proofs JHEP_092P_1215 xxxxxxx ???, 2016 Springer : doi: March 1, 2016 : y February 25, 2016 December 11, 2015 : : setting. Conserved Published Accepted Revised Received e dimensions 3,4,5,6 and 10. d super-twistor spaces for 3,4 nhuis bracket. Superconformal erconformal Killing tensors are Uppsala University, cles and Poisson brackets are used Published for SISSA by [email protected] , b,c . 3 1511.04575 The Authors. c Extended Supersymmetry, Superspaces, Higher Spin Symmetr

and U. Lindstr¨om The notion of a Killing tensor is generalised to a superspace , a [email protected] E-mail: The Strand, London WC2R 2LS,Department U.K. of Physics andSE-751 Astronomy, 20 Theoretical Physics, Uppsala, Sweden Theoretical Physics, Imperial College London, Prince Consort Road, London SW7 2AZ, U.K. Department of Mathematics, King’s College London, b c a ArXiv ePrint: Keywords: Killing tensors in flatThese tensors superspaces are are alsoand studied presented 6 for in dimensions. analytic spacetim Algebraic superspacesalso structures an associated briefly with discussed. sup quantities associated with these are definedto for define superparti a supersymmetric version of the even Schouten-Nije Open Access Article funded by SCOAP Abstract: P.S. Howe Notes on super Killing tensors proofs JHEP_092P_1215 1 3 6 25 26 27 28 30 31 10 18 24 29 ctor (1.2) (1.1) invari- g ate basis by the nstant up to a Lorentz in the momentum whose he Hamiltonian formalism for hat leaves the metric t up to a Lorentz transformation , is a symmetry of the metric, i.e. a K , , b b a a ˜ L L = = b b – 1 – K K a a ∇ ∇ denotes the standard flat components of the metric ab η is a metric covariant derivative; for a conformal Killing ve , where ∇ S ab η and + 2 ba L ab L − a Killing vector (KV) satisfies = 4 = 6 = 3 a = = = 0. With respect to an orthonormal basis related to a coordin m D D D e ab ab g ˜ L L K L 6.1 6.2 6.3 6.4 Algebras ant, vector field that generates an infinitesimal diffeomorphism t In ordinary Lorentzian spacetime a Killing vector, 1 Introduction 7 Comments on higher spin 8 Concluding remarks A Supersymmetric SN bracket 6 Components of superconformal Killing tensors 3 Superparticles 4 SCKTs in flat superspaces 5 Analytic superspace Contents 1 Introduction 2 Killing tensors in superspace in an orthonormaltransformation frame. while a In conformal Killing othertogether vector with is words constan a a scale transformation. Killinga We vector massless note is particle, also that, a co Poisson in CKV bracket t with is the a Hamiltonian function vanishes on weakly. phase space linear (CKV) we have where where vielbein proofs JHEP_092P_1215 r eter. (1.5) (1.4) (1.3) curly rest. An ]. One feature 13 – = 5 there is an ex- 10 are given by finite- D ssed above to superspace, antities. This turns out to Given this Poisson bracket ess and obeys Killing tensors (KTs). These neral case we shall apply it d two-form on the even tan- e then turn to a discussion of 1 entations can be reducible but onsidered in a similar way. r, so that any spacetime that stor spaces in sections 4,5, and conformal symmetry in super- s. ed product of two (conformal) ralgebras [ roups, in g tensors that extends the even = 1 while in ten dimensions the ]. We also discuss the first three s lead to higher-order constants f massless particles moving along , 9 uperspaces of the superconformal , cally in dimensions 3,4,6,5 and 10. educible) Killing tensors but no Killing , sions of Killing vectors defined on super- } ) N ets of odd spinorial coordinates. n n n a ...b ...b ]. In this paper we shall focus on superspaces 2 2 8 ,b ,b 1 1 . . . p b b ( 1 { a a a p ˜ L L n = = – 2 – ...a 1 n n a ...b ...b K 1 1 b b = K K (4), only for the case a a K This is not entirely straightforward due to the presence F ∇ ∇ 2 ]. 7 ] and references therein. We discuss examples of this in supe – 1 15 is covariantly constant with respect to suitable time param , p 14 now includes a trace part on the first two indices and where the is antisymmetric on the first two indices and symmetric on the n n ...b ...b 2 2 ,b ,b 1 1 ]. ab ab 25 ˜ L L In flat superspaces superconformal Killing tensors (SCKTs) In this article we shall generalise the basic concepts discu These relations can readily be generalised to higher-order The opposite is not true, there areIn the cases mathematics where literature there one can are find (irr supersymmetric ver 1 2 th rank conformal Killing tensor (CKT) is symmetric, tracel Killing vectors is aadmits the second-rank former (conformal) will also Killing have tenso (conformal) Killing tensor It is straightforward to see that the (traceless) symmetris where the momentum of the motion defined by brackets denote traceless symmetrisation. Ingeodesics the context in o the given spacetime, conformal Killing tensor where dimensional representations of the corresponding Lie supe n where Minkowski space, analytic superspace and also in super-twi as commonly understood in physics. These extend spacetime by s group for which the local description resembles spacetime c that is not present in theindecomposable, purely see even case [ is that some repres superconformal Killing tensors in flat superspaces,In specifi the first three cases there are classical superconformal g cases in analytic superspaces. These are particular coset s to conservation laws forgent superparticles bundle where to we define usebe Poisson a well-defined brackets close even for these thoughstructure we conserved the can qu two-form derive a isSchouten-Nijenhuis bracket bracket itself for (see superconformal singular. below) Killin to the super case. W building on earlier discussionsspace, of see, various for aspects example,of of [ constraints super in supergeometries. Having discussed the ge are symmetric tensors obeying vectors [ Riemannian spaces equipped with ortho-symplectic metrics [ compensating scale parameter is required to be constant [ ceptional superconformal group, proofs JHEP_092P_1215 ] V 18 etc. and y as , (2.1) A and 17 ) below B α, β algebras S Ω B 2.16 that relates E ss separately A + M A a bundle carrying E dE V concluding remarks. This then reduces the := 4 A pinorial) indices r realisations of supersymmetry. generates the even tangent ymmetries of the Laplacian nents), and correspondingly symmetry that reduces the DE le of this is given by a SCKV 1 ce was introduced in [ ension-zero torsion is flat, onstraints on the torsion. In ate additional constraints are taking its values in the Lie al- runs over the dimension of the ce of sub-representations that group. T ice to reduce it further to a di- = ). The structure group is taken me parts of the super-vielbein. even and odd so that the struc- is maximally non-integrable. In α B etc and the product of the corre- α run over all the odd indices, but A 1 A on of SCKTs given in ( . The basic structure is a choice of T α T ,E A a, b a , 1) and which, in addition, may carry M E E − αβ ) = ( M c We denote the superspace coordinates by is a spinor bundle and ,D A dz (Γ 3 , where now i E S ]. = − αi – 3 – 22 A = , and we shall always make such a choice in what – 0 E -dimensional spacetime by a number of odd coor- c T 19 where is an R-symmetry index, corresponding to D ]; in other words, αβ i T V 22 by a pair ⊗ α S = 1 , the tangent bundle, such that ]. Such algebras may have applications in higher-spin theor T T . The various components of the connection can be determined B 16 ⊂ C 1 Ω ], for a more formal discussion. T C 24 by Lie brackets [ A 1 4). In section 6 we also comment on the possibility of defining | + Ω (4 ). There is no super-metric but there is a super-vielbein = 4 supersymmetry where one has to impose super-tracelessne ] where superspace was introduced in the context of non-linea T/T µ B psl 23 A , θ = ,D Ω m 0 d x T = 4 = = ( See also [ See, for example, [ B N 3 4 A M bundle Finally, throughout this paper, we shall assume that the dim in terms of theaddition, vielbein we if can we impose impose further suitable constraints conventional that c specify so R odd tangent bundle gebra of the structure group (sodefine that the there torsion are and no curvature forms mixed in compo the usual way: We use a two-step notationwhen for necessary, the we shall odd replace indices. We let coordinate and preferred frame bases by respectively. We then introduce a connection one-form Ω appropriate spin representation and to be the Lorentz group acting on even (vector) indices sponding spin group and any R-symmetry group acting on odd (s algebra to and where one is still left with an additional one-parameter structure group to a triangularagonal form one but by it an is appropriate standard choice pract of z and generalised to the curved case in [ is not always sufficient, andimposed furthermore, there that are when invariances appropri thatcannot be correspond removed to due the to indecomposablity. presen in A familiar examp follows. Given such a choice,ture the group tangent does bundle not splits mix into them. Thus we have 6 respectively. In particular, it turns out that the definiti addition, we suppose that associated with SCKTs in ain similar the way purely to even those case that [ arise as s a representation of an internal R-symmetry group. Superspa dinates that transform as spinors under Spin(1 the fundamental representation of the internal R-symmetry 2 Killing tensors inWe superspace consider superspaces that extend we briefly comment on in section 7. We end in section 8 with a few proofs JHEP_092P_1215 ) 2.6 (2.2) (2.5) (2.3) (2.6) gives two ) implies b ]. In other However, as K 2.2 30 , nvariant tensor, 29 . It then follows from γ acting on etry mean that the full rmined in terms of the atisfied, and sometimes posed, but it will not be αβ ponent of this torsion is T α . ], but we shall not consider etime-dimension-dependent e would seem to be a vector imension one. In addition, lf component of the Lorentz ∇ f the tangent space into odd on the right, where the tilde y corrections it may be that 27 formation [ ng to the dimension one-half = 0 , rsion ]. r-spinor and a spinor, and ( B ucible representation of the spin o be zero so that ( c ), where the spinorial component A 26 B 28 , α βb ˜ , L A T 7 [ ) holds. In addition we shall impose L ,K ). The dimension-zero torsion takes = 0 αβ a = ) 2.1 c K γα B α, β . ) ) with b , from the lowest-dimensional component a = ( CA (Γ = 0 (2.4) 2.5 = 0 T K A γ c c C – 4 – K ab iK αb K T T − + ), namely b = 11 supergravity [ B K 2.5 D K α A ∇ = 0 and (Γ ∇ ] bc [ α T . The vector field b 5 K ) are imposed. The spinorial derivative 2.3 is symmetric on the joint indices c denotes an element of the Lie algebra of the structure group. ) and ( B denotes a product of the appropriate gamma-matrix and an R-i that satisfies A c 2.1 A L K The natural generalisation of a Killing vector to superspac For other discussions of this topic in curved superspace see 5 (i.e. dimension