arXiv:1612.03532v4 [hep-th] 2 Apr 2017 opnnso hs ed r eoe by denoted are fields in these fields of type Rarita-Schwinger components to relates note This hr Γ where definition. en ti em fteuulfil teghacrigto according strength field usual the of terms in it define qain.W eoetecmoet ftemdfidfil strengt st field field modified modified the a of of components const the definition not denote suitable are We components a whose equations. is fields note type this Rarita-Schwinger for of purpose main The D/ ne and index rimply or htmyo a o epeet(u ontcnandrvtvsof derivatives contain not do (but present be not may or may that hscntanteuulfil teghwoecmoet are components whose strength field usual the constrain thus oal niymtie rdc ftregamma-matrices. three of product antisymmetrized totally aebe upesd(swl eotndn eo)adΓ and below) done sp often hold), be equations will field (as the suppressed whenever equality been have (i.e. on-shell equality onepr fteWy esrfrRrt-cwne type Rarita-Schwinger for Weyl the of Counterpart ntttfu hoeicePyi,LinzUniversit¨at H Leibniz Physik, f¨ur Theoretische Institut 1 .Tefilsaesbett edeutos(qain fmto)wh motion) of (equations equations field to subject are fields The 2. euecnetosa n[1]. in as conventions use We epnigrsl o rtodrdrvtvso ia yes type Dirac of derivatives order presented. fiel first usual for the supergra result with several responding on-shell in coincides fields free strength gravitino field for for modified as and such fields cases, gauge various type In m by tensor. a related Weyl provides variantized) strength result field the constrai gravitino theories not variant) are In components whose equations. constructed is fields type mn ndmnin agrta oie edsrnt o Rarit for strength field modified a 3 than larger In ∂ m α W Γ = ψ mn 1 = n Γ mnr [ = m , . . . , Γ D D n + − − ] 1 3 2 n Γ and . . . T ⌊ D/ mn ≈ 2 T ⌋ 06 anvr Germany Hannover, 30167 − mn mnrs sasio index, a is hr,bt eeadblw lissdnt terms denote ellipses below, and here both where, 0 ( α D reeanBrandt Friedemann 2( = − 1)( D Γ = ∂ − D m 3) Abstract − ψ fields [ 2) m n α Γ T 1 n − r [ Γ m ∂ r ψ Γ Γ n m n ψ s ] ] α r ⌊ m esalnwcmeto this on comment now shall We . D/ − α where + D ( 2 D ⌋ . . . − iesoswith dimensions 1)( en h ags integer largest the being 1 m noe,Apltae2, Appelstraße annover, D − 1 = 2) mnr 1 tegh cor- A strength. d io ed salso is fields pinor Rarita-Schwinger iytere,the theories, vity dfid(superco- odified T ote(superco- the to e ytefield the by ned h edequations field The rs D , . . . , Γ = andb h field the by rained Γ a-Schwinger rs by h ψ mn ITP-UH-24/16 [ m m ), Γ , > D nrindices inor savector a is n W ≈ Γ mn c read ich r denotes ] .The 3. rength sthe is α and (2) (1) ≤ Rarita-Schwinger type fields are particularly relevant to supergravity theories (see [2] for a review) where they are the gauge fields of supersymmetry transformations and are termed gravitino fields. The supersymmetry transformation of a gravitino field is

α α δξψm =(Dm ξ) + ..., (3) where ξ is an arbitrary spinor field with components ξα (‘gauge parameters’), and α (Dm ξ) are the components of a of ξ defined by means of a nr (supercovariant) spin connection ωm ,

1 nr Dmξ = ∂mξ − 4 ωm ξ Γnr . (4)

Accordingly, the (supercovariant) gravitino field strength Tmn = Dmψn −Dnψm +... transforms under supersymmetry according to

1 rs δξTmn = [ Dm ,Dn ] ξ + ... = − 4 Rmn ξ Γrs + ..., (5)

rs rs rs where Rmn = ∂mωn − ∂nωm + ... denote the components of the (supercovari- antized) Riemann tensor. This implies that the supersymmetry transformation of the modified gravitino field strength defined according to (2) in terms of the (super- covariant) gravitino field strength is

