JHEP03(2021)157 and a ξ Springer March 16, 2021 : January 11, 2021 January 29, 2021 , : : and a Published Received Accepted Published for SISSA by https://doi.org/10.1007/JHEP03(2021)157 , which proves to generate extended su- α  , supergravity backgrounds [email protected] Emmanouil S.N. Raptakis , 0) , b,c Dedicated to Jim Gates on the occasion of his 70th birthday d 0) supergravity-matter systems in six dimensions may be = (1 , N = (1 . Ulf Lindstr¨om, 3 N a 2012.08159 The Authors. c Extended Supersymmetry, Higher Spin Symmetry, Supergravity Models, Su-

General , superfields. The former parametrise conformal isometries of supergravity back- ) [email protected] n ( a ζ Box 516, SE-751 20School Uppsala, of Sweden Mathematics andSt Physics, Lucia, University Brisbane, of Queensland Queensland, 4072,E-mail: Australia [email protected] [email protected] Department of Physics M013,35 The Stirling University Highway, of Crawley WesternDepartment W.A. Australia, of 6009, Physics, Australia Faculty of06800, Arts Ankara, Turkey and Sciences, MiddleDepartment of East Physics Technical University, and Astronomy, Division of Theoretical Physics, Uppsala University, b c d a Open Access Article funded by SCOAP Keywords: perspaces ArXiv ePrint: tries of the hypermultiplet.multiplet By we studying the introduce higher theproblem symmetries of concept of computing a of higher non-conformal symmetries aand vector for demonstrate Killing the that, conformal tensor d’Alembertian beyond superfield.limited in the class curved first-order of space We case, backgrounds, also including these analyse all operators conformally the are flat defined ones. only on a the geometric symmetries ofnotion supergravity of a backgrounds. conformalperconformal Killing In transformations. spinor particular, Among superfield we itstensor introduce cousins the are the conformalgrounds, Killing which vector in turn yieldthe symmetries conformal of Killing every superconformal tensors field of theory. a Meanwhile, given background are associated with higher symme- Abstract: described using one ofgravity: the (i) two SU(2) fully superspace; fledgedrigid and superspace supersymmetric (ii) formulations field conformal for superspace. theories conformal in super- With curved motivation space, to develop this paper is devoted to the study of in six dimensions Sergei M. Kuzenko, Gabriele Tartaglino-Mazzucchelli Symmetries of JHEP03(2021)157 38 16 19 16 | 6 M 26 23 29 38 6 44 11 – 1 – 27 14 26 42 41 35 9 11 40 6 0) superconformal algebra , 14 2 35 = (1 34 N C.1 Conformal Killing vectors C.2 Conformal Killing spinors A.1 Spinors in sixA.2 dimensions The 7.1 Superconformal vector7.2 multiplet Supersymmetric Maxwell theory 3.2 Conformally related3.3 superspaces Isometries 2.1 SU(2) superspace 2.2 Conformal superspace 3.1 Conformal Killing vector superfields D From conformal to SU(2) superspace B The conformal Killing supervector fieldsC of Bosonic backgrounds A Conventions 8 Maximally supersymmetric backgrounds 9 Conclusion 5 Higher symmetries of the conformal6 d’Alembertian Higher symmetries of the hypermultiplet 7 Higher symmetries of the vector multiplet 4 Conformal Killing spinor superfields and their higher rank cousins 3 Conformal isometries Contents 1 Introduction 2 Conformal supergravity in superspace JHEP03(2021)157 - ], 6. N 9 ≤ = 2 (4). , (1.1) (1.2) F d 8 is the of the N ] for a and an bc α 10 ) the local defined by α ˆ is realised µ M N = 4, , θ δ N bc | δ d m | d A d x Ω M ], see [ , 1 2 ] superspace is a 9 M = ( I ]. It is a powerful J N 6 dimensions using = M I 13 ; . z – ]. Both formulations d Its structure group is [ K A ϕ ≤ 2 R 16 K 11 + d , G 9 = bc ≤ ϕ M . K bc (with suppressed Lorentz and δ -symmetry subgroup of the ], as a natural generalisation of A K 1 R ϕ obey standard reality conditions Φ 1 2 ) is I − = 6 realisation of the general ap- + K 1.2 A = 5. We denote by B d Ω d By definition, D ] is the B 1 − 0) supergravity in six dimensions was for- and N ξ , ; A fermionic dimensions. cb d E -symmetry connection. The index ˆ [ := = 8 for K R R N 1) is the double covering group of the connected = (1 . – 2 – K δ N d , − G ]. More recently it was further developed [ ) = δ 1 7 ˆ N α M the – = − D 2 ], superconformal algebras exist in spacetime dimensions d dimensions. , I bc a J 17 d I ] is a particular D K ]; and (ii) conformal superspace [ A , , 15 ] A 15 = ( ξ bosonic and A 4 and 6, and A , d D ] has found numerous applications, in particular the explicit ] for a related discussion. These limitations are avoided by D , 1 = 3 denotes the inverse superspace vielbein, Ω 1). This means that the differential geometry of K 14 , d ], where Spin( -extended conformal supergravity in 3 1 is, in general, composite; it is comprised of a spinor index 6 dimensions. M = [ ] superspace, where , with N N ˆ − α for ; N d A N . Without loss of generality, we assume that the zero section of = 2 supergravity [ D d δ ( | [ d < ; N N D 0 d ∂/∂z c d δ R 2 | = 1 cousins, it has two limitations. Firstly, it does not offer tools to [ K N d M δ G d/ R M b A M G × N E 0) conformal supergravity and its general off-shell couplings to supersymmetric = 2 1) = 4, , , = N 1 d δ A = 5, − = (1 E = 5 case is truly exceptional, for it allows the existence of the unique superconformal algebra ] and a higher-derivative extension of chiral gauged supergravity [ d d 8 The SU(2) superspace of [ The tensor calculus [ d According to the Nahm classification [ Here N 1 2 = 0 corresponds to the spacetime manifold -symmetry indices), its transformation law under ( -symmetry index. The supergravity gauge group includes a subgroup generated by local ˆ µ R The coordinates for θ transformations where the gauge parameters but are otherwise arbitrary. Given a tensor superfield Here Lorentz connection, andfermionic Φ operator = Φ R extended superconformal group in supermanifold Spin( Lorentz group SO in terms of covariant derivatives of the form matter: (i) SU(2) superspacehave analogues [ in proach to formulate the so-called describe off-shell charged hypermultiplets. Secondly,of it is view rather impractical of fromsupergravity, constructing the point see, nonlinear e.g., supergravityresorting actions [ to such superspace as techniques.for invariants There for exist conformal two fully fledged superspace formulations pedagogical review. construction of off-shell curvature squaredapproach supergravity to actions formulate [ supergravity-matter systems.and However, similar to its The superconformal tensor calculus for mulated by Bergshoeff, Sezgin andthat Van for Proeyen in 1986including [ the construction of theity complete [ off-shell action for minimal Poincar´esupergrav- 1 Introduction JHEP03(2021)157 ] ˆ β ϕ 4. ] is N w D = 1 ; This (1.3) (1.4) N d ], the [ ; ]. The N 1 3 = 3 and ]. This 4 N ≤ R d [ G 26 R N , 29 , G in 25 28 × C = 3, d 1) AB , 6 dimensions, first in ] who put forward T 1 stands for a linear − ˆ α 19 d < , d , D , I 18 σ ], then in J δ = 2 supergravity-matter I , where the parameter ··· 34 = 3 realisation of , N AB + σϕ d 21 ], building on the concepts of R ϕ ˆ α ] is that it offered off-shell for- w 21 D − = 6 realisation of the universal 6 dimensions, which is based on 15 = 2 [ σ = cd 1 2 d ≤ -models can be carried out using the tech- ]. The N ]. ϕ M σ = 4 by Howe [ d σ = 27 δ cd 36 ]. d , ˆ α -models coupled to supergravity. = 4, σ = 2 conformal supergravity, respectively. 35 D AB . The ellipsis in 31 d , σ K R N 30 J 1 2 0) superconformal tensor calculus of [ , − – 3 – and , δ C D = (1 cd ] is a particular C ··· = 2 case [ = 1 and ], followed by M N 16 + N 4) supergravity [ AB N , ] and superconformal projective multiplets [ a 33 T , D − 24 = 4, σ – 32 ) superspace. Its geometry was developed in [ ] provided a natural extension of the earlier construction of = (4 = 6, , . d is an arbitrary real superfield. The ellipsis in the expression = N 22 ϕ d 21 = 27 N } σ a B ] to construct off-shell supergravity-matter couplings for D D = 2, = 1 [ 29 , σ δ d ). In many dynamical systems of interest, matter superfields may be A N -extended superconformal group, of which Spin( D D [ ) superspace. In particular, he introduced the U(1) and U(2) superspace , N N N = 5, δ | d d -extended conformal supergravity in ], which provided a unified description of the off-shell formulations for ], corresponding to = 4 were constructed in U(2) superspace in [ N M 20 19 d includes, in general, a linear combination of the spinor covariant derivatives ], and finally in a 29 D σ The superspace formalism of [ As compared with the The approach sketched above was pioneered in In order to describe conformal supergravity, the superspace torsion δ The component reduction of these locally supersymmetric The concept of covariant projective supermultiplets was introduced earlier in 3 4 = 4 [ a subgroup. This approach, known as conformal superspace,niques was developed originally by developed Butter in for the the framework of N mulations for general supersymmetricwas nonlinear achieved by making use of the concept ofapproach covariant to projective supermultiplets. gauging the entire the SU(2) superspace formalism insuperspace five is dimensions known [ asformalism SO( was used in [ important advantage of the SU(2) superspace approach [ in the book [ supergravity and their couplingssystems to in matter. Generalrigid off-shell projective superspace [ four-dimensional results of [ is the super-Weyl weight of the concept of U( geometries [ Howe’s analysis was purely geometricconstructing in supergravity-matter the actions. sense The that full he power did of not U(1) address superspace the was problem revealed of for and the structure groupcombination generators of the generators ofbe the denoted structure ( group. The resultingchosen curved to superspace be will primary under the super-Weyl group, invariant under super-Weyl transformations of the form where the scale parameter must obey certain algebraic constraints,torsion-free which constraint may be in thought gravity. oftry, in as A generalisations order fundamental of to requirement the describe on conformal the supergravity, superspace is geome- that the constraints on the torsion be JHEP03(2021)157 ], ]. ]. F ]. 2 39 16 16 51 F ] is that 0) super- , 16 ]. More re- = (1 38 , 0) conformal su- 0) conformal su- N 37 , , 0) superconformal 0) supergravity in , , 0) supersymmetric , = (2 = (1 = (1 = (1 N = (1 N N N = 1 old minimal supergrav- N ] for the technical details. In ] superspace can be obtained The latter is at present much N 40 N 6 0) conformal supergravity [ – 0) and ; , , d 37 [ ], and analogous formulations have = 4, R ] that the d G 48 = (1 = (1 ], and further developed by Kugo and -extended conformal supergravity [ 0) conformal supergravity was briefly 45 , ] offered the first superspace description N N N 42 , 48 41 = (1 = 3, = 6, d N – 4 – d ] was the first construction of the locally supersymmetric ]. Ref. [ 16 49 ] within the harmonic superspace approach. 