minus one-half) of ( the dimension-one-half Bianchi identity that this must als we shall see, thevector constraints field that is we determined have by imposed its on even part, the geom this form inoff-shell. supergravity, when In the thethis presence equations is not of of the higher-order case, motion string for are example or in s M-theor connection, while the secondand allows even. one to These fixsymmetric, two the traceless constraints splitting and gamma-traceless. imply o This thatgroup is that the an is remaining irred not present com in the other dimension-one-half to The first of these allows one to solve for the dimension one-ha where states that thespinorial former derivative of should vanish. The spinor is then dete some conventional constraints that donature not of depend the spinors. on the At spac dimension one-half we can take this possibility here; we shall always assume that ( if needed (Γ field indicates the inclusion of an appropriate super-Weyl trans We can also choose where Γ when ( is determined in this fashion, then satisfies ( as a conventional constraintwe for shall the assume Lorentzand that connection one conventional components at constraints of d the correspondinecessary R-symmetry to connection be have explicit been about im these in this paper. representations of the spin group, a gamma-traceless vecto proofs JHEP_092P_1215 ), = A 2.6 (2.7) (2.8) (2.9) is the (2.12) (2.14) (2.11) (2.13) (2.10) DT M A E . ) β , C ] ↔ ndard constraints l constraints that n one-half, (by setting appro- . (where α A,B hi identity, [ C B B φ A D + ( L variations of the super- H ) (i.e. those for which the M,A + 2 ive to both sides of ( D For a diffeomorphism ac- S, Ω γβ S, S C β ) δ 6 2.5 ] b AB β b α st vanish up to a local frame sion must also vanish. Now a B α T δ n. M | S, ave . δ f ( δ = 0. The dimension-zero com- γ A D γ )(Γ | − + ) b + γ on is E ∇ T β + 2 β α C ) ] β D α b -grading throughout the paper. c βγ Dα H 2 α a B := ) A | L [ B T Z a L (Γ L D C = 0, in which the scale parameter is a A | γ D H = (Γ α T = 2 L = S ( K β C D A,B β α b iA α − − H A φ a α + ˜ [ L 2 ∇ ). This equation is now formally identical C ˜ ˜ B = L L γ L DA,B D − β K = = 2.4 CA c R – 5 – H α d + 2 b T γ Da D D S C ∇ T H C is determined in terms of K b K Dα a a D K AB αβ δ T C AB + ) ∇ T K c + D ) and ( T ) = ( C b + + 2 + A,B D B K B b 2.3 φ C β Ca ∇ ] a K L + T D B A A K L γ 0 = (Γ C , and connection, a H ∇ ∇ K B = K ∇ K A ) holds), [ α b M = = = a + ∇ ∇ ˜ 2.1 L b C C B δE A =0. Since K = 2 M AB a B A,B H A C A φ ∇ δT E ( H AB ) reduces to αβ := ), we have used ( ) δT a ) are either both even or both odd) are satisfied if B 2.9 2.12 A (Γ ) defines a superconformal Killing vector (SCKV) when the sta is a constant. Depending on the theory (i.e. on any additiona H is a local scale parameter. Explicitly, A, B , to get (when ( 2.6 A S A ). We can then use a similar argument to show that, at dimensio B Another way of deriving this result is to consider arbitrary To see this directly, we apply a second odd covariant derivat 1.2 By graded (anti)-symmetrisation of indices we shall mean R 6 B where the square brackets denote graded antisymmetrisatio constant. E inverse supervielbein). The induced variation of the torsi where have been imposed) it might be the case that ponent of ( companied by a local structure-algebra transformation we h to ( indices ( given above are imposed. take the graded commutator and then make use of the first Bianc suppose that we make a diffeomorphism but only set priate parts of the torsion to zero), the variation of the tor where vielbein, words, ( For a Killing symmetry therotation variation so of that the supervielbein mu This equation tells us that the dimension-zero components o where, in ( proofs JHEP_092P_1215 . n ), n ...A ...b 2.6 (3.1) 1 1 b (2.15) (2.17) A indeed K K β a to be irre- H γ 1 − n . Now consider c ). Given this one ) and making use 1), both of which ...b 1 ne-half connection b − } : it can be defined to 2.16 n 2.16 n K will also vanish up to a d background is given ...B 2 tensors (SCKTs) straight- and the angle-brackets de- hout the rest of the paper B take ,B ism that preserves the odd A 1 se can be studied when the A s purely even tensor ents of a full SCKT . It is easy to see that setting is object should satisfy out of ( B H E = 0 (2.16) { btain a SCKT we simply have g this out explicitly gives ( , we conclude that A αb,c irreducible spinorial representa- S γα ˜ L T ion 4, we shall work out all of the ) ation preserves Γ rivatives to ( } = n b . } , although is not straightforward to n (Γ a B γ ˙ {} z 1 = 0 a ) and thus that we have a superconformal − ˙ CA i z n b 1 T − C ...b 2.11 1 1 e ,E ] b − 1 2 { – 6 – n ,K = ), as shown above. ...B α vector indices, and one with ( 1 inK is, or, equivalently L E B n [ { − h 2.11 X n nK ...b 1 . But since the only other non-zero terms in the equation b + β n ) and ( a K α H is odd if ...B ∇ 1 2.10 B X K K ), ( L A 2.9 ∇ , as well as a,bc ] φ denote basis vector fields dual to the basis one-forms 32 A , E = 0 allows one to solve for the Lorentz part of the dimension-o This result means that we can interpret a SCKV in a simpler way The above results can be extended to superconformal Killing 31 αbc has the same form as the right-hand side of ( variation contributing to this component involves the derivative of T note the standard pairing between forms and vectors. Writin are symmetric-traceless on these andto gamma-traceless. set To the o larger representation to zero. Thus we have the variation of the dimension-one-half torsion component when the standard constraints are satisfied. Note that we can be a vector fieldtangent K bundle, that i.e. generates an infinitesimal diffeomorph scale transformation since a structure-algebra transform Its covariant spinorial derivativetions again of contains the just Lorentz two group, one with ducible because any gamma-trace term it could contain drops of the Ricci and Bianchi identities. One would expect that th would then expect to be able to construct all the other compon forwardly. Thus a SCKT is determined by a symmetric traceles and this leads to ( Killing vector. Here This clearly implies that the dimension-zero components of for some appropriate definition of the brackets systematically by applying further spinorial covariant de 3 Superparticles The Lagrangian forby [ a Brink-Schwarz superparticle in a curve verify this explicitly in thecomponents general of case. SCKTs Later in on, flatwill in superspaces. be sect The on emphasis the throug scale superconformal transformations case, are but omitted. the non-conformal ca proofs JHEP_092P_1215 e are (3.3) (3.5) (3.4) (3.2) (3.6) t o the nor and ). We have = 0. The . Varying a on-shell, and 2 z p } 2.16 n . is symmetric and ...a n is conserved if and 1 th rank symmetric, a n ce constraints given ...a , K 1 K b ond term on the right a n { e true. Thus the second a on K ∇ is an ve parametrised by orldline of the superparticle . . . p K 1 n tensor satisfies ( b a ) kappa-symmetry transforma- p = 0 on shell ) n , , n . . . p b 1 , , where ...a b , 1 n , . β p a = 0 = 0 a α γ κ . . . p b c κ n K ) holds. When it does then the first term 1 p p α = 0 = 0 β αβ b b c ˙ ) ...b z . . . p p a n 2 ∇ p a 1 b b Ca p βα Bα iη β · 2.16 a p t ) T 2 z T p γ K is invariant under kappa-symmetry. We have p gives the mass-shell constraint ∇ C n n – 7 – B − . The equations of motion are · + ˙ ˙ z . . . p αγ z = 0 along the worldline. For this expression to e A K ) ...b 1 n ...a = (Γ = , the first term involves 2 1 b (Γ + ˙ 1 a M b b a a p β α p a ...a ˙ p t E δe n z 1 K p (Γ K a t δz γ ∇ β M ...b ∝ ) defines a SCKT. z 1 κ ∇ = K κ b a b n a z αβ K K ∇ := ˙ Γ ) ...a b α 1 p 1 ), are satisfied. The equations of motion simplify to b z A a · ∇ z 1) tensor indices. Clearly the former will not give zero in th (Γ α K 2.4 = ( ˙ (Γ z th rank symmetric, traceless, gamma-traceless tensor-spi − is the pull-back of the superspace covariant derivative ont κ n δ in iηn n t ]: dt dK = = = is the derivative of the Lagrangian with respect to ˙ ∇ 32 ) and ( a K ˙ z κ 1 2.3 δ − e ), and we set ˙ t ), ( 1 depending on the dimension. ( ) holds, i.e. if contains only two irreducible representations because ± = M 2.2 n z = a 2.16 p ), ( ...a η ) vanishes as well: since ˙ 1 a Now consider the function We can also show that the function 2.1 3.5 K β traceless. These are an a similar object with ( The superparticle action istions invariant introduced under in [ the (fermionic traceless tensor. We claimwhen the that equations this of motion is are satisfied conserved provided along that this the w this vanishes for a SCKT. So we have the result that the functi where we have used the fact that where be zero the∇ two terms have to vanish independently. In the sec only if ( in ( Here the coordinates of the superparticle moving along a cur the action with respect to the einbein above equation and henceterm must gives be a set zero to contribution zero if in and only order if for ( it to b worldline. Fromby now ( on we suppose that the standard superspa covariant derivative where given by proofs JHEP_092P_1215 , ], L n a 34 (3.7) (3.9) (3.8) , (3.10) (3.14) (3.12) (3.11) (3.13) and . . . p 33 1 K a p n . The basis racket obeys a , is zero. The ...a e m , 1 p e } on bracket with a n phase space is q m K ...a dx 0 with the Hamilto- m n Schouten-Nijenhuis a := = ∼ = K structed from nless zero-mass particle e a b . K p e -shell, is another such function, a ∇ is the vielbein, the momen- | ∂ b | , satisfies a L ∂p e to a time translation. 1 f m , r-embedding approach [ − e ) , m := b . p This new tensor defines the even a ...a b a g , ). If 1 ∂ a = 0 e a 7 )˜ { ω α ↔ 1.4 f σ . ) a , where f − f κ a ∂ mL X ) a a ( ι b m − Γ g df e − ∂ b dp − g c X ( p } m ι a p = a b a q c x e ∂ − σ ) f∂ (Γ . This is consistent as kappa-symmetry can be ...a – 8 – f = ˙ ) where a f α m,b n α := a ˙ e X a a z z a ω ι ) = e a x L , ∂ ) also has a weakly vanishing Poisson bracket with iη b + a ) = ˜ f, g e ∇ Dp = (˜ = | m ( a b ∂ | f a e K,L ( 1 ], f, g p where ˙ X ( − for two vectors is the Lie bracket. κ m = ) is (˜ n δ a a , corresponding to a function a a ] ˙ e σ x f ...a K,L a 1 ˙ X x a := , Dp 1 K,L { a . The Poisson bracket of the constraint while the momentum associated with the einbein, a − 1) factors of the momentum. 