1 rs δξWmn = − 4 Cmn ξ Γrs + ..., (6)

rs where Cmn are the components of the (supercovariantized)

rs rs 2 r s s r 2 r s Cmn = Rmn + D−2 (δ[mRn] − δ[mRn] ) − (D−1)(D−2) R δ[mδn], (7)

n rn m where Rm = Rmr and R = Rm denote the (supercovariantized) Ricci tensor and the (supercovariantized) Riemann curvature scalar, respectively. The modified gravitino field strength is thus related by supersymmetry to the Weyl tensor but not to the Ricci tensor. Notice also that, like the Weyl tensor, the modified field strength (2) vanishes for D = 3. In order to further discuss the modified field strength (2) we write it as

α β rs α Wmn = Trs P mnβ , (8)

rs α where P mnβ are the entries of the

rs D−3 r s 1 D−3 r s s r 1 rs P mn = D−1 δ[mδn] + (D−1)(D−2) (δ[mΓn] − δ[mΓn] ) − (D−1)(D−2) Γ mn , (9) where 1 is the 2⌊D/2⌋ × 2⌊D/2⌋ unit matrix. This implies

rs D−3 r s 1 D−3 r s s r 1 rs Pmn = D−1 δ[mδn] − (D−1)(D−2) (δ[mΓn] − δ[mΓn] ) − (D−1)(D−2) Γmn . (10) These matrices fulfill the identities

rs mnk mnk rs P mnΓ =0, Γ Pmn =0, (11)

2 rs kℓ rs P kℓP mn = P mn , (12) rs ⊤ rs −1 P mn = CPmn C , (13)

rs ⊤ rs −1 where in (13) P mn denotes the of P mn, and C and C denote a charge conjugation matrix and its inverse which relate the gamma matrices to the transpose gamma matrices according to Γm⊤ = −ηCΓmC−1 with η ∈{+1, −1} and are used to raise and lower spinor indices according to

−1 β α αβ Wmnα = C αβ Wmn , Wmn = C Wmnβ . (14) The first equation (11) implies that the modified field strength identically fulfills mnr WmnΓ = 0, i.e. indeed the components of the modified field strength are not rs constrained by the field equations. (12) implies that the matrices P mn define a projection operation on the field strength as they define an idempotent operation. Hence, this operation projects to linear combinations of components of the usual field strength which are not constrained by the field equations. Owing to (13) the components of the modified field strength with lowered spinor index are related to the usual field strength according to

rs β Wmnα = Pmn α Trsβ . (15) The second equation (11) thus implies that the modified field strength with lowered mnr spinor index identically fulfills Γ Wmn = 0. Moreover, the modified field strength coincides with the usual field strength on-shell mnr whenever TmnΓ vanishes on-shell:

mnr TmnΓ ≈ 0 ⇒ Wmn ≈ Tmn . (16)

mnr 2 This is obtained from (2) by means of the following implications of TmnΓ ≈ 0:

mnr ·Γr mn ·Γr m ·Γs m TmnΓ ≈ 0 ⇒ TmnΓ ≈ 0 ⇒ TrmΓ ≈ 0 ⇒ Tm[rΓs] ≈ −Trs , (17)

mn ·Γrs mn m (17) mn TmnΓ ≈ 0 ⇒ TmnΓ rs − 4Tm[rΓs] − 2Trs ≈ 0 ⇒ TmnΓ rs ≈ −2Trs . (18) In particular, (16) applies to free Rarita-Schwinger type gauge fields satisfying mnr TmnΓ ≈ 0 with Tmn = 2∂[mψn]. 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- mnr pergravity in D = 11 [6], where the equations of motion imply TmnΓ ≈ 0 for the usual supercovariant gravitino field strengths Tmn. Accordingly, in these cases the modified gravitino field strength fulfills on-shell the same Bianchi identities as the usual gravitino field strength, such as D[mWnr] ≈ 0 in N = 1, D = 4 pure supergravity for the supercovariant derivatives of Wmn. mnr r r If TmnΓ ≈ X for some X , one obtains in place of equations (17) and (18) the following implications:

mnr r m 1 m D−4 TmnΓ ≈ X ⇒ Tm[rΓs] ≈ −Trs − 2(D−2) X Γmrs + 2(D−2) X[rΓs] , (19)

2See also, e.g., equations (35) of [3].