44 ] is universal, for in principle it may be generalised to ], and Several months later, these invariants were reduced to 53 40 5 ] within the framework of ]. In accordance with the above discussion, the local supertwistor 53 50 ]. ] is a gauged fixed version of the conformal superspace developed in [ 47 1 , ], which resulted in the first tensor calculus description of the conformal 46 45 ]. Unfortunately, this approach has not been pursued for over thirty years. = 2 supergravity theories in four dimensions by Butter [ 0) Weyl supermultiplet, which was originally formulated using the supercon- , 52 N ]. Their fundamental property is that they offer a universal setting to generate ]. = (2 16 43 , N 15 action coupled to conformal supergravity. In Minkowski space, the The present work is devoted to new applications of the supergravity formula- Recently, local supertwistor formulations for Conformal superspace is an ultimate formulation for conformal supergravity in the One of the important achievements of the conformal superspace approach [ A simple by-product of the analysis in [ Both constructions are based on Cartan connections, first discussed in the superspace context in [ = 1 and F 5 6 = 5 conformal supergravity [ 2 ity. The approach describedcurved in backgrounds associated [ with any supergravity theory formulated in superspace, see F action was described for the first time in [ symmetric theories that were originally constructedoff in from terms of a ordinary fields, superfieldsupersymmetric may theory be field read upon theory elimination into of a determine the the given auxiliary (conformal) supergravity fields.was isometries background, developed of In long one the order ago needs background [ to a superspace. develop formalism Such a formalism the harmonic superspace formulationdescribed in for [ tions [ off-shell supersymmetric field theories in curved space. In particular, all been proposed in diverse dimensionsof the [ formal tensor calculus [ formulation should be equivalent tomore conformal developed superspace. and is thus the one favoured in this paper. We should also mention that from a partial gauge fixingthe of conformal case superspace, of see six [ dimensions,tensor it calculus was of [ demonstrated in [ pergravity in six dimensions have been constructed [ metric completion of severalsix curvature-squared dimensions invariants [ for sense that any different off-shellby formulation partially is either fixing equivalent the to gauge it freedom. or is In obtained particular, from it Uehara [ it provided the firstpergravity ever in construction six ofcomponents dimensions. all in the [ invariants for supergravity actions. Conformal superspace has also been used to describe the supersym- cently, it has been extended tod the cases of Conceptually, conformal superspace isfor a conformal (super)gravity superspace as analogue theby of gauge Kaku, theory the Townsend and of famous van the formulation Nieuwenhuizen (super)conformal [ group pioneered N JHEP03(2021)157 ] = 71 d – (1.5) of the 63 ϕ ]. The = 0, where 57 ] dimensions, ϕ – , which allows 56 B O ξ ] devoted to the 55 , 75 53 . b ξ ], mostly in Minkowski . ]. Higher symmetries of ), the first-order operator 74 ] and four [ – 3 D ). Following the discussion 62 0) supergravity background , 55 , – taking solutions to solutions, 72 D . For every solution N , ) δ ϕ 58 | parameters. For any dimension n N d ( ζ δ = (1 | σ ]. d D ) is called conformal Killing if M ξ [ N D M σ , , on ( N = δ | A d σ = 0 ξ ]. One of the goals of this paper is to work A M 6 in several publications [ 0) conformal supergravity in six dimensions. D 57 , ) ). One may show that it is finite-dimensional. ] and – 5 – σ on ( D ξ δ [ d < , and super-Weyl I B = (1 + N -order operators δ I E K | K th d are uniquely determined in terms of B N δ K n ξ = ( = 2 supergravity in three [ σ M I = N is uniquely determined in terms of K ξ obeys a closed equation that contains all information about ˆ and β b ], ξ ξ I ξ [ is also a solution. It is of interest to study higher symmetries K bc ϕ , -symmetry ] superspace formulation, there exists a universal description of all K bc (1) ξ R N 0) case, the proof will be given in section = ). So far there has been only one publication [ , K ) is a Lie superalgebra with respect to the standard Lie bracket. This D ; , ]. In particular, this approach has been properly generalised to study is a symmetry of any supersymmetric wave equation bc d bc D [ ,D ] , 54 R K ξ K [ N N = (1 δ σ G δ | | δ d d N M M ] + 0) case will be studied in this paper. By construction, the set of conformal Killing ξ , [ 6 and any conformal supergravity, the following general properties are expected = 6, K = 1 supergravity in five dimensions [ d ≤ The spinor component The vector component the conformal Killing supervector field. All parameters us to write ], a real supervector field = (1 = d N Given a conformal Killing supervector field Within the • • • 54 is the kinetic operator for some matter supermultiplets N ≤ (1) ξ higher symmetries of supersymmetric waveThe equations present in paper curved is supergravity aimed, backgrounds. supermultiplets in in part, at a a background studyTheir of of non-supersymmetric the higher analogues symmetries are ofwith also several them on-shell examined new in insights for diverse the dimensions, supersymmetric and story. bring for instance inrelativistic the wave context equations have of extensively beenand higher-spin studied references superalgebras in therein. the [ In literature,higher the see, e.g., supersymmetric symmetries [ case, of however, the thediverse program dimensions so-called of has super-Laplacians studying been the andsuperspace initiated related ( only geometric a structures few in years ago [ D O mass-shell equation, of supersymmetric wave equations, The properties have been established6, for vectors on ( is the superconformal algebraIn of the ( 3 to hold: conformal isometries of ain given [ curved background ( for some Lorentz supersymmetric backgrounds in and out the structure of (conformal)in isometries six of dimensions. a given the discussion in [ JHEP03(2021)157 , ) ]. θ R 16 , re- (2.1) (2.2a) (2.2b) 7 SU(2) ) and the × and ab . The name 6 M 1) 2 , , , α ψ β γ = 1 δ ı − , ) γ j , we detail a formalism ψ ) and eight fermionic ( χ β i α x ( 4 and C δ k , 1 4 ε . 