2 e ˜ e e a nK 1 2 ˙ − x ep 1 2 1 = − m := e = q L + = symmetric traceless tensor, then the fact that the Poisson b ...a H n 1 a a ˙ x ] m ∂L ∂ so that the Hamiltonian is weakly zero. The symplectic form o is the standard torsion-free connection one-form and = b H K,L a a [ − p ω We shall now discuss the supersymmetric extension of the eve A CKT can now be defined as a function on phase space whose Poiss The minus sign is so that [ 7 by virtue of the equation of motion for ˙ explicitly A Hamiltonian vector field thought of as extended world-line supersymmetry in the supe multiplied by ( With respect to a covariant basis is easily seen to be it is easy to see that this constraint is precisely ( We define the Poisson bracket of two functions by Schouten-Nijenhuis bracket [ nian is bracket for CKTs. Wethe Lagrangian briefly is review the bosonic case. For a spi so that the commutator of two such transformations gives ris where The variation of the momentum can be ignored as it vanishes on of vector fields dual to ( tum is Hamiltonian is the Hamiltonian vanishes weakly. Writing such a function as with a rank the Hamiltonian. We can write (minus) this as a new tensor con the Jacobi identity implies that ( proofs JHEP_092P_1215 8 σ ]. 40 (3.15) (3.17) (3.16) , 39 nsequence ) as a constraint. = 0) as will become , 2 2.16 can be replaced by an , defined by symmet- α p g. sonic particle straight- b ˜ L β E , then, demanding that ough not quite straight- ˜ ∇ E  tions defined by SCFTs. x ch brackets in the physics gent bundle coordinatised onic momenta, and, as is in the denominator of the f tor fields as before, with αβ and β 2 ) ˜ p e anti-symplectic two-form on 2 E p ell, but this singularity cancels p ss-shell ( ound in, for example, [ K ] given by · does not allow a simple covari- rested in the even bracket which αβ , while the odd bracket can be involves symmetric contravariant 43 ) there are two brackets attributed metry on the worldline rather than the (Γ 2 p p . M f · 1 depending on the dimension. The ∗ as well as α a p ˜ ± T even (Γ E be weakly zero leads to the constraint θ a , involves anti-symmetric contravariant iη = T iη (i.e. the fibre coordinates are taken to be H of the same form as in the bosonic case,  ssions of generalised Killing tensors in this context, η odd + ) + M a + K g ∗ ]. The even bracket, as noted above, is related a T ˜ Dp – 9 – E 38 ↔ a ) f f E a depends on ∂ − ( n indices respectively, the g a ) and where − Σ := ...a b 1 m a ∂ a f∂ c ∂ a ) p K ˜ c E f and a ˜ M,b E n with the Hamiltonian ) = ( = ( K ]. The one we refer to here as + Ω f f, g 37 ( ]. M X – ∂ 42 ( 1. In a coordinate basis the covariant derivative , 35 = 0, which is formally the same as the bosonic case. This is a co . There is a natural closed two-form Σ [ − M a } 41 p A n E + +1) n ( m = and ...a A µ = 2 ) but does not imply it. We are therefore obliged to impose ( ˜ a E q , θ K m We cannot repeat the arguments given for symmetries of the bo The foregoing can be extended to the superparticle case alth 1 2.16 In the context of particles it is also possible to have supersym x a 8 { well-known, the constraint structureant of discussion. the superparticle If we start with a function forwardly because the phase space does not include the fermi odd). Discussions of this topic and other variations can be f with the difference that now the Poisson bracket of ∇ apparent below. However, wereplaced can by still Σ, and define we Hamiltonian find vec derived in a similar way fromthe the Grassmann-flipped anti-bracket defined cotangent from bundle th Π although it is not symplectic because it is singular on the ma literature in the context of Killing tensors [ tensors, while the other,tensors which (or could multivectors). be Henceforth called wewe shall shall only refer be to inte as the SN bracket. This is the term used for su Poisson bracket is ordinary partial derivative. Noteto that, these in authors the [ literature, If we compute theric, Poisson traceless bracket tensors of with two such functions, forwardly. In the superby case the phase space is the even cotan to the Poisson bracket on the contangent bundle ambient space; this is called asee, spinning for particle. example, For discu [ of ( where It is clear from theseout formulae when that we Σ compute is the not Poisson invertible on-sh bracket of two conserved func where proofs JHEP_092P_1215 ) = N = 5 | ]. In D ). A (4.1) 2 9 , (3.18) D N ). The = 10 it ( = 10 on u N ) D ( D becomes a s SU(2 O , ), but it can } ) In 5 &10. In the q 1) symmetric K , = 3 to emphasise ra ( N 9 6 | ...a − 2.16 , D ). For the case in ras, while in m ˙ (2 4 n i α a , (4). ¯ ation D K F ( b SpO = 3 obeying the constraint stant scale transforma- ∇ αi | therefore conclude that b n D | oup D deed define a new SCKT. rm. Except for 1 epresent a spinorial deriva- − ivatives in a simple fashion. ...a nal symmetry algebra for a 1 . m a s in oup and the constraints on a δ ulus. This was introduced for epresent spin representations. 1)th-rank tensor given by the } t is straightforward to compute ...a K q 1 ty, so does the super-SN bracket − ivative a { ...a { m +1 ) of supersymmetries, while in n + mL a N n L − } γδ -supersymmetry transformations, we shall q ) n n S 2 }| a ) and the R-symmetry group is ...a n R , while its derivative has (2 a (Γ , n γ L 2 – 10 – 1 b − n ...α ∇ | 1 b | α produces a symmetric, traceless tensor-spinor which z ...a 1 1 − K a ∼ αi n { is 6 there are superconformal groups, D , guaranteed that this tensor also satisfies ( ...a K K 1 4 2) but we write it with the symplectic factor first in a | , { N imnK not indices, = 3 nK + n = 1, for which one can have superconformal transformations [ D := N q ...a 1 a ] = 1. In = 3 the spin group is SL(2 N D ), and the SCKVs represent the corresponding Lie superalgeb ] where the tableaux were for the internal symmetry Lie algeb K,L [ ) is the same as OSp( N | 44 In N 1) vector indices and is also gamma-traceless. In spinor not | (2 − n = 1, the superconformal group is the exceptional Lie supergr We emphasise that it is SpO = 3. = 4 in [ 9 = 10, where there are no conformal boosts or ,N spinor indices. The tableau for symmetric spinor with 2 has ( that applying an odd derivative top component of a SCKT is a symmetric traceless tensor 5 and OSp(8 that this refers to the spacetime conformal group. only consider first three cases we shall consider arbitrary numbers ( there is only one case, is simpler to use spinorD notation and the Young tableaux calc D 4 SCKTs in flatIn superspaces this section we give the details of SCKTs in flat superspace be verified directly thatthe supersymmetric it SN does, bracket defined asMoreover, above since shown for the in SCKTs Poisson bracket does the obeys in and the appendix. thus Jacobi identi we We have a Lie algebra structure on the space of SCKTs. single box with a dot (cross) then represents a covariant der SCKV mean that it differstion. from a In non-conformal the SKV following we only shallall by go a the through con each components case of in turn. a I SCKT starting from the leading even te the other hand, there is no corresponding superconformal gr bracket cancels so we find a new function with an ( supersymmetric (even) Schouten-Nijenhuis bracket. D hand, however, it is more convenientAgain to one take can the place tableaux to eithertive r a and cross then or one agiven dot can component inside read obtained a from off tableau the the top to representations one r by of applying the odd der inter proofs JHEP_092P_1215 (4.4) (4.2) (4.3) (4.7) (4.5) (4.6) in the ). The dotted + 1) in ij = 3, the K n m D m ,K α i steps we get ∂D anti-commute, , ,K supersymmetry m { αβ S , namely that the Ds t must vanish, and K K n all of the internal . The leading compo- . Thus and pacetime derivative to ij K Q n must sit in the second row. 2 }| s of he moment, we only want to l Killing vector in D . cetime derivatives of all of the , { , are in the same row, it follows that = 0 . m · − ) in the first two and 2 in the second. z n 2 Ds +2 = 0 n m , m ∼ ) ...α − ...i 1 m n n 1 +2) boxes on the first row and +2 { i α 2 2 }| n · ) = 0 m n 2 ]. The components are ( i ...α K -expansion. So any pair of ...i , say, i.e. one satisfying · 3 κ i 1 ...α 45 θ } and antisymmetric in m α i 3 κ b ) · – 11 – α ∂ D x ( a K +2) boxes in the first row, one with (2 K · { in a n 2 m z n γ + 1) symmetrised boxes in a row must vanish. No ∼ . The spacetime constraint on 2 }| α 1 D ) parameter. K ∼ n N ( α K ( 2 N vanishes, becomes m α ∂ ( K ≤ 1 m D α so ( m ∂K D ∂ z is and × , is constant in n ij · ) 2 DK K ≤ ) vector indices must be antisymmetrised. Thus, after 2 ∼ N D m ( o DK , is a conformal Killing spinor in the representation corresponding to the first of these tha DK ) tableau can have more than two rows, so any further R ∂K , As a simple example consider a SCKV [ we get three tableaux, one with (2 all the internal and since all of the spinor indices associated with the with symmetry onindices. all Clearly of the spinor indices and antisymmetry o second, first component obeys the standard constraint for a conforma or, in indices, in other words, the tableau with (2 SL(2 Moreover, we are not interestedfind in the spacetime independent dependence components at of t while the third, ( symmetrised traceless part of and the constraint on This constraint then leads tospinorial derivatives, similar constraints on the spa This can also beK represented by Young tableaux. If we apply a s nent thus has spacetime conformal parameters, the second ha representation specified by the tableau with (2 It is the first row and one in the second, and one with 2 parameters and the third is an this constraint then descends to all of the other components boxes on the second must vanish. proofs JHEP_092P_1215 s n K ...a 1 p D (4.9) (4.8) a (4.12) (4.13) (4.11) (4.10) K with KT K -dimensional N . antisymmetrised . p     . . The reality of = 3 case this means { = 3. The spin group         N   { { D { is D the box with a cross to uched. Similarly if we , , or antisymmetrised internal q K q n ...j = 0 1 n ) ,j }| . q q n n +1 }| }| − n q }| n × ˙ − α = 0 n ... ˙ ) α × 1 ), while leaving the undotted indices ˙ ... α +1 , 2 are set to zero. Acting on N n × p z ˙ α ( α ) ˙ − u × , n z z ... 2 z +1 p { ¯ , , p DK α ) index in the fundamental ...α , ) ˙ − ...i { { 1 n 1 { N i α +1 ) n ) dotted indices and and ...α dotted spinor indices, symmetrised on both sets. – 12 – q 2 K ...α q α n 2 ) − p ¯ α D DK n n }| p K · K q n n to anti-commute for these purposes, it follows that the }| }| D 1 n ¯ ) symmetrised undotted indices and ( }| D · α ¯ ( ˙ p p D 1 ). The spacetime constraint on ∼ · α D − ( ( K 1 p ∂ = 4 is roughly speaking the square of K n · and α q z ¯ ( ˙ D · ¯ 1 D q D z z     D undotted and α p z ( D         n carries an anti-fundamental index. As in the ( ∼ ∂ D   ¯ ∼ ∼ ∼ will have the form D K ∼ q K ¯ K D K also carries an internal U( q ¯ q p DK DK must both be less or equal to the smaller of ¯ ¯ D D D D q p p D D and The situation in ) and we use two-component spinors. A symmetric traceless SC s we will get a tensor with ( ¯ C p D , q ) fundamental indices while leaving the dotted indices unto = 4. N ( Similar constraints hold for all the descendants, where the box witha a dotted dot corresponds one. to an undotted spinor and It can be represented by a pair of tableaux, where the internal indices have been suppressed. representation, while becomes an object with is SL(2 Clearly D The constraints are implies that that the larger spin representations in will then give a tensor with ( indices in the anti-fundamental representation of untouched. Since we take u apply or, in tableaux form, components proofs JHEP_092P_1215 - ) ε ). 