenhuizen, “Supergravity,” Phys. Rept. (1981) 189.[3] E. Cremmer and S. Ferrara, “Formulation of 11-dimensional supergravity insuperspace,” Phys. Lett. (1980) 61.[4] D. Z. Freedman, P. van Nieuwenhuizen and S. Ferrara, “Progress toward atheory of supergravity,” Phys. Rev. D (1976) 3214.[5] S. Deser and B. Zumino, “Consistent supergravity,” Phys. Lett. (1976)335.[6] E. Cremmer, B. Julia and J. Scherk, “Supergravity theory in 11 dimensions,”Phys. Lett. (1978) 409.[7] J. Wess and J. Bagger, “Supersymmetry and Supergravity,” 2nd ed., PrincetonUniversity Press, Princeton, USA (1992).[8] D. Butter, S. M. Kuzenko, J. Novak and G. Tartaglino-Mazzucchelli, “Confor-mal supergravity in five dimensions: New approach and applications,” JHEP1502