0) maximally supersymmetric = = ··· , . jk , k γ J 9 χ ψ jk β , is defined in accordance with the ij = 1 ]; and (ii) conformal superspace [ = (1 β α A , we introduce the notion of a confor- α µ Φ being the inverse superspace vielbein, 15 4 reviews the SU(2) and conformal su- connection. The Lorentz ( N M . In appendix − 5, 2 M R , bc ), have the form A B i α E M ··· . D , bc , . For further details regarding our spinor con- 1 – 6 – a A in addition to the spin group. Therefore, the , ab Ω D A.1 . In section R 1 2 M the SU(2) = 0 ,M β 3 = ( ij − α γ ) A m A parametrised by six bosonic ( ψ A ab β α 8 D E | δ γ 6 ( 1 4 ,J = 1 4 ] M b − − A V β D a [ ), where = c ψ µ ı η γ α β δ α , θ is the frame field, with is devoted to the study of m = = 2 M x 8 c γ M ), V ψ ∂ ) generators are defined to act on Weyl spinors, vectors and isospinors as = ( β ab ij M -symmetry group SU(2) α J A M A.19 M R M z E , we detail how to ‘degauge’ from conformal to SU(2) superspace. = recounts our conventions. We review the conformal Killing supervector fields of D we review the higher symmetries of the conformal d’Alembertian and present 0) Minkowksi superspace in appendix 0) conformal supergravity and its couplings to supersymmetric matter. In the lit- , , A 5 A E the Lorentz connection, and Φ The main body of this paper is accompanied by several technical appendices. Ap- This paper is organised as follows. Section = (2 bc = (1 -symmetry ( A where the Lorentz generatorgeneral with rule spinor ( indices, ventions we refer the reader to appendix Here Ω R follows: “SU(2) superspace” derives fromincludes the the fact thatsuperspace its covariant derivatives, structure group, Spin(5 Since both approaches willthese be formulations. used in the present paper,2.1 in this section we SU(2) briefly superspace We review consider a supermanifold coordinates 2 Conformal supergravity inAs superspace discussed in the introduction,N there exist two fullyerature fledged they are superspace referred formulations to for as (i) SU(2) superspace [ pendix N for the study ofappendix supersymmetric backgrounds from a superspace perspective. Finally, in section some new observations pertinentthe to higher the symmetries supersymmetric ofspectively. the story. hypermultiplet Section Following and this, vectorbackgrounds. we multiplet Concluding in study comments sections are given in section perspace formulations forsuperspace conformal are supergravity. then studied in Themal section Killing conformal spinor isometries superfield,By of which a a systematic generates study, fixed extended itand is superconformal shown tensors, transformations. that which among generate its conformal cousins are isometries the and conformal higher Killing symmetries, vectors respectively. In JHEP03(2021)157 := (2.7) (2.3) αβ (2.8c) (2.5a) (2.6a) (2.8a) (2.8b) (2.6b) (2.5b) W , , ij -symmetry J = 0 R αβ k bc ] ) M the a β abc γ αβ N ( kl j ) , ) j . δ a l γα [ γ ε , abc  kl D AB ) ( δ J a  abc  γj R N β j k γ kl ) abc ) (2.4b) ( ( γ 3 8i W γδj γ b 1 2 cd ρ − C AB W − γ − W ( ij R W γj − C 1 6 ε = i kl , j ρ − acd J 2i W δ γ k − ) ) b N def cd k αβ 1 − 12 ( ) have the following form: cd ) N , a β γ − W M a γ bc )  C . ( j δ δ γ N j i i cd ( (dimension-0)(dimension- (dimension-1) (2.4a) (2.4c) b M C α  j ] D 2.5b ( βγ ) d kl βγ δ a l ) γ γδ AB ) αβ βγ η ε abcdef β ) a ] ) C ) ) c ) ε a d R [  γ a a a ij δ k γ γ γ 1 2 cd 1 γ 3! ε ( abc 2( ( ( γ β , γ i c β γ δ [ , and their covariant derivatives. The 3-forms a ( ( 5( . ) involve several dimension-3/2 descendants of − a ( , − k kl δ – 7 – ji := ( a C ij ε a J + 6i C − ρ β . acd 3 + 2 C δ  C kl , D + + 4 bc 2.5b = 0 b a = 0 α W αβ j β − ] γδ C ) δ = ) ) , a c 1 4 β ) γjkl M k the Lorentz curvature, and γ k abc b βγ β ( C R + 2i ) β ij cd N . j a i AB abc β α a γj αβ − ( ) αβ N a γ a a ) cd k γ ) T [ cd ) C C β βγ − γ k δ c i ( cd β a ) a k D ( ( a N = − D ) and ( γ a γ γ γ D k h cd AB 4 ( ( ( ( γ δ abc W ε αβ 1 4 + (  h h = ij R β ) and α M def N ( ) γ ε a β abc ,T,T + 2.5a + − − − − ] ] } 1 6 δ cd γ N ab α j β N ( B γ abc = = δ + ijk a [ N ( ij ij D := = 2i = 0 = 0 C γ k ε ε R , k [ N kl cd c c γ k αβ γδ 1 2 A bj 4i 2i j β j β j β abcdef D αβ β ) j a a b i = a ε α γk , defined by D i a  − C − − α [ N T T R γ R 1 3! T 2 3 W αβ = abc ] = N = ( = = } W are self-dual and anti-self-dual, respectively, and j ] β j β , is the torsion, ] D 15 D αβ αβ αβ abc , a ij , C ) and a abc N [ i α C W W D abc [ AB γk W γk αβ γk {D γ T D D (˜ N D = and , The curvature tensors ( The covariant derivatives are characterised by graded commutation relations abc abc abc a ij W 1 6 C They are equivalently described in terms of the symmetric chiral rank-2 spinors The algebra of theW covariant derivatives is determinedN by three dimension-1 real tensors, where the curvature tensors in the second line of ( Their general solution is given by the relations where curvature. In orderconstraints [ to describe conformal supergravity, the torsion must obey certain JHEP03(2021)157 ). 0) , if it (2.9) 2.11 (2.12) (2.10) w (2.14c) (2.13a) (2.14a) (2.11a) (2.13b) (2.14b) (2.11b) = (1 , N kl kl J ) J ) σ . ] is l σ β , l β  D . 15  D k α . kl