5) N , nor N wise ( (2 x are (4.15) (4.16) (4.14) SU(2) sp su ∼ = r whether (4) Young ∗ su ish. obtained from that ) (i.e. Sp(1) KT constraint. This K N m = 0. In terms of Young } l. Thus we have ∂D , (4) representations, but the l axis. Thus we get +1 aised or lowered using the { ∗ n cible representations of ting below those of the original escendants: the component of su . ...a 2 { a K 1 a p { n }| ∂ · q ) representations by removing the sym- + 1) columns and two rows must vanish. is · · . (4) representations but they also give the N n n we get two ( ∗ -extended case is Sp( }| · · · q · · K sp N su K · · · · p · · – 13 – z with ( ) that can be further decomposed into irreducibles to N D z ∂K (2 ), and the sum is over all possible tableaux compatible X , again ignoring spacetime derivatives for the moment, su N ∼ m ∼ 2 D In the preceding subsections we have computed the com- K n, K = 6 is a little more complicated. The spin group Spin(1 corresponds to a single box with a dot in it to represent the m D D 6. αi -expansion when the top component satisfies the standard con min( , . Applying θ ) tableaux are obtained by rotating the bottom two rows clock D 4 ≤ N , N 2 p th rank symmetric traceless tensor corresponds to the (4), a non-compact form of SU(4), for which the Young tableau (2 ∗ = 1) case). A vector may be written as an antisymmetric bi-spi n ≤ ... = 3 su N SU columns and 2 rows, D = 1 m, q n i ). One might ask whether this leads to an irreducible object o = ). The The situation in q N by appending one extra box in each of the first two rows must van 2.16 ( + in the representation corresponding to the tableaux for p sp K ) index K m m = 6. N D ( ponents of a SCKT in a straint ( larger one withimplies the that extra when box wethe in act resulting with the Young tableaux first will havetableau, row extra but vanishes dotted of boxes by course si the there cannot SC be more than 4 rows overal Remarks on in the minimal ( similar, while the R-symmetry group in the sp tableaux this means that the one in under plectic traces. The spacetime constraint on is isomorphic to tableau with which can be decomposed into irreducible where Instead they are representations of The spinorial derivative of internal symmetry ones, although not immediately as irredu diagrams of the form D through ninety degrees and then reflecting about the vertica with these rules. These tableaux give the ∂D This constraint implies similar constraints for all of the d (note that pairs oftensor), anti-symmetrised so spinor that indices an can be r proofs JHEP_092P_1215 4) | (4 (4.18) (4.17) psl ) symmetrised q ( p , so that its trace n above can clearly ; differentiating this ˙ k βj ion does not contain , connected by arrows α i n := K ˙ ≤ α pinorial; we might refer to α es are not in the non-super = 6. This result is related ebras can be reducible but is actually the parameter of gher-up in the diagram. A K p, q ) standard SCKV constraint for

ise for higher-order SCKTs. In k N . ❅❘ αi ); ✠ ˙ = 4 SCKT in a diamond-shaped β ❅ D . Now this could be the leading β )) antisymmetrised lower (upper) ˙ ˙ i α D α q -supersymmetry parameter, so that p, q d be imposed. It is well-known that K α ¯ S D k (0,1)

− βi

− ❅❘ ✠ ❅❘ D n ˙ ˙ i ✠ α β ❅ ❅ α ¯ D is the ∂ ) (( (0,0) (2,2) αi ˙ p

❅❘

β 4) β D ✠ ❅ ✠ ❅❘ − − – 14 – K = 4. In this case ❅ n βi N ( (1,0) (2,1) (1,2) N D ˙ β

) therefore represents a tensor with ∝ = 2, we have the diagram: α ❅❘ ✠ k ∂ n acting respectively to the left or right down the diagram, ❅ p, q αi = 4 SCKV because the superconformal algebra is (2,0) (1,1) (0,2) D D ] which does not form a part of the superconformal algebra. -components of a general or 47 θ ,N ¯ D 1); it represents a tensor of the form = 4 , = 0 only when D k algebra [ we get 0). The vertex ( Y 4). Consider the quantity ( , D | (1) (4 u sl = 4; it must be imposed separately. ]. We shall come back to this point in section 6. ,N 14 It is possible that similar additional constraints could ar The second comment we wish to make is that the definitions give = 4 the so-called over the internal symmetry indices gives a vector it is possible to set component of aany SCKV terms in involving the spacetimesimple case derivatives computation that of shows its components that supersymmetry this hi can variat happen only for to the factindecomposable that even some though tensor thecase representations corresponding [ symmetry of typ Lie superalg Consider the vertex (1 with respect to On the right-hand side the object rather than general we can arrange the Notice that this additional constraintD is not implied by the this is the case for a higher-order constraints (in fermionic derivatives) coul internal indices. For example, for undotted (dotted) spinor indices, and ( array with each vertex labelled by a pair of integers, ( be generalised to the case where the leading components are s representing the action of and ending at (0 proofs JHEP_092P_1215 and (4.19) (4.24) (4.23) (4.21) (4.20) (4.22) γα ]. One 9 = 4 one , η N, αβ equation to D 2 η , . = 1 [ ) implies that , ) βγ
) ... η ith N S , γk βj 4.24 e β ρ αβ α = 1 ↔ δ η αβ ). Differentiating this η derneath the first two h + i, j αi N = 1, ( + β its spinorial counterpart, α k boxes in the first row and ) N ble superconformal group, β L β ) + ( ( α n ects is, but clearly they exist ρ γ j i L ommutator we find nfused with Killing spinors in α β 4 : ( δ lgebras. γ L ations, spinor, gamma-traceless . t difficult to verify that the last η α + 2 j ,... D i = 0 , jk L i ) ij ) η β η 10 = 1 γj γ αk α ζ ζ − δ λ β ki ]. j ( ) + ( η = i α = 46 S αi η η + δk ) as σ βj βj ρ γi δ D = 2 = = 2 K ζ γ – 15 – , α, β ik 4.21 ) αi S jk jk η β a βγ i (undotted,dotted) indices again subject to similar η D γ αi L L K ( + n + D αi αi the Lorentz parameter, ik αβ αβ D ) ) b D 6= γk L a a a (4). It is easy to confirm this by a direct calculation. A ρ ; γ γ βj L ) parameter. Differentiating the second of these equations ij S F ( b , m i η D N ) a ), and subject to similar constraints, while in ( = ( ( − δ γ = 6 one could have objects with = 3 we could consider objects with an odd number of indices α sp 4.1 = γk δ m, n + 2 D D K b βj a a = ( ∂ K L γ k is the ij satisfies the standard constraint αβ ). In η = ) K ji a a a b L αβ ∂ γ K ) ( K 4.10 a = a αβ ij γ ) ∂ ( ij a iη i = 1 SCKVs were discussed previously in [ γ L denotes the symplectic “metric” on spinor space. The terms w In five dimensions, superconformal symmetry only exists for ( is the symplectic “metric” for the R-symmetry group Sp( = is the scale parameter, ij ,N 1 this equation has no non-trivial solution and one can use th a αβ ij iη η S η K = 5 1) in the second again with the derivatives required to sit un must vanish separately, from which we find, from the terms wit αi D = 5. − N > D 10 βγ n η If where There are three possiblevector-spinor spinor and representations gamma-traceless in this tensor-spinor.two equ It must be is zero no so we are left with just simple spinors. We set and using the supersymmetry algebra again we find constraints to ( where show that all of the spinors must vanish. On the other hand, if putative SCKV with respect to another odd derivative and taking the anti-c can understand this fromthe the exceptional fact Lie supergroup that there is only one possi ( while we can write the left-hand-side of ( and where rows. It is not clearand what should the give geometrical rise meaning to of representations such obj ofD superconformal a with similar tableauxcould to have ( tensors with ( where supergravity). Thus for such objects as superconformal Killing spinors (not to be co proofs JHEP_092P_1215 (4.25) (4.31) (4.28) (4.29) (4.30) (4.26) (4.27) oost. Hence we 1, this analysis aceless. We can . We have N > n we can raise and lower k (2), because the internal ableau calculus. For the For is symplectic-traceless, so sp two spinor representations tableau, with a single box, . e represented by a single-box αβ } n spinors in the tableau. K , ...a tion of 2 { a . So in this case there is one spinor i (2) index, but does represent the ρ (2), and the constraint implies that K sp 2 1 · sp we get αβ = ) , 1 , D . In other words, the derivative removes a ∗ ∗ ∗ { = 1 as anticipated. , αi n + ρ, σ γ }| { ( 2 3 D ∼ ∼ N ∗ · ∼ in − a K – 16 – − 2 ∼ = DK K D = λ ) and replaces it with an asterisked box which only n z (2). In spinor indices DK n }| ...a ∼ sp 4.26 1 a K K αi D z ∼ equation then gives DK (1) content. Applying a second βγ is traceless, and the tensor-spinor on the right is gamma-tr η sp is constant which in turn implies that there is no conformal b n S = 1 we can also have SCKTs obeying the usual constraint ...a = 1 we have 1 a N N K , and the This can be generalised very easily to SCKTs of arbitrary ran For ζ (1) internal symmetry doublet carried by = that the tableau has thetableau trace with removed. a Applying dot, a we derivativ find indices have the opposite symmetrisation properties to the analyse the components ofcase such of an object by making use of the t conclude that one can only have SCKVs for as far as the spin group is concerned. We therefore write sp the larger representation mustbut be in absent. fact theyspinor We are therefore indices the with have same, the symplectic as “metric” one we can only easily need check. one Because where shows that which implies which represents a Lorentz scalar in the triplet representa represents the where the box with an asterisk does not carry an The second diagram decomposes into the 4 + 16 in as a tableeau of the spin algebra which can be identified with the S-supersymmetry parameter. ρ a box from the original diagram ( proofs JHEP_092P_1215 c ∼ en for 16). α x (4.36) (4.35) (4.32) (4.33) (4.34) D 0). So n, , is 0 , , with the 0 K } min( property is , 5 0 , 4 1) then . , ≤ n, , 3 0 m , , m 2 , 0 , ...β { , 1 1 ∗ 0 β } , n ∗ 0) representation is set ∈ { (0 , escent must stop at level sily enough using Dynkin ...a k 1 , and there are space-time d that th rank SCKT , , x ∼ n +1 the same row are antisym- 0 an be computed in terms of , K , m bout the representations in a itself. The expansion in α n 0 a β 3 while the two 16-component }| 0); (31010); (30020)+(22000); θ . } , is a corresponding set of gamma n K K 2 n, { , ∗ ∗ ). m ...a has Dynkin labels ( β ] 2 ; they are symmetric on their spinor ∗ ∗ a m = 1 2.16 K αβ α K ) ) k -supersymmetry transformations or conformal t ( -indices must also be antisymmetrised z th slot, where a S m αβ k β γ a ) n + 1 γ = 6. There are seven levels. The represen- }| , ( a { { { n ∗ ∗ ∗ ∗ 5. If we take γ th descendant of ... ( , 1 m ∗ – 17 – in β -forms for 1 -expansion depends on k = 4 α = θ [ z th rank tensor ) 1). The constraint on an n k 1 n , a . n n ∼ , and the antisymmetrisation on the right-hand side }| }| { 0 ] ...a m , 1 γ n a 0 − K ( α 4 , n 2 K 0 ∼ D , x α n is symmetric traceless on its even indices and antisymmetri 1) D ...D 2 ...a z z m − 1 α a = 10 supersymmetry the spinors are Majorana-Weyl with sixte n ∼ ∼ D ...