3 mn 1 m 2(D−3) TmnΓ rs ≈ −2Trs − D−2 X Γmrs + D−2 X[rΓs] . (20) Using (19) and (20) in (2) gives in place of (16):

mnr r TmnΓ ≈ X ⇒ Wmn ≈ Tmn + Xmn , (21) 1 r D−3 Xmn = (D−1)(D−2) X Γrmn − (D−1)(D−2) X[mΓn] (22) which implies D[mWnr] ≈D[mTnr] + D[mXnr], relating the on-shell Bianchi identities for the modified gravitino field strength to those for the usual gravitino field strength. mnr r I remark that Xmn fulfills XmnΓ = −X (identically). Hence, Wˆ mn = Wmn −Xmn mnr r fulfills Wˆ mn ≈ Tmn and Wˆ mnΓ = X , and may be used in place of Wmn as a modified field strength that coincides on-shell with the usual field strength in the case mnr r mnr mnr r TmnΓ ≈ X . Moreover, Tˆmn = Tmn + Xmn fulfills TˆmnΓ ≈ 0 if TmnΓ ≈ X , rs i.e., alternatively one may redefine Tmn to Tˆmn and use Wmn = TˆrsP mn which then coincides on-shell with the redefined field strength, i.e., Wmn ≈ Tˆmn. mnr rs (16) and the identities WmnΓ = 0 also show that the matrices P mn remove pre- cisely those linear combinations of components from the usual field strength which mnr occur in TmnΓ . For instance, in N = 1 pure supergravity in D = 4 the modified gravitino field strength Wmn contains precisely those linear combinations of compo- nents of Tmn which, using van der Waerden notation with α =1, 2, 1˙, 2,˙ are denoted by Wαβγ | and W α˙ β˙γ˙ | in the chapters XV and XVII of [7] but no linear combination mnr occurring in TmnΓ . The modified gravitino field strengths are thus particularly useful for the construction and classification of on-shell invariants, counterterms and consistent deformations of supergravity theories. We end this note with the remark that there is an analog of the modified field strength (2) for first order derivatives of Dirac type spinor fields ψ which are subject m to field equations which read or imply TmΓ + ... ≈ 0 with Tm = ∂mψ + .... This analog is defined according to

D−1 1 n n n D−1 n 1 1 n Wm = D Tm − D TnΓ m = TnP m , P m = D δm − D Γ m . (23)

n The matrices P m fulfill identities analogous to (11)-(13):

n m m n n r n n ⊤ n −1 P mΓ =0, Γ Pm =0, P rP m = P m , P m = CPm C . (24)

m In particular Wm thus identically fulfills WmΓ = 0. Furthermore, analogously to m m n (16), TmΓ ≈ 0 implies Wm ≈ Tm because TmΓ ≈ 0 implies TnΓ m ≈ −Tm. Hence, n the matrices P m remove precisely those linear combinations of components from m n Tm which occur in TmΓ . For Wm with lowered spinor index one has Wm = Pm Tn m and Γ Wm = 0. Added note: After this paper was published the author learned that D = 5 versions of the modified gravitino field strength (2) and of the projection operator (9) were published already in [8].

4 References

[1] F. Brandt, “Supersymmetry algebra cohomology I: Definition and general struc- ture,” J. Math. Phys. 51 (2010) 122302. [arXiv:0911.2118 [hep-th]].

[2] P. van Nieuwenhuizen, “Supergravity,” Phys. Rept. 68 (1981) 189.

[3] E. Cremmer and S. Ferrara, “Formulation of 11-dimensional supergravity in superspace,” Phys. Lett. 91B (1980) 61.

[4] D. Z. Freedman, P. van Nieuwenhuizen and S. Ferrara, “Progress toward a theory of supergravity,” Phys. Rev. D 13 (1976) 3214.

[5] S. Deser and B. Zumino, “Consistent supergravity,” Phys. Lett. 62B (1976) 335.

[6] E. Cremmer, B. Julia and J. Scherk, “Supergravity theory in 11 dimensions,” Phys. Lett. 76B (1978) 409.

[7] J. Wess and J. Bagger, “Supersymmetry and Supergravity,” 2nd ed., Princeton University 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,” JHEP 1502 (2015) 111. [arXiv:1410.8682 [hep-th]].

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