k α kl J D σ )  J ( D
σ ( βk kl l αβ β σ ) ) D αβ K j β D ) a ) a γ σ D )( + (˜ γ ) k α σ i (˜ 8 σ cd k α D i i 8 ( α − D M ( D − cd ab + 4( , 2( αβ , the curved superspace introduced ab ). ) K , σ M  a − 1 ]. ) 1 2 M ij ij γ ) βk σ 2.4 of Weyl weight (or dimension) (˜ σ J b J σ ) + D 76 ) ) b j β 4 , 3i D k α σ σ C ( D D U. 15 i D ( − D αj αj ( α K − C wσU . D − D D cd αβ = βk K – 8 – ) 4( = 4( primary M βk D ) U αβ = ) − − abc D U ) σ K ) ) σ γ σ a β β δ K (˜ δ σ k α βk γ α α (˜ k α i D D i 32 8 M ( 2.11 M D ) ) )( ( − σ αβ σ + σ , ). ) i β i αβ β ] k , α a ij ) (with its indices suppressed), its transformation law with A D D abc D D a a γ , ( (˜ abc is unconstrained. The crucial feature of these transformations γ D is said to be 8 N C 2( 2( U i (˜ | 2 , 6 αβ i σ W   2 − ) K − U σ σ σ − M i α i cd α − a a = [ a D D γ = e = e = e D  (˜ σ A D σ i  8 1 2 ij σ D 1 2 proves to be a primary superfield of dimension +1. It is the 0 0 0 σ a abc abc − K C = = N δ W abc ) is = e = e a i α W a D i α D 0 2.9 0 σ σ δ D δ D In what follows, we will need a finite form of the super-Weyl transformations ( A tensor superfield In SU(2) superspace, the gauge group of conformal supergravity is generated by three Such a transformation acts on the dimension-1 torsion superfields as follows: Direct calculations lead to transforms homogeneously under ( The torsion supersymmetric extension of the Weyl tensor [ where the real parameter is that they preserve the supergravity constraints ( An infinitesimal super-Weyl transformation of the covariant derivative [ Given a tensorrespect superfield to ( above will be denoted ( types of local transformations:transformations; (i) and general coordinate (iii) transformations; super-Weyl transformations. (ii)the structure combined An group type infinitesimal (i) transformation and of (ii) acts on the covariant derivatives as In accordance with the general discussion in section JHEP03(2021)157 = N (2.21) (2.19) (2.20) (2.17) (2.15) (2.16) , (2.18a) (2.18b) ) is the is the di- = 6 2.1 D d . kl J βi . kl , . ρ W ) B AB δ αβ ) K αβ = 0 J ) W . W ( a ) } j ρ AB γ γ R k F γβ ∇ ] , i 0) superconformal algebra are W − , ( β = ( − , , W }  , k γ ∇ α ] for the component analysis) the cd D αβ ) connections, where k αγ i α γ α C ( [ A M 45 W {∇ δ ∇ ∇ W = (1 , AB B K , cd − k β a F = − , N ∇ AB = = 8i } {∇  ) kl ABC αβ ) β ) J γδ γδ α γj M . W kl ) K ( A ( A W W D W ab ] (see also [ R P ) , , γ Φ R j β δk i – 9 – 2 1 ( a ( γ 16 , ∇ i 8 − ∇ − 0) Weyl multiplet in conformal superspace, one i ∇ − ( α k γ , {∇ 2 operator valued in the superconformal algebra. − bc D C ∇ ∇ αβ β / α ) are the special conformal generators. The complete = 0 ) δ β α M α ∇ i a AB δ 1 4 ] = ) bc = (1 C γ αβ b 1 4 ( ,S A D = a ∇ ( ij W Ω AB − , N ε } A K 1 2 a R ) T 2i βγ ∇ K − βj − − [ − = ( ) and special conformal ( W A = W A A = , γk E } i B } ( α K B ∇ . j β = k α ∇ ∇ {∇ A , ∇ , A i α A.2 ∇ ∇ [ {∇ ) of the following form α i ∇ is a primary dimension 3 , a αi ∇ W By solving the Bianchi identites, one obtains To describe the standard precisely as if they were the generators = ( 0) super Yang-Mills theory , A A and the additional constraints Here Moreover, one imposes that the∇ structure group generators act on the covariant derivatives Additionally, we require that the(1 algebra of covariant derivatives resembles a which satisfies the constraints to be completely determined in terms of the super-Weyl tensor given in appendix constrains the algebra of covariant derivatives The difference compared withpresence the of SU(2) dilatation superspace ( latation covariant generator derivatives and of ( list of graded commutation relations defining the In the conformal superspace approachwhole of superconformal [ algebra is gauged∇ in superspace by introducing covariant derivatives 2.2 Conformal superspace JHEP03(2021)157 , and (2.22) (2.24) (2.25) (2.26) σ (2.23c) (2.23a) (2.23b) (2.23d) D αβ . W B i β K ∇ , is generated by B , 1 8 G k , dilatation ) . + ji , β + Λ ρ ) ij ) X D δ βj J α ]. σ = Λ ( γ βk X δ αβ 45 i + ij ( α X ]. − k ρ Λ W jk ∇ 16 ∇ J 5 2 βj R if αβ , associated with the following γ coincides with its super-Weyl ( α jk ∇ − δ w W 1 2 . k γ 1 6 U = is + Λ a + ∇ −  i U 4 , SU(2) ) K α bc ρ wU .
) β ba − γj δ M βδ Λ = M X bc αk ). The covariant derivatives transform as i := U. − = 0 projection of the previous superfields. ( γ Λ W βγ βi i α U X K i δ 1 2 θ ∇ S k ρ Λ D = αβ , associated with a local superdiffeomorphism W , β α ∇ = k γ ∇ i γ a δ γα + α γ ab γ – 10 – , δ 1 4 ( β ∇ U B W δ 1 2 K 1 4 and of dimension − ∇ δ 1 6 βγ ,X = (Λ ) = 0 − B − − ξ ∇ A βj γ αβ U i 2 βα ], the multiplet of superconformal field strengths of the β γδ = A X , i W k + W ( α 45 M ) γ k i β K K i γ β β j primary ∇ X αβ ∇ ∇ X S k γ  ), these superfields are the only independent descendants of i α W b 5 10 2 k ( ∇ αγ i γ , 1 4 ∇ ∇ − − for a description of the degauging procedure. ] 2.18 ∇ W γ A i γ β D ) := := := := + ∇ ∇ , ab i β j β Y αi γδ K γ βij is then constrained to be standard superconformal transformations ∇ ∇ ( X α is said to be αβ Y 1 1 αi αβ Y = [ 16 12 U