β K 1 1 β α m ,D [ is understood to be totally symplectic-traceless, and this K K m 3 2 D − ...α n K D D 1 = 1 0) + (( α , := ...a 1 N 1 D , -indices only. It implies that the a 0 1. Each component in the m α , In K 0 ...α 0). A symmetric, traceless The basic representations have a 1 in the 1 , n > n, α 1 will therefore go up to , 11 D 0 = ( We give an example of such a SCKT for , K We use the term SCKT here even though there are no 0 = 10. , m 11 tations are, starting from the top: (60000); (50001); (4010 is over the Dynkin labels fairly easily. The representations at each level need not be reducible but c on its odd indices. The descendants will be non-zero provide Here and hence to zero. We can iterate this to obtain the DK constraints that follow from the usual CKT constraint on which in terms of Dynkin labels means that the larger ( D (0 indices. Spinor indices cannot bematrices raised with or lowered upper so indices. there SCKT It using Young is tableaux, no but instead longerlabels. we useful can work to them think out a ea The diagram for and, finally, spinor representations are given by metrised there cannot be morefour than for two of them, and hence the d D inherited by the descendants. Since the internal indices on other four labels being zero. They give boosts. This is because we still use the same basic constrain components. We denote the gamma-matrices by ( proofs JHEP_092P_1215 . ]. ]. a 60 61 K – 58 . Thus ], while 4 ] in four x ]. All the 63 53 , 57 4. For exam- , 52 , 56 , = 3 24 ] where it was found D 55 is also constant. This , but otherwise only one. expansion of the leading arameters are no longer α mpared to Minkowski su- Minkowski spaces consid- 54 Lorentz group is extended ets of the superconformal n some open subset in the K K , 4 coordinates is maximal, and 6 anifolds [ e there are conformal boosts even, for s than the associated conven- D one in which local coordinates e even directions). The spaces dinates. They were first intro- ki space as a component of the ese spaces correspond to fields ] and as harmonic superspace n [ al and odd ones. We shall be in- ct to the odd derivatives. These N 64 se the notion of chiral superfields. istor geometry, see, for example, [ ] and later developed in [ is the standard conformal one, but 62 a 4 and 6 in the context of analytic su- , -supersymmetry transformations in K ] or projective superspaces [ S 51 = 3 – D 48 They have additional even sectors, cosets of the – 18 – 12 = 4 superconformal field theory see, for example, [ although we do not have a complete proof. and uses the defining constraint one easily finds that n N α = 6 superconformal field theory in an analytic superspace D . It is a quotient of the complexified conformal group by the D 4 ]. = 4 complex Minkowski space can be regarded as the Grass- C whereas for the true conformal case it would extend to 66 2 D x = 3 harmonic superspace was introduced in [ -supersymmetry. For example, for a second-rank tensor, the ]. A detailed study of S For an SCKV the derivative constraint on Let us recall that D One can consider these spaces as supersymmetric versions of tw 65 = 6 was first discussed as projective superspace in [ 12 tional (Minkowski) superspaces, such thaton superfields Minkowski superspace on satisfying th constraintsso-called with Grassmann- respe (or G-) analytic superfieldsTypically generali they also depend onduced in additional the internal physics even literature coor as harmonic [ perspaces. These are superspaces with fewer odd coordinate component goes up to For the higher rankconstant SCKTs but obey the stronger generalisedand constraints scale than and in Lorentz the case p wher 5 Analytic superspace In this section we shall discuss SCKTs in the only difference between theseby objects a scale and transformation. SKTs is that the both the scale andconfirms Lorentz that parameters there are are constant no and conformal that boosts or if one differentiates this with This seems to be the case for all in [ setting can be found in [ mannian of two-planes in terested in those for whichwe the shall reduction in also the restrict number our of attention odd to the simpler cases of D purely even part, andered indeed as there cosets of isR-symmetry the a conformal groups, formal groups. and resemblance reduced to perspace. number of The analytic odd superspace coordinatesare formalism co employed we for shall all use of is the coordinates including the intern dimensions. More general treatments wereconvenient later to developed i work ingroups complexified with parabolic superspaces isotropy definedfields groups: as are these taken cos are to flag bespacetime superm sector holomorphic, as and the we cosetswe shall themselves shall usually are consider compact work all (in o contain th standard complexified Minkows (21010); (20100). So there are two representations at level ples of this formalism applied to proofs JHEP_092P_1215 = are and rnal ′ (5.4) (5.5) (5.3) (5.1) (5.2) ′ , i.e. 2 AA αβ aα ) ξ x ′ M X | γ 2 ) | ′ n M | and 2 ...β ′ 2 ). This can be odd directions. C ,β K, αa are internal indices. n ξ · ′ M ) are super-indices ∂ ′ ...β ′ 2 β a, a p between Minkowski β ′ , a 1 ′ α β 4 matrices that consists δ α β K × α ′ er case. ′ δ and odd to can then be represented by = ( γ αγ ) 1 2 symmetric matrix ′ ′ n b ∂ of 4 ′ 1 × ′ β ...β ( α ′ 2 The super-coordinates ,A ) + δ indices. ′ ) ′ , γ,β + βγ ) , is a 2 β n ′ n 13 α, a ! K . x ! ′ ...β ...β X x 2 ′ , 1 M αγ and = ( β ∂ 1 0 1 1 0 1 ′ γ,β ! β K ) β A ′ ′

n 0 α γγ δ ...β ∂ → → L LL = 4 dotted spinor indices, while – 19 – 2 ′ 1 + β = 3 where β x

( ′ X D ′ K ∋ α D ′ γβ . Here ) matrix, and such that the full matrix is an element ∋ as follows, δ ′ are odd. for = 4, analytic superspace is the super Grassmannian 1 γα ′ K M ′ n M β α ′ are the even spacetime coordinates, ∂ | M ( AA in two-component spinor notation. A conformal Killing 1 D α ′ a, a γα ′ run form 1 to β ...α (2 X δ ( α ′ 1 ∂ ) with isotropy subgroup consisting of matrices of the form δ ′ in αα n αα β × ,α ( odd. Analytic superspace is the coset space of the complex b x x α ) n M n δ 2 a M + ( | M are internal coordinates representing (a patch of) the inte M 4 antisymmetric matrix. A, A ...α instead of ˙ 1 | , complex flat superspace with 4 even and 2 1 ′ = a ′ α × α M = 2 ′ n aa 2 = | K y 4 ′ ...β N ′ 1 C ), where ββ 2 matrix ′ ,β is a 4 n 2 while aa K × ′ , x ) is replaced by ...β , y ′ αα 1 denotes a (2 β ∂ satisfies are constants that can be computed by consistency (see below 5.1 aα to be even indices while = 1 L ′ K 1 ′ For even ′ , ξ ′ ′ and the conformal group can be represented by odd, 2 even and αα , b is the 2 in ( αα )-planes in 1 α, α αa K ∂ a ′ x M M 2 blocks with a zero block in the top right-hand corner. The ma α, α M = 6 where , ξ | ′ αα The above constructions generalise rather easily to the sup One has similar constructions for × In section 5 only we use = 4. x D αα 13 x D It is actually more convenient to change the ordering of even of (2 where each even, so in where the parentheses apply separately to the an element of the group of the form of the superconformal group. A point in analytic superspace superconformal group SL(4 vector generalised to a CKT where space of 2 where with ( odd coordinates and isotropy group that preserves a two-plane; this is the group We take proofs JHEP_092P_1215 ). ) M that ′ (5.6) (5.7) (5.9) (5.8) 2 s this C | (5.10) (5.11) ) does ) z ′ n (4 a, b ) can be 6= 4, one sl ...B . ′ 2 5.6 N ′ . (These are ,B n CC M = 3 and 6 cases. , K while the full di- ...B ′ ′ is then consistent − 1 t D B , graded symmetric BA CC 6= 2, i.e. X ′ n ∂ K ′ ′ = 2 X B B ...A t M ′ ′ , ′ 1 AC A ′ B ∂ ,A δ c peralgebra to be C ′ .e. 1 ′ n ) B B ′ n tensors. A SCKT is a tensor ch are necessary to maintain ( A ′ AB ight-hand side of ( ...A δ parameter. We shall therefore ). For A 1 ...B 1 δ X ′ 2 conformal transformation on the A b Y , M r-traces, so that the zeroes cancel to be covariant and we do not include + 2 + K 2 | C,B ) ′ (1) ′ n ) ) + 1 is. n A u (4 ′ n t ′ , ( term give factors of ...B gl B t ′ 1 BC ...B 1 ! d − 2 ′ a indices. As well as the constants K B leading to the observation that str( ′ C,B = ′ ) 1 + AB c d a b n K d n n ′ B AC − b − X − ∂ ′ ...B n CC

+ 2 – 20 – and B ∂ . Similar remarks apply in the ′ ′ B and ′ = 0.) It is evident from these formulae that the 1 2 = = a A t B n K δ BA ( = 1. In this case a super-conformal Killing vector is 1 z n B ′ t ′ = 0. This is not a real problem because the partial t M A X + ). Differentiating with respect to n δ , n CA = ′ . Since analytic superspace is defined as a coset, we can B t 1 ∂ n A 1 B 5.6 M ( CB a a 2 = 4 B ( A C K δ + A ′ N ′ δ n and satisfies the differential equation ( b CA ′ n ) does act and defines the AA ∂ a z + b AA B = = A K δ ′ ′ n ) = 0 as an additional constraint in this case. We shall discus ( -planes in z 1 = 4 analytic superspace of the form AA a ...B term gives a factor of M ′ 1 1 = D ,B b δX ′ n The constants are given by BB ...B 14 1 K B ′ ) because the partial divergences in the K = 4, however, str( ′ AA 5.8 -extended ∂ N AA Throughout the paper all tensorial equations are understood N ∂ 14 being the super-trace over the primed or unprimed indices, i further below in the context of reducibility problems. The r have to impose str( For not act on the coordinates. So in these cases we can take the su given by the right-hand side of ( coefficients are singular whenever can take out the supertraces from both corresponding to the element also valid in the bosonic case when divergences themselves contain factors involving theout. supe An example is given by where of the Lie superalgebra which we could take to be straightforwardly work out the effect oflocal infinitesimal coordinates; super it is given by with ( t where the bracket refer to the sets of identified as a SCKV Grassmannian of on primed and unprimed indices, obeying the constraint As in the bosonic caseon this can be generalised to higher-rank appear in the abovecovariance. equations there are also sign factors whi vergence in the Grassmann sign factors explicitly; it is always possible to do th proofs JHEP_092P_1215 1 in X X X s. As (5.14) (5.13) (5.12) (5.15) ), and  ′ es and, X B M ′ | A ) (as men- ′ D R-symmetry N ′ | . By isotropic C d (2 M ′  2 ′ | 4 B ). This consists of BD ′ C SpO th row corresponds D X ′ k M th rank SCKT) that ′ 2 n | he local coordinates for DC vanishing scalar product AC (2 d ′ X s. For example, in the e indicated dimensions. Al- spo ditional minus sign when the DA + we can define (complex) ana- ′  )-planes in X ′ ′ B M CB ′ M | A BD ′ X ). The solution can be represented C C = 2 X ′ ′ , B 5.9 , ! d N ′ BA AC , ! ′ ! ′ M = 0 0 C X AC 0 I )-matrix with only non-zero element b M 0 which we can take to have the form ) indicates (for an st ′ DB 2 + X 0 J L M 1 ′ 0 1 ), but where now the | J 2 X AC p, q 0 1 only one of which is independent. + ′ J 1 −

