Physics Letters B 767 (2017) 347–349 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb Counterpart of the Weyl tensor for Rarita–Schwinger type fields Friedemann Brandt Institut für Theoretische Physik, Leibniz Universität Hannover, Appelstraße 2, 30167 Hannover, Germany a r t i c l e i n f o a b s t r a c t Article history: In dimensions larger than 3 a modified field strength for Rarita–Schwinger type fields is constructed Received 20 December 2016 whose components are not constrained by the field equations. In supergravity theories the result provides Received in revised form 19 January 2017 a modified (supercovariant) gravitino field strength related by supersymmetry to the (supercovariantized) Accepted 23 January 2017 Weyl tensor. In various cases, such as for free Rarita–Schwinger type gauge fields and for gravitino Available online 16 February 2017 fields in several supergravity theories, the modified field strength coincides on-shell with the usual field Editor: M. Cveticˇ strength. A corresponding result for first order derivatives of Dirac type spinor fields is also presented. © 2017 The Author. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. This note relates to Rarita–Schwinger type fields in D dimen- fields of supersymmetry transformations and are termed gravitino sions with D > 3. The components of these fields are denoted by fields. The supersymmetry transformation of a gravitino field is ψ α where m = 1, ..., D is a vector index and α = 1, ..., 2 D/2 m α = α + is a spinor index, D/2 being the largest integer ≤ D/2. The δξ ψm (Dm ξ) ... , (3) fields are subject to field equations (equations of motion) which α where ξ is an arbitrary spinor field with components ξ (‘gauge mnr read or imply ∂mψn + ...≈ 0 where, both here and below, el- α parameters’), and (Dm ξ) are the components of a covariant lipses denote terms that may or may not be present (but do not derivative of ξ defined by means of a (supercovariant) spin con- contain derivatives of ψm), ≈ denotes equality on-shell (i.e. equal- nr nection ωm , ity whenever the field equations hold), spinor indices have been mnr [m n r] 1 nr suppressed (as will be often done below) and = = − Dmξ ∂mξ 4 ωm ξnr . (4) is the totally antisymmetrized product of three gamma-matrices.1 = The field equations thus constrain the usual field strength whose Accordingly, the (supercovariant) gravitino field strength Tmn − + components are Dmψn Dnψm ... transforms under supersymmetry according to T α = ∂ ψ α − ∂ ψ α + ... (1) mn m n n m =[ ] + =−1 rs + δξ Tmn Dm , Dn ξ ... 4 Rmn ξrs ... , (5) The main purpose of this note is a suitable definition of a mod- where R rs = ∂ ω rs −∂ ω rs +... denote the components of the ified field strength for Rarita–Schwinger type fields whose compo- mn m n n m (supercovariantized) Riemann tensor. This implies that the super- nents are not constrained by the field equations. We denote the α symmetry transformation of the modified gravitino field strength components of the modified field strength by Wmn and define it defined according to (2) in terms of the (supercovariant) gravitino in terms of the usual field strength according to field strength is D−3 2(D−3) r 1 rs = − [ ] − 1 rs Wmn D−1 Tmn (D−1)(D−2) Tr mn (D−1)(D−2) Trs mn , =− + δξ Wmn 4 Cmn ξrs ... , (6) (2) rs where Cmn are the components of the (supercovariantized) Weyl [ ] [ ] where mn = mn and mnrs = mnr s . We shall now com- tensor ment on this definition. C rs = R rs + 2 (δr R s − δs R r ) − 2 R δr δs , Rarita–Schwinger type fields are particularly relevant to super- mn mn D−2 [m n] [m n] (D−1)(D−2) [m n] gravity theories (see [2] for a review) where they are the gauge (7) n rn m where Rm = Rmr and R = Rm denote the (supercovariantized) E-mail address: [email protected]. Ricci tensor and the (supercovariantized) Riemann curvature scalar, 1 We use conventions as in [1]. respectively. The modified gravitino field strength is thus related http://dx.doi.org/10.1016/j.physletb.2017.01.083 0370-2693/© 2017 The Author. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 348 F. Brandt / Physics Letters B 767 (2017) 347–349 by supersymmetry to the Weyl tensor but not to the Ricci tensor. In particular, (16) applies to free Rarita–Schwinger type gauge mnr Notice also that, like the Weyl tensor, the modified field strength fields satisfying Tmn ≈ 0with Tmn = 2∂[mψn]. Hence, (16) also (2) vanishes for D = 3. applies to linearized supergravity theories. Moreover (16) applies In order to further discuss the modified field strength (2) we to several supergravity theories at the full (nonlinear) level, such write it as as to N = 1pure supergravity in D = 4 [4,5] and to supergravity in D = 11 [6], where the equations of motion imply T mnr ≈ 0for α = β rs α mn Wmn Trs P mnβ , (8) the usual supercovariant gravitino field strengths Tmn . Accordingly, in these cases the modified gravitino field strength fulfills on-shell where P rs α are the entries of the matrix mnβ the same Bianchi identities as the usual gravitino field strength, D ≈ = = rs D−3 r s D−3 r s s r such as [m Wnr] 0in N 1, D 4pure supergravity for the P mn = δ[ δ ]1 + (δ[ ] − δ[ ] ) D−1 m n (D−1)(D−2) m n m n supercovariant derivatives of Wmn . mnr r r 1 rs If Tmn ≈ X for some X , one obtains in place of equations − − − mn , (9) (D 1)(D 2) (17) and (18) the following implications: D/2 × D/2 where 1 is the 2 2 unit matrix. This implies mnr r Tmn ≈ X rs D−3 r s D−3 r s s r m 1 m D−4 Pmn = − δ[ δ ]1 − − − (δ[ ] − δ[ ] ) ⇒ [ ] ≈− − + [ ] D 1 m n (D 1)(D 2) m n m n Tm r s Trs 2(D−2) X mrs 2(D−2) X r s , (19) 1 rs mn m 2(D−3) − − − mn . (10) ≈− − 1 + [ ] (D 1)(D 2) Tmn rs 2Trs D−2 X mrs D−2 X r s . (20) These matrices fulfill the identities Using (19) and (20) in (2) gives in place of (16): rs mnk mnk rs mnr ≈ r ⇒ ≈ + P mn = 0, Pmn = 0, (11) Tmn X Wmn Tmn Xmn , (21) − rs k rs X = 1 Xr − D 3 X[ ] (22) P k P mn = P mn , (12) mn (D−1)(D−2) rmn (D−1)(D−2) m n rs rs −1 D[ ] ≈ D[ ] + D[ ] P mn = CPmn C , (13) which implies m Wnr m Tnr m Xnr , relating the on-shell Bianchi identities for the modified gravitino field strength to those rs rs where in (13) P mn denotes the transpose of P mn, and C and for the usual gravitino field strength. I remark that Xmn fulfills −1 mnr r ˆ C denote a charge conjugation matrix and its inverse which Xmn =−X (identically). Hence, Wmn = Wmn − Xmn fulfills ˆ ˆ mnr r relate the gamma matrices to the transpose gamma matrices ac- Wmn ≈ Tmn and Wmn = X , and may be used in place of m m −1 cording to =−ηC C with η ∈{+1, −1} and are used to Wmn as a modified field strength that coincides on-shell with mnr r raise and lower spinor indices according to the usual field strength in the case Tmn ≈ X . Moreover, ˆ ˆ mnr mnr r T = T + X fulfills T ≈ 0if T ≈ X , i.e., alter- −1 β α αβ mn mn mn mn mn W = C W , W = C W . (14) ˆ ˆ rs mnα αβ mn mn mnβ natively one may redefine Tmn to Tmn and use Wmn = Trs P mn which then coincides on-shell with the redefined field strength, The first equation (11) implies that the modified field strength ≈ ˆ mnr = i.e., Wmn Tmn. identically fulfills Wmn 0, i.e. indeed the components of the mnr (16) and the identities Wmn = 0also show that the ma- modified field strength are not constrained by the field equations. rs rs trices P mn remove precisely those linear combinations of compo- (12) implies that the matrices P mn define a projection opera- mnr nents from the usual field strength which occur in Tmn . For in- tion on the field strength as they define an idempotent operation. stance, in N = 1pure supergravity in D = 4the modified gravitino Hence, this operation projects to linear combinations of compo- field strength Wmn contains precisely those linear combinations of nents of the usual field strength which are not constrained by the components of Tmn which, using van der Waerden notation with field equations. Owing to (13) the components of the modified = ˙ ˙ | | α 1, 2, 1, 2, are denoted by Wαβγ and W α˙ β˙γ˙ in the chapters field strength with lowered spinor index are related to the usual mnr XV and XVII of [7] but no linear combination occurring in Tmn . field strength according to The modified gravitino field strengths are thus particularly useful rs β for the construction and classification of on-shell invariants, coun- Wmnα = Pmn α Trsβ . (15) terterms and consistent deformations of supergravity theories.