and W − − W A ). A ∇ = ξ K δ αi 2.12 W . As described in detail in [ We conclude by mentioning that SU(2) superspace is a gauge-fixed version of the In conformal superspace, the gauge group of conformal supergravity, It is convenient to define the following αβ It is importantweight ( to point out that theconformal superspace dimension geometry of and thusthe their reader physical to multiplets are appendix equivalent. We refer The superfield While the transformation law for a tensor superfield covariant general coordinate transformations parameter local superfield parameters: thespecial Lorentz conformal Λ transformations Λ Due to the constraintsW ( standard Weyl multiplet isA described reduction by to the components is straightforward and discussed in [ Results at mass-dimension higher than 3/2 can be found in [ The operator JHEP03(2021)157 ) ]. = ξ 77 , of 3.3 , ] (3.6) (3.4) (3.5) (3.1) (3.2) ξ [ (3.3c) (3.3a) (3.3b) (3.3d) 16 σ δ , , b . ξ + -symmetry a ] = 0 ξ R ]), dilatation [ D ξ A β K [ δ ] = 0 α ]), D ) jk ] ξ A ξ [ [ ab ∇ σ bc γ , ) is the fundamen- δ ( K B 4 1 + 3.4 K −  ] ξ ). Such transformations A [ = D B β , α 2.11 jk ) -symmetry (Λ + Λ J bc ] R , ξ γ D [ ] . , and eq. ( )( ξ ]), a b jk [ ξ ξ ξ . [ = 0 σ abc K i β c bc N + ξ D + abc c β jk α D bc We will say that a real supervector field + 2 ) J δN ] , ab is defined in ( M ab ξ ] η a 7 [ = abc γ ξ ξ 1 6 A [ ( ) jk , j W – 11 – 1 5 bc D abc ( β a = σ a ξ − K ) δ D i ξ + Λ b i β 1 2 δW ( ξ 2 1 = α bc a D ( a = + + D ]) parameters such that ξ M D 0) Minkowski superspace in six dimensions were studied in [ αβ ] B ξ ij i , a α )  [ ξ αβ [ a ) D γ i D B γ a ξ δC bc B (˜ i = (1 γ γ , ]) parameters such that ξ Λ (˜ ξ i a  D [ 1 2 12 N ξ i β α σ 48 a = δ − + D 1 4 − A ) is conformal Killing if there exist Lorentz ( B = 1 6 D = − D ) and their extensions. ∇
, ) ] = βα ) is: β i D 8 B α j ξ | ξ ξ [ , ξ ξ α i 6 i i σ 8 i α | ξ 3.1 β ( δ 6 i α α D M = [ D +  D D M i A ] 12 1 2 1 4 1 4 : ξ [ ∇ a on ( ] K ξ ξ = δ obeys [ ) is parametrised by the vector superfield B ). For instance, the latter implies the usual conformal Killing equation for the ] = ] = ] = K α a i δ ξ ξ ξ E D is conformal Killing if there exist Lorentz (Λ ]) and super-Weyl ( ξ ξ [ [ [ = , B 3.4 ξ β σ ij 8 [ | ξ B While the analysis above was carried out in the SU(2) superspace setting, equivalent α ]) and special conformal (Λ A 6 The conformal isometries of K jk ξ E 7 [ D K = δ B σ M K ξ ( superfield results can be derived using the conformal superspace approach. Here we will say that We have shown that( every infinitesimal superconformaltal transformation, constraint defining thisand transfromation. ( All other conditions are implications of ( where and thus are said to be superconformal. 3.1 Conformal KillingThe vector solution superfields to ( where the super-Weylrender transformation the superspace geometry invariant, in particular In this paper abackground central ( role is playedξ by the conformal isometries( of a given supergravity 3 Conformal isometries JHEP03(2021)157 (3.9) (3.7) (3.8f) (3.8c) (3.8e) (3.14) (3.10) (3.11) (3.12) (3.8a) (3.13) (3.8b) (3.8d) ). An 1, 3.8 − βγ , U. W B , (1) ξ αγ ξ K , ] D βγ ) ξ + w [ W γδ b B . i γ ξ = a W to a primary superfield ∇ U ∇ δi + Λ ), and the remaining pa- = 0 β αβ w ∇ (1) ξ ξ α D  ] ) A D βγ 1 ξ 3.10 16 [ ab ξ D ∇ ( σ γ , i α − ( . + 1 4 a , . ∇ (1) ξ . a ξ − a jk ξ b D αβ ξ  . J ) , − ∇ ] = 0 = β i = 0 a c ξ a i α is primary and of dimension [ γ of dimension ξ ξ ∇ = U (˜ c ) ∇ βγ a αβ β jk is superconformal and of dimension 0 in , j a 1 8 ξ U α ∇ (1) ξ β 1 ξ ⇒ a ) W 12 W ξ = ] ab − D D ∇ i (1) ξ ab + Λ i ξ η αγ b β A [ ( = γ D ξ ξ 1 6 α ( bc K b ∇ K – 12 – b 5 1 βγ + ∇ ∇ = M ξ = , αβ ] a − i β ) )  ξ W αβ b [ a γ ) i ∇ i γ ξ ∇ αβ = generates a conformal isometry, γ ξ a bc a 4 3 ⇒ β = 0 (˜ ∇ ( i γ , γ a Λ α = W (˜ a i ξ a ] ) ∇ ∇ (1) ξ = 1 2 αβ 12 ξ ξ ξ i α i [ β α ξ ab 48 a D B K δ − βi ∇ + γ δ 1 ∇ 1 4 ( − 16 K Λ i α = i 1 6 1 5 wU α = − ∇ − − ) is ). The operator ] ) α = i βα β ∇ i α j = ξ ξ ξ ξ [ ξ ), then = α i i ) is: i 3.9 3.8 i α αβ ξ σ β ( U + ) a α α i α i ∇ a ξ 3.9 b D ∇ 3.6 γ i α  ∇ ∇ ∇ ∇ i b 1 1 2 12 2 1 4 4 1 ∇ , ξ = = ] = 2(˜ ] = ] = ] = ] = = 0 α i ξ ξ ξ ξ ξ ξ [ [ [ [ [ (1) ξ U a β α i σ ij D A α Λ Λ is characterised by the superconformal properties ( Λ obeys the conformal Killing vector equation Λ K a a ξ ξ is a solution to ( In what follows, we will often make use of the first-order operator The solution to ( a ξ If where rameters are given bythe ( sense that itof takes every the primary same superfield dimension, These relations determine theimportant superconformal corollary of properties ( of the parameters in ( where This equation is conformally invariant provided Weyl tensor invariant, Since this transformation preserves the superspace geometry, it must also leave the super- JHEP03(2021)157 (3.22) (3.17) (3.18) (3.20) (3.21) (3.19c) (3.19a) (3.15a) (3.16a) (3.19b) (3.19d) (3.15b) (3.16b) , , , γ k ij c 2 ab a ξ T b βi ξ σC D b 1 − + ξ , . ) i . k α ) j D abc a = 0 abc = 0 N αβ C ) A  k i b σN , bc ( D , , + 2 a ] i