(4 ′ – 21 – X DB 2) but we have written it in the reverse order to into two halves each with dimension (2 d | +

CA = 4.18 × X + CB ′ N M X ) = = J ′ 2 J | X and L ′ I 4 J CA M CD | DB A a ′ C X C ′ D X are understood to be symmetrised. For each power of ′ ′ )) covariant indices that are contracted with the C q ′ AB BD b CA ′ a B − ′ X  X ′ ) satisfying A n BD + ′ X CD )) belongs to the super Lie algebra 5.7 ′ = 2: ) (( B CDD , while the vertex ( ′ ) free, symmetrised contravariant unprimed (primed) indic and p k A AB 5.5 n q AC ( a a − X p  X  AB AB,A n b + + + = 3 the spacetime conformal group is symplectic (Sp(2)), the = ) and the superconformal groups are orthosymplectic, ′ D B N of the form ( ′ ( In O L AB,A K It is rather easy to solve the SCKT equation ( = 3. the top right (as in ( we mean that any pairwith of respect vectors to belonging the to super-symplectic such tensor a plane have where the non-zero entriesternatively, are we can the say identity that matrices the with (4 th where we have split thewhere full space indicate that Sp(2) islytic the superspace spacetime as part). follows: When it is the space of isotropic (2 where the indices terms, there can be traces in both tioned above, this is the same as OSp( group is D except the zeroth and fourth, there are redundant parameter an example, consider the parameter has to the terms with by a diamond structure of the type ( correspondingly, ( where the super-transpose is thefirst matrix index transpose is with an odd ad and the second even. This then implies that t matrices proofs JHEP_092P_1215 t is G = 3. (5.17) (5.18) (5.21) (5.16) (5.19) (5.20) ), and D the odd N , | αb ξ g from 1 to CD ) n 2 . Analytic super- ...B 3 , B are given by n 2 K , a, b ) satisfying ...A CD 1) ∂ +1 5.7 superconformal groups are 2 m for which there are indecom- ssary to subtract out super- − B ants A e indicated dimensions. The 2 n m 1) A M δ − 1 th rank tensor satisfying ...B + 2 B 1 n n = 2 ( t B 1 (2 )( a A N , but now the internal and spacetime , where the symmetry is graded. We , n n δ m , ! 2 N B n of the form ( 2 BA | b ! m N ! = 3 with the difference that the roles of + 2 8 and = 0 L 1 0 A X t N + 0 I C = 3 case. The ortho-symplectic metric 0 n ( − st D C = 0 J L ) − D 4 4 0 1 0 n 1 1 G 2 ...X

= – 22 – into two halves each with dimension (4 AB 1

+ = n B ...B = 3 case but with the internal and spacetime even X b 1 N 2 are the complex spacetime coordinates, G = = 2 G A B | L D 8 I J αβ X K ) planes in C ) is x n C , are N ) ) this does not cause any difficulties for SCKTs in n -indices are totally graded-symmetrised. Note that in this | 2 =0 =2 M A n A X M m 5.16 m | 4 ∂ is an element of this super-algebra with non-zero upper righ + 2 1 t (2 = = 2 B ( X gl ) consists of matrices n 1 ), where = 2 N A is a totally graded-symmetric 2 ab N ( n | δ ...A a N , y 1 2 (8 n the internal coordinates, with the internal indices runnin A a αb ...A ab K osp 1 , ξ = is again the supertrace. y A n αβ 2 M K x = 6 is similar in some ways to ...B − 1 = ( D B ), where the symplectic groups are now the R-symmetry groups = 2 K AB N 2 t | A X 1 The solution to equation ( A = 6. .. A SCKT ∂ co-ordinates and where have where it is understood that the elements only. This is similar to the case, there are no redundanttraces. parameters Thus, so although there that could it be is values of not nece analytic superspace, for posable representations of where the brackets denote graded symmetrisation. The const and the coordinate matrix D M where we have split the full space OSp(8 the symplectic and orthogonal groups are interchanged. The super Lie algebra where the non-zero entries are the identity matrices with th where space is the space of isotropic (4 sectors are interchanged compared with the proofs JHEP_092P_1215 r ). ), Q ab | , y P (5.24) (5.22) (5.23) s, and ( αb gl , ξ αβ ) is to derive the ′ x ,  = ( A, A factors of the unit ,... 2 2, because both the p AB D 1 X 1) terms) N/ ,D vanishes if it is graded- ] 2 − ndices ( = = 0). In other words the C riefly discuss this problem K 1 n graded) symmetry structure C ect to a given pair of indices. ,... ensor representations with M + cyclic) 2 + ( K dinates , C . · ,... 1 2 ,... raded antisymmetrisation. It can ) does not lead to an irreducible M ∂ ,C 2 e which have C 3) 2 2 terms take care of themselves. We ) (5.25) 1 ) Lie superalgebras. The formula C − B p B 1 C ] ] − a ), but where the symmetrisations are 2.16 2 1 M pairs, and where the sum in the third t | , with graded antisymmetry on each K − n A ,B D,C · n δ ] = 2 − (2 2 ,... 2 1 + 4.14 2 ∂ 1) terms altogether. In the expression on A t ] gl B B t 1 C [ 2 + 1) 1 1 ) + 1 A − B . When this equation is satisfied one can in = 1 [ K ] s n A ,C 2)( [ 2 n ( D 2 p Q 3 K ] δ ( = 4 Minkowski superspace we saw that there A + 2 B − − n δ 1 − − − – 23 – A r 1 2 1 n D ∂ s 4 ,B X n B P 1 [ 2 satisfies the constraint + 1 B + A  + In [ t = 1 A = 4. Here 1 [ r terms, while the t ( = 2(( A K t δ n A [ ( b b D + 1 δ K = = N 6 n n n b n n a b a − + = ( , are graded antisymmetric, so that we indeed have six even ,... 2 ,C 1 covariant indices, taken to be totally symmetric on both set ,...N ), but which remain super-traceless. It is straightforward ,C ) holds for the 2 s B r, s 1 = 1 4.14 B K 2 min( , a A 4 1 ≤ A is the relevant super-trace, p ∂ ,... t Let us consider first the case A SCKT now has the form = 1 primed and unprimed indices are acted on by principle have indecomposability problems. following formula for when this can happen: becomes where matrix, totally super-traceless, by looking for tensors of this typ contravariant and be checked that these termsof are the necessary tableau to ( ensure that the ( the third line for each selected pair of pairs there is total g system and that furtherin constraints can ASS. be In imposed. all cases Here the we Lie b superalgebra that acts on the ASS i understood to be graded. A SCKT where in the second line the cyclic sum is over the line is over all distinct pairs of pairs, i.e. We can study reducibility problems for finite-dimensional t have used the dot notationThe to coefficients denote the are divergence given with by resp α dimensions interchanged. Thus, in this case, the local coor Reducibility problems in ASS. spacetime coordinates. can be cases where the standard SCKT constraint ( pair and symmetry under the interchange of pairs. Moreover, antisymmetrised on any three indices (e.g. symmetry structure corresponds to the tableaux ( proofs JHEP_092P_1215 z 2) 2) , , ) = (6.1) (6.2) ). In ) = 0, ), and 4). For r, s z | ,D ) = (2 (4 (2 5.25 o ) cannot be , which has r, s psl ). z 5.6 N f the conformal | ly. For the (1 (4 sl we found a problem ntation of n then explicitly show ). For example, when , so that there are two ints to get rid of half of d e does not have an anal- ,D mposable representations parameters have ( his is not surprising given ebra is (2 = 6, while for ( and algebras act linearly. d r Minkowski space to get rid O l traces are removed. Thus 4). This additional constraint N a | (4 and sl with constant coefficients which can a . , = 1 to get a solution to ( x { p -dimensional spacetime as a surface in = 0 D = } s ), the +1 n = n }| r 5.12 ...a 2 – 24 – a = 4. , and this implies that the algebra is = 6. These reducibility problems correspond to K X 1 N ] N a -box tableau that represents the CKT in spacetime { ) as a representation of the conformal group. z 16 ∂ n 6.2 parameters, but we are then left with a residual invariance only one of which is independent. Since the unit matrix in d th rank conformal Killing tensor on flat spacetime satisfies = 2. However, we can impose the constraint that str( = 6. The problem can be resolved in a similar fashion to the n X = 1 so that there is a problem for M and N p ), so that the full superalgebra is a = 2. In equation ( the super-traces can be removed from 5.7 n N = 2 and still have p 2) with respect to the unprimed (primed) indices respective = 4. This stems from the fact that the matrix parameters in ( , transformation. Once this has been done, the unit matrix in (1 N , Y th rank CKT has components that fall into the representation o 2) is given by ( n , (1) + 2)-dimensional space with two timelike directions. One ca 1) cases we have u z , = 1, i.e. SCKVs, we must also have = 2 case discussed previously in super Minkowski space where (2 , D In the following section we shall address the issue of indeco This picture does not generalise to the super case because on Now consider n n 1) = 4 case we have just discussed. We can impose further constra = 1, the components of a CKV together form the adjoint represe , that an group given by the Young tableau [ flat ( n the general case one can use the representation of 6 Components of superconformalIn Killing the tensors purely even case an for SCKTs in super-twistor spaces where the superconformal the super-traces in the similar to the projective symmetry in be assembled into a representation of the conformal group, This equation can be solved as a finite power series in we can choose N and (2 the of this sort precisely for (2 where where the representationthe is original taken (irreducible) to one-row be irreducible, i.e. al ogous superconformal embedding of super Minkowski space. T this implies made super-traceless for determines the two-row tableau ( For is equivalent to the extraof condition the that was imposed in supe vanishing super-trace, does not act on all other values of “scale” transformations of does not act, we can simply set it equal to zero, so that the alg proofs JHEP_092P_1215 - ) ) n a N ani- | 4.1 ] are (6.5) (6.4) (6.3) 14 can be th rank er Young n K and the . K (2). We know efined by (
bled into a rep- sp ) and OSp(8 m (2) is indeed the 2 N | sp ...A = 6. These are the 1 r on the appropriate ) as follows: . Again the SCKT is A D SL(4 can be assembled into a M N e. We can split a super- , d out into even and odd ) K for nes are antisymmetrised. = 2 N mal algebra | components of ,...,K N N 2 n (2 a tableau with one row and fact that (anti-)symmetrisation | rmal spacetime constraints and 2 8 we complexify the (super)spaces , th descendant will correspond to ). Putting all this together, we C { ...A 4&6 (apart from the exceptional m N SpO , ( +1 o , m (5). ) A = 3 o (2): A 4 and m , sl D ,Z n ...A 2 }| 1 A indices (when = 3 A Z ) = 3 case for arbitrary – 25 – M D | = ( (2); in fact the (2 ) boxes. In addition this descendant will transform D A sp C for m ,...,K Z z n − 2 N | ∼ ] and have been used in the discussion of superparticles in a m n 4 (2), and we know the supersymmetric descendants from 2 ...A -box tableau in C 2 sl K ]. n A ). Solving this we find that the components of 1 68 A , 4.5 K 67 -extended superconformal algebra. Super Young tableaux [ into a pair of = 3. Let us start with the purely even case again. An (2) with (2 ) → N D sp N | n 2 ). (4 A = 3 C ) given by exactly the same tableau but now considered as a sup ... 1 D 5.22 N (5) and it is easy to check that this representation in | A = 3 case no reducibility problems arise; the super-tensors d o ) as a diagram in (2 K 15 D ∼ = 4.1 = 5 case). However, the components of a SCKT can still be assem spo = 3 th rank totally antisymmetric tensor under (2) m ,D D sp It is straightforward to relate this discussion to the ASS on In the Now consider the supersymmetric Super-twistors were introduced in [ = 1 15 tableau for the Clearly there is a one-to-one correspondence between these as an parameters in ( CKT is given by the one-row 2 then the representation of twistor index for tensor of the discussion given in sectioncan be 4. computed All as of representations of these satisfy confo can see that all of the components of the SCKT determined by festly superconformal context [ is replaced by super-(anti-)symmetrisation.columns represents For example, acomponents, graded the symmetric even tensor; indices are when symmetrised expande while the oddare o all irreducible. interpreted in a similar way to non-super ones except for the satisfying the constraint ( represented by the same diagram,that but this time for the confor 6.1 The simplest case is that there are only superconformal groups in given by ( same as the irreducible two-row Young tableau in N resentation of the appropriate superconformalinvolved, algebra. we If can representsuper-twistor the spaces which components are of a SCKT as a tenso fundamental representation spaces for the supergroups respectively. proofs JHEP_092P_1215 ′ ]. A 14 = 4 (6.9) (6.8) (6.6) (6.7) N the epsilon tableaux to . This example B A factors of the unit K super-index, and p N ucibility problems [ | 4 ndex. The super-trace of uffice to describe all rep- C = 1. In this case the unit ors that are reducible but th former corresponding to ox by placing a bullet in it, ) super-algebra is given by a , , n super-traceless. { N according to the formula = 0 the latter would be a one- n | In more detail, this means that all super-traces between contra- rder to describe irreducible rep- B tions that cannot be removed in , corresponding to contravariant n = 4 (4 ... N 1 16 sl B N n ) (except for the special case n ) it can be seen that problems of this A }| ) which contain p , N ... . | 1 6.9 6.8 − = 1, has one index of each type and can N A (4 . = 4 case the only problem is the one just . (So for n − K sl is a contravariant N th rank SKCT: such an object is given by a z | + 3 N { 4 n ∼ = A • = 4 n • C n – 26 – B • n, t = 2 ... 1 • = = 4 are B n , N }| s n ′ • D A = • ... r 1 ′ A • with one upper and one lower index and so has the tableau th-rank SCKT is a tensor of the form z n K ) for N | 4 = 4 Lie superalgebra, so that the unit tensor has to be modded , where an upper 4 and that for the 5.24 C B N A ≥ K N 4). | 4)). Because there is no straightforward generalisation of | (4 , or ]. Instead it is necessary to introduce two basic single-box th-rank tensors of the type of ( B (4 ′ n psl 14 A denotes the number of unit tensor factors. psl , has vanishing super-trace but cannot be subtracted from K n B = 4 A ≤ δ p D is taken to vanish. An The above discussion is not complete because there can be red Reducibility problems can occur for higher values of This is a special case of ( B , while a general tableau will have left and right sections wi A 16 • tensor in the superresentations case [ a single type of tableau box does not s out to obtain where it is just corresponds to the where totally graded-symmetric on each set of indices and totally a covariant one which canK also be written as a lower unprimed i tensor, one can have type only arise for This means thatindecomposable in because some of cases the existencea there of manifestly are sub-representa covariant super-traceless way. The tens simplest example is A superconformal Killing vector,be for written the case This can be immediately generalised to an and covariant indices for the vector space denote fundamental and anti-fundamental representations tableau of the form tensor but which remain super-traceless. From ( and co-variant indices are removed. An element of the covariant indices and the latterresentations to it contravariant is ones. necessary In to o impose the requirement that The super-conformal algebras for 6.2 super-traceless tensor in column three-column tableau). We distinguish a covariant b proofs JHEP_092P_1215 z ∼ ns ′ = 1 D p t. An (6.12) (6.10) (6.11) C then we AB 4). Note, | K (4 a, d sion. We set ), but where , but then one aceless tensors psl ), with those in 6.2 K ) representations. 5.12 M ′ | , we have D 1 ′ (2 dices from bove type that are not C X ′ gl tations of , and such that the tensor B ′ , N ⊕ .4. A 2 ) | ) of the form ( ′ ′ 8 K ion it is not a straightforward A D , but although this space does D C M ′ al group, it is not a straightfor- ′ | ss condition on N C C s mean that we can add a ,Z 2 ′ ) and so can be set to zero. This ′ | (2 A 8 parameters in ( B B ′ ′ gl C Z A d A ′ D arts one can run into these problems 5.12 C K K ), for various numbers of supersymme- are . = ( AD B ′ ′ and N in ( )) | D A A ′ . For example, at CD ′ d CD ′ k K B ′ a, b B AB ′ for a vector in X str( = 2 one can extract the traceless part leaving ABC A ,Z K A − – 27 – K K AB,A ) , n ) ′ D ′ a K A C ′ = 4 (the minus sign orginates from the conventions for ,Z B ′ ′ ′ (str( A N D A ′ CD = 0 in super-twistor space. If it is not possible to remove Z C C ′ K B ′ ′ ) by opposite supertrace terms because this will not change B = ( A AB D AD A ) K A K d = 1 object which is itself indecomposable, although this tur a, d K Z n ∼ CD ′ = 0 case. Nevertheless the generalisation is not that difficul D ′ AB N C ′ K = 2, the components of B = 1. In particular, this implies that higher-order super-tr n A n K = 2 SCKT without losing super-tracelessness. = 4 case it is also possible to make contact with the ASS discus n D and = 6 ′ D . This acts linearly on the super-twistor space = 6 the super-conformal groups are OSp(8 )). If we can remove the super-traces from over the central in D C = 4 are projectively invariant and so correspond to represen th row corresponding to the terms with N In the These components can be matched to the D k 5.7 AB N a th-rank super-conformal tensor is given by a super-tableau however, that when onesuper-traceless attempts into their to irreducible decompose super-tracelessmore p tensors than once. of the For example, a for will still be free to adjust ( have an orthosymplectic invariant under theward super-conform extension of super-Minkowski space,super-version so of that the the situat can see that the combination does not contribute to the ASS SCKT n in term to the the unit tensor times an 6.3 For the super-trace free condition. In super-twistor space thi now each box corresponds to a super-index corresponds to setting (str the tries − the super-traces then one can impose the this super-tracele discussed with For example, for which allows us to decompose any tensor in terms of in ( out not to be a problem in the algebraic context discussed in 6 proofs JHEP_092P_1215 ], ith 16 (6.13) (6.15) = 6 and of the form denotes the D ), where the K Y N ) factor to the ( ⊚ N sp 2 in ( ove), and are are X ⊕ sp uct are determined by under the interchange (8) three indices vanishes. N > o ts of any CKT are given emove the super-traceless at, when the super-traces ymmetry properties of the r = 2. This tensor represen- t an algebra, known as the er, the product of two such rary CKT representations: i etries of the Laplacian [ nd the n e orthosymplectic metric on ] which is generated by ontained in the product. In maps solutions to Laplace’s metric on any pair of indices oduct can be described in Lie 69 D , we can describe the Eastwood because there are no non-trivial X,Y g h c a constant and − c ] , being considered as an odd index. ,... with 2 N ∆ (6.14) X,Y [ B 2 δ g 1 2 1 = 6, starting at B , = 2 has a leading term given by a CKT, and this − ,... D A D – 28 – 1 Y D th-rank SKCT are given by a tensor ∆ A = 1 n ⊚ that preserve the Laplacian in the sense that i K 2) it is just given by the two-row two-column tableau X D − modulo its Joseph ideal [ − g ,D Y (2 ⊗ o X = g is the Killing form on th rank tensors we simply get the traceless two-row tableau w i m = 3, so that we again have a reducibility problem. X,Y h , N g ∈ pairs of graded-antisymmetric indices, graded-symmetric is another linear differential operator. Clearly, th rank and n n δ X,Y columns. In other words, the antisymmetric terms in the prod 4 in D=4 (except for the algebra itself as we have discussed ab Note that the only problems of this type that occur have Reducibilty problems can also arise in m + (8) factor corresponds to the spacetime conformal algebra a In the purely even case conformal Killing tensors define symm therefore of limited interest insuperconformal a field physical theories context. that This exceed is these bounds. 6.4 Algebras N > tation has the gradedare symmetries removed, one of would the expect Riemann toWeyl find tensor. tensor, a so However, tensor one th withRicci the can graded tensor show s in that it is not possible to r o internal symmetry algebra, the index Furthermore, the trace withvanishes. respect to Such a the tensor orthosymplectic can be expanded out into tensors unde having of pairs, and such that graded-antisymmetrisation over any algebra as the tensor algebra of that is, linear differential operators equation to other solutions. Each such this space. Thus the components of an represented by such a tableau is traceless with respect to th algebraic terms. Denoting the conformal Lie algebra by can be extended in asymmetries natural defines way a to third lower-order (modulo terms.Eastwood the algebra. Moreov Laplacian) and In so flatby we ge space, representations of as the we have conformal seen, algebra, the so componen that this pr where where which is also traceless.for an The Cartan product extends to arbit Cartan product which isthe the conformal case highest with weight representation c n proofs JHEP_092P_1215 is +1), (6.16) ]. The n lgebra, and the 92 (2), there g , ]. Further sl is work also 81 78 – , 79 77 n with CKTs. A ]. ] is relevant in this 16 owski spaces cannot ]. These in turn can 84 – per-Laplacian more or 87 82 th rank conformal Killing 6. In the context of super ra of Minkowski space in . This algebraic definition papers on quasi-conformal n elated algebraic structures discuss in the next section. , e [ tor [ lated field on the boundary et of differential operators is 2 m gauge fields, parametrised . tions involved, while the first +1) ] and the theory of them was − etry, see e.g. [ i hat the symmetry algebra of = 3 ld be feasible to generalise the a similar type carrying a linear D rm in this product is just given symplectic metrics, Eastwood’s is bracket [ D 76 e, for example, [ ss tensor current of rank ( en with reducibility. However, in 4( ise in this situation (and studied D itter spacetime and the algebraic X,Y h c 2 ]. For the classical cases, bar − 71 Y , ⊚ 70 dimensions is X of its indices with an – 29 – 2 n D − YX such that the quotient algebras are infinite-dimensional. c + XY ]. ]. Although this is not what we are directly interested in, it 86 ]. 