) take the form , ∂ a ] K + γ ξ 3 a i α ∂ 1 [ (˜ σ d abc ∂ ξ ] σ δ [ βi  i + 2 k γ bc 48 ,D θ W K a + ( , D ij N ] ∂ ] β , αβ k + b d β α 3 2 ) ξ a b ξ δ C ξ [ a [ γ 2 [ b 1 2 = ( γ K ξ β , a K = 0 , ) K δ ] i( βi ] A h  − K ξ bc ξ [ i ] α [ D − D γ + 3 b ξ σ 1 = + ( σ 3 [ ξ ) α i 1 2 σ ,D σ ij k i α ] := α αβ ∂ abc a A k a α 3 ∂ ∂θ D ∂ D + ξ N D C [  j ij D ) upon degauging. In particular, it is trivial to k γ – 13 – ( k γ  ε and β i = αβ γ jk K . i 1 ) D δ ε b D 3 σ 3.1 i α a , γ k − γ k ξ δ C ξ γ ξ ) is (˜ αβ βj + + i ∂ + ξ ] = 0 one can derive the following relations for the second ] = 2 ] = 4 ] = ] = 0 . 48 3.4 1 ,D γ ξ ξ ξ ξ ij ξ ij [ [ [ [ 2 β [ ε abc a a ) β − σ σ abc σ K jk ∂ 2i b N α j β C a ). 1 A ab d ∂x ∂ c , δ ξ K − γ K δN D i 2 D α b D i D α k γ i σ ) and ( α 3.4 = c d A 1 = D D δ D ξ ξ ξ + ( b D 2 D a ) reproduces ( } ξ ) that the commutator of superconformal transformations must 3.3 ∂ + σ j β − = = ] − 3.6 2 1 βk 3.1 ξ ) and one can prove that applying any number of covariant deriva- σ σ [ σ ,D a 2 = 0 and ) D σ σ ξ i α K j δ A δk b δ C ij βγ D D D  D a D i ) D { k γ ( b A γ 1 2 a ξ ξ γ D D δC (˜ , σ, i γδ = = γδ ) degauges to ( 2 ) ) jk 3 a a 3 = 3.9 ξ σ γ abc (˜ ,K γ γ k i 8 (˜ ab ξ a i − . Here the superspace covariant derivatives 32 The proof above readily generalises to curved superspace. For example, by analysing The statement above is most easily proven when working in Minkowski superspace, It is clear from ( Since SU(2) superspace is a gauge fixed version of conformal superspace, it must nec- D 8 ,K | 6 B ξ Another implication of ( Thus, our claim holds for this geometry. the invariance spinor derivative of and satisfy the algebra In this context, one may readily derive the following constraints M This analysis impliesmensional that the super conformal Lie( Killing algebra. supervector fields Thetives generate independent to a Υ parameters gives finite are a di- linear set combination of of Υ. superfields Υ = From this we may extract the form of essarily be true thatsee ( that ( result in another such transformation JHEP03(2021)157 , it ) if A D e (3.23) (3.24) E , (3.26a) (3.25a) (3.25b) (3.26b) 8 A | = 0 one e 6 ξ . 0) super- , β k abc = M . kl ab A J 1 T δW ) = (1 E ρ , A αβ l β ) ξ N  a D γ k α = kl , ( J b , D ξ ) ( ] ξ ρ . The two superspaces ξ i 3 [ l β ρ = 0 αβ ) K D − a ), see section ) γ abc )( dimensions where conformal = D (˜ ρ , d i 8 k α abc kl N σW J δ D N | ] − ( d e ξ 6 dimensions, the Weyl multiplet + [ ab , d are related by a finite super-Weyl αβ + 2 ] kl M ) ≤  replaced by M bc a A K ) d ij γ abc ρ σ (˜ W D J b + d ) W 4 ≤ 3i a ( D [ ρ bc δ ( k ξ − K αj and M − ] D δα cd e ξ ) with ) A [ + 3 βk 4( – 14 – e M D bc ) abc D ] superspace ( − abc ) ρ 2.14 γ K ρ ( β N 1 2 ) is said to be conformally related to ( W βk i k α 6 α ; k γ e D d D ξρ . D + [ , M D ( − ) )( 8 R γ . The torsion superfields are then mapped from a curved k | B − ρ ρ 6 ξ ρ αβ ] e G i D k β α ) ξ [ a αδjk + B D M D γ e ξ σ ( C (˜ 2( i abc 2 δj αβ ξ ) − ) plays a fundamental role in the study of supersymmetric space- W − ) prove to have the same conformal Killing vector superfields. In ] := ] = 3 d cd i α 4i e e ξ ξ a a [ [ D D D γ , σ + d D 3.22 (˜ K  8 ξ | ρ i β  8 k 6 1 2 ρ ξ − M αβ = e = e D i 3 a i α e e D D − ) and ( = e D σ , 8 | We also note that by imposing the invariance of the super-Weyl tensor αk 6 D M of conformal supergravity hascompensators are to required be for coupledgravity theories in to with six eight some dimensions. supercharges compensatingsuitable such to The multiplets as start conceptual with Ξ. a setupsupergravity general is is Two discussion actually of described the universal, using situation which in is why it is 3.3 Isometries In order to describe Poincar´esupergravity in 3 fact an efficient way tosupervector analyse fields conformal from isometries one is superspace bycase to mapping its of their conformally conformally related conformal flat one Killing superspaces. —may see be Given for shown such instance that the a supervector field for some super-Weyl parameter superspace to the other( according to ( transformation, tries are in general constrained. 3.2 Conformally related superspacesBy definition athe superspace corresponding ( covariant derivatives obtains which hints at the fact that superspace backgrounds admitting non-trivial conformal isome- Note that equation ( times. which implies the following expression for the spinor derivative of the super-Weyl parameter JHEP03(2021)157 . 2 | Ξ Ξ σ | Ξ (3.32) (3.30) (3.31) (3.27) (3.28) e Ξ and (3.29a) (3.29b) w = Ξ (Ξ) that is a σ δ f , ), = Ξ D w , Ξ N δ | -symmetry singlets d R M ), and the compensators -symmetry group. The compen- ]. It is called a Killing supervector R . ξ 3.25 [ . , σ = 0 Ξ) are conformally related if their co- . = 0 , . Ξ , Ξ D A ] and ]) , Ξ , ξ σ ])Ξ = 0 ξ D [ ρ [ ξ Ξ N ) I Ξ [ ] δ σ | ξ w w σ [ ] = 0 d Ξ) is a Lie superalgebra. The Killing equa- K = 1 Ξ σ Ξ ξ , – 15 – [ δ ], w w Ξ M D σ ξ [ Ξ = , + + e Ξ = e σ ] bc N ] + ξ δ B δ [ | ξ K [ d K D δ 6= 0, K B ( ( M ξ Ξ e Ξ) and ( ( , w e D , N δ ) are super-Weyl invariant in the sense that they hold for all | d ) that M 3.29b Ξ) if the compensators are invariant, be a conformal Killing supervector field on ( , Ξ). The notion of conformally related superspaces, which was introduced B D , 3.29b , E ) reduces to D N B , is naturally generalised to the case under consideration. Specifically, two , δ | ξ ) and ( N d δ | 3.30 = 3.2 d M ξ M 3.29a Using the compensators Ξ we can always construct a superfield Let Once Ξ has been fixed, the off-shell supergravity multiplet is completely described in In general, the compensators are Lorentz scalars, and at least one of them must have Then eq. ( (ii) it is nowhere vanishing; and (iii)It it follows has from a non-zero ( super-Weyl weight The super-Weyl invariance may be used to impose the gauge condition The set oftions Killing ( vectors onconformally ( related