8 , 72 ], the algebra can be viewed as the the universal enveloping a while the symmetric terms are determined by the trace in 85 2), modulo the two-sided ideal generated by g 16 − in the conformal case in ], although from a somehwat different point of view to ours. Th ,D c 75 (2 – o 73 := g ]) should be related to superconformal symmetry for In addition, super Joseph ideals have featured in a series of The AdS/CFT correspondence can be used to make the connectio Alternatively [ In the supersymmetric case, as we have mentioned, super Mink 8 , of g say. Such a current can be contracted on extends to simple complex Lie algebras, see [ methods [ tensor, so that for each such current there is a conserved vec Fronsdal higher-spin gauge fieldwhich in couples the naturally to bulk a gives conserved, rise symmetric, tracele to a re couple to fields in the bulk so that we arrive at a set of one-for natural setting for higher-spin gaugestructures theories that is anti-de arise S AdS reflect spacetime this; is in isomorphicone to dimension particular, lower the the implies conformal thathigher-spin fact symmetry the context t algeb AdS/CFT [ correspondenc 7 Comments on higherHigher-spin spin fields were originallysubsequently introduced developed by in Fronsdal a [ seriesdevelopments of have papers included by the Vasiliev, incorporation se of supersymm has applications to higher spin and AdS/CFT, which we briefly nevertheless the case that thein algebraic [ structures that ar U is a unique value of the constant to a follow-up paper [ Minkowski space a different notionmore of natural. a super-Laplacian We as shall a postpone s a discussion of this and the r less straightforwardly [ be presented in terms ofaction higher-dimensional of superspaces the of appropriate superconformalpurely Lie group, algebraic but approach, provided it that shou duethe care context is of tak super-Euclidean spacesLaplacian equipped with symmetry ortho- formalism can be extended to a natural su The constant the Lie bracket in symmetrised traceless product. Theby leading the (symmetric) te highest weight in(antisymmetric) the term decomposition coincides of with the the representa Schouten-Nijenhu proofs JHEP_092P_1215 ) is ]; in would 2.1 eorem 72 ion, it 5 gebra is S cted rather × 5 AdS iscussion of compo- = 4 superspaces was given hese symmetry algebras er-spin structures in the elds are free, but one can e able to exhibit SCKTs ve. (In this context it is se ideas further in [ ,D structures. The question y CFTs has been clarified rs subject to the constraint hen one differentiates with s should couple to currents ction of currents in terms of tructures on the space of all It should be mentioned that le, i.e. generated by KVs, as = 1 on-abelian terms in the field = 4 super Yang-Mills theory ormal Killing tensor in curved uperspaces as well as analytic etries. inations of analytic superspace N of the general discussion is that N to zero. We then discussed these urally to placing the emphasis on ns of the appropriate superconformal ], and a more recent general study was 88 – 30 – ) (as well as some conventional ones). Since ( = 2 supersymmetry in 2.1 N = 4 analytic superspace. These currents are constructed in 17 ]. ,N 87 = 4 ]. It seems likely, but remains to be proven, that a similar th D 95 – ] that the currents on the boundary can be explicitly constru ] where it was shown that in most dimensions of the bulk this al 93 91 = 4 theory on the boundary in the way described above. In addit , 89 87 N ] it was argued that the massless limit of IIB string theory on 90 ], where a connection to superstrings was also conjectured. Although we define SCKTs in curved superspace, the detailed d In [ A study of higher spin superfields and 92 17 ordinary KTs in spacesdetailed of in constant [ curvature are all reducib nents, etc, we have given here is limited to flat superspaces. terms of two free field-strengthderivatives superfields acting with on linear them comb infree a scalar similar fields manner given to in the [ constru correspond, via the AdS/CFT correspondence, to a free holds for superspaces of this type. can then be related,bulk. via As we the mentioned AdS/CFTparticular, in correspondence, we the shall text, to study we high shall super-Laplacians develop and some their of symm the objects in the contextsuperspaces of superparticles and and super inexplicitly twistor various as flat spaces. tensors s carrying irreducible Ingroups. representatio the We also last indicated caseSCKTs how we for this wer a should given lead dimension to and algebraic number s of supersymmetries. T invariant under local scale transformationssuperconformal this Killing leads tensors. nat The mostSCKTs significant can point be defined asthat purely even the traceless smaller symmetric tenso ofrespect to the the two spinorial spin covariant derivative representations should be that set arise w straightforwardly in presented in [ in [ 8 Concluding remarks In this paper we have givensuperspace a subject general to definition the of constraint a ( superconf attempt to introduce interactionsstrengths. in the In bulk order byof including to the n do uniqueness this offor one higher-spin the has case theories of to in three-dimensional introduce AdS boundaries and algebraic in boundar [ by the CKTs, in AdS. In this picture the underlying boundary fi was shown in [ on the boundary, inin the the sense free that massless string gauge field unique and determined byknown the as the Eastwood Eastwood-Vasiliev algebra algebra). discussed abo proofs JHEP_092P_1215 K ], a (A.3) (A.2) (A.1) 96 also two ]. When 98 ry. δ } b . This gives rise L ssociated with L ) does indeed sat- γδ ) a or { β 3.18 } cation of (C)KTs is to b (Γ K y non-trivial higher-rank γ β f ( L , but there are also terms particle mechanics [ ach of the first two terms β tes in the Hamilton Jacobi K αβ iK β ) β,a 2 dα ouncil through VR grant 621- α a αβ α ) T { . Some examples of this include ) . These come from the torsion ) in the first term and using the − a uperspace. K L oup at Imperial College, London, ab (Γ ( ( ac β i ab (Γ ˜ ˜ } S S i M L b L c = = = 2 = M to ∇ a a c β are respectively a SCKV and a second- ab α αβ ) K K L Dα Dα L ∇ a α α { T T − , which is fine, and there are two other terms ∇ ∇ D } b (Γ = 2, but it would be easy to extend the proof b βD i and ] and the Perry-Myers black hole [ K K L K c – 31 – c 97 , m + K + = 2 ∇ ∇ β | β cβ c ab | = 1 K a L Dα α { M n T αβ α L ∇ ) . Since a ∇ ) some terms are in the required form straight away, but aD aβ = 2 L (Γ i A.2 M + ab ] aβ to ( L K,L α α [ ∇ ∇ − := ] for a recent discussion of this in the context of string theo in section 2). ab 99 ˜ L ). It turns out that the sum of all such terms vanish. There are functions are the generalised Lorentz and scale functions a M ) we get a term 2 ˜ S A.3 ). We do this for the case A.3 (called 2.16 L It would be an interesting challenge to see if one could find an When we apply Apart from the higher-spin connection, the other main appli rank SCKT, we have and where the for some (gamma-traceless) to the general case. We want to show that third of ( satisfies others need some work. For example, applying Here we demonstrate thatisfy ( the function on the left-hand side o as well as partial financial support2013-4245. by the Swedish Research C A Supersymmetric SN bracket SKTs in non-trivial supergravity solutions formulated in s Acknowledgments UL gratefully acknowledges the hospitality of the theory gr situations where a given spacetime admitsthe an irreducible case KT of aparticle fourth-order with tensor worldline supersymmetry foundirreducible [ in KTs the exist, they contextequation. may of See be two- [ used to separate coordina to torsion and curvature terms. The torsion terms involve like this. Whenit the has odd to derivative be hits taken the past second the factor even in derivative so e that it can act on terms in ( like this coming from differentiating the third term in proofs JHEP_092P_1215 , (A.4) (A.5) , which ommons M β γδ (1987) 1848 T (1995) 1823 (1978) 55 aδ L 12 , Germany M¨unchen γ D 35 K e use of the identities and the derivation of B 132 , − β β ]. the Poisson bracket of two cd tity. In order to show that cd T redited. T e dimension one-half Bianchi AE-PTH 32-76, Germany d s the general result. d Phys. Rev. , sum up to ng the third term in K , which is of the required form. , K SPIRE c M ]. bδ } Nucl. Phys. ac b IN L , L L γ Proceedings, Quantum Theory and The Class. Quant. Grav. − αβ K SPIRE ) , β a , in aβ IN ) is itself conserved. This method can be { [ γδ L T b (Γ , i ∇ αβ 2 K,L b ) with respect to time, and then make careful β,a − K γ (Γ) – 32 – ) i ]. = 3.5 − L } ( (1981) 106 (in memoriam Werner Heisenberg) b β ] ˜ Ideas and methods of supersymmetry and supergravity: S Cohomologies and anomalies in supersymmetric theories 2 γ K c,d (which arise from differentiating the third term and c , which arise from the other terms with even derivatives SPIRE [ K α IN ∇ d,e [ A superspace survey R c ), which permits any use, distribution and reproduction in − β,c B 179 bc | ) , IOP, Bristol U.K. (1995) [ d α L β ) L γ ( K ) L + c ˜ ( | S a a ˜ K S { ( K ˜ L S b and 4 (1985) 458 aγ ∇ d and CC-BY 4.0 Nucl. Phys. c L ) bβ , ]. β The conformal group in superspace L This article is distributed under the terms of the Creative C Supercurrents and superconformal symmetry + K α ( ) B 252 = ˜ of the type discussed in ( S Supertwistors and conformal supersymmetry K Construction of the minimal superspace translation tensor ( SPIRE aβ ]. ]. ]. ˜ S IN M )) to K,L . The final result is SPIRE SPIRE SPIRE A.3 IN IN IN the supercurrent (1977), pg. 241 and Max-Planck-inst.(1976) M¨unchen Phys., [ MPI-P Structure Of Time and Space, vol. or a walk through superspace [ [ Nucl. Phys. [ An alternative way of proving this result is to differentiate K,L [1] M.F. Sohnius, [6] P.S. Howe and G.G. Hartwell, [7] I.L. Buchbinder and S.M. Kuzenko, [3] W. Lang, [4] L. Bonora, P. Pasti and M. Tonin, [5] K.-I. Shizuya, [2] A. 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