superspace geometries. singlet under the structure group and has the properties: (i) it is an algebraic function of Ξ; for uniquely determined parameters field on ( terms of the following(ii) data: the conformal (i) compensators.preserve a both Given superspace of a geometry these supergravity inputs. for background, conformal This its leads isometries supergravity; us should and to the concept of Killing supervector fields. variant derivatives related to eachΞ other are according connected to by ( the same finite super-Weyl transformation, should be strictly positive.choices Different of off-shell supergravity Ξ. theories The correspondtriple ( superspace to different corresponding toin Poincar´esupergravity is section identified withcurved a superspaces ( a non-zero super-Weyl weight They may also transform insators some are representations required of to the be nowhere vanishing in the sense that the JHEP03(2021)157 , a 0) ξ , = 4 (4.2) (4.1) is exists G (3.36) (3.33a) (3.34a) α (3.33b) (3.34b) w 1 = (1  − ] consists 1 N G ] . 79 0) supermultiplets , , , . to study supersym- G 78 C , is defined to satisfy the = (1 = 0 15 α A  N Ξ = 0 ) to the case of  D ] ] ξ ξ [ [ σ σ . The former is a primary real 3.29b δ Ξ : either Φ or ji (3.35) + w Ξ G and appendix 0) Poincar´esupergravity [  + , . . A 8 = , . kl ) and ( γ D .  Φ = 0 J ij , α exists. The latter is a real SU(2) triplet ] i
γ = (1 ξ = 0 kl G [ 1 σ D ij J αβ  − β = 0 α 3.29a kl 1 2 N kl ]Ξ = 0 C δ A ) take the following form: ) ξ − K 1 4 [ D K jk , – 16 – K + ] = + 4i G = + ξ i [ ) ( α α B 3.29b β j β bc   K D D σ D  i α M δ i B ] ( α D ξ ξ 0) supermultiplets. [ D  , ) and ( bc ] = 2, which obeys the constraint [ K ), which is a primary superfield of super-Weyl weight Φ 78 = (1 1 2 , kl w 3.29a G + 15 N jl B ε D ik ε B ξ . There are two natural choices for =  ij ij G G ij G = 1 2 ij q G ). Some of them can be used to describe extended superconformal transformations := 3.4 In SU(2) superspace, a conformal Killing spinor superfield The above formalism will be employed in section Now we specialise the Killing equations ( G This equation is super-Weyl invariant provided the super-Weyl transformation of eq. ( (conformal Killing spinor superfields) and(conformal higher Killing symmetries tensor of superfields). constraint and higher symmetries of 4 Conformal Killing spinorIn superfields this and section their we introduce higher various rank cousins cousins of the conformal Killing vector superfields The linear compensator is requiredfor to be nowhere vanishing in the sensemetric that spacetimes in the superspaceof (conformal) setting. Killing Now vector superfields we to will the turn case to of (conformal) describing Killing the tensor extension superfields and is nowhere vanishing(that in is, the sense thatand Φ obeys the constraint [ The most convenient set ofof compensators a for tensor multipletscalar Φ of and super-Weyl weight a linear multiplet supergravity in six dimensions. The equations read and the Killing equations ( JHEP03(2021)157 = . In (4.5) (4.7) (4.8) (4.3) N α (4.4a) (4.4b)  ). 2; and (ii) symmetric / 1 2.8 − , γ  describes extended  i ˆ , α  β satisfies the conformal i αβ,γ  a (4.6) ’s. αβ ξ − C ∇ i N . ), for more details. Given an ij b ) are defined in ( , we find that their ) ε traces n n 4.1 αβ,γ B ) is a conformal Killing tensor ). The latter is also a conformal a n − N ...a 4.4b ) ,  2 4.8 ’s with n into a symplectic Majorana spinor a + 16i δ a n A.12  , ζ D 0) superspace, β α + 8 i ) β  , , , . . . , ξ  to ( σγ n δ 1 ( D β 2  a ij 1 . . . ξ αβ a }  β W  ξ n ], although the SU(2) superspace formulation was 1 α C = (1 a ) and ¯ αβ γi ( 1 73 1 ) ...a C nσζ ξ a 1 a α N αβσδ ( a  − b γ ε 16 { ) that the symmetric and traceless product of ( + 16i := γ ζ defined by ( – 17 – α 1 ( = − β − } )   index. Such objects naturally arise from an 4.8 ) α n δ =  n a γi  n ) by replacing = ( n 1. As follows from ( + 4 δ αβ 4.1 R ] a n a W β − ζ D n 4.1 ξ are the only independent component fields. Making σ . . . ξ ...a ij D 5i δ 12 1 ε 1 = α [ a ) and ( a  =0 ) 3 γ ζ { 8i 1 θ n δ ξ | ( 4.6 − 6= SU(2) a 2 3 α is required to (i) be primary and of dimension  αβγδ ζ = ). As was shown in the previous section, these generate the − ε i = α i α ) α  n D β 3.4 D ( = =  a is its conjugate ¯ j β ζ γ γ   α D  i γ and i α 0) superspace reduction, see appendix αβ D be two conformal Killing spinor superfields. Associated with them is a , D ) leads to ) imposes significant restrictions on the component content of D =0 α 2 αβ θ  | 4.3 4.1 D α = (1 product  conformal Killing vector superfields It is clear from ( and n 8 α 1  0) superconformal theory realised in , −→ N Given Let Associated with Equation ( that carries a new SU(2) Our definition is equivalent to the one proposed in [ 0) = (2 8 i , ˆ α We will say thatsuperfield. any solution not used. has the following super-Weyl transformation law and satisfies the constraint and traceless which is primary andKilling of vector dimension equation ( conformal isometries of superspace.  (2 N superconformal transformations. vector superfield where the torsion superfields on the right-hand side ofKilling ( spinor superfield. We can combine implies that further use of ( obey the constraint obtained from ( particular, the following corollary of ( In conformal superspace, JHEP03(2021)157 (4.9) (4.13) (4.12) (4.14) (4.15) (4.10) (4.11) } 1 is called − , n } c + n } b 1 m } ) is the usual − ...a 1 ], where it was n 1 , ...a − a + n ) n 4.8 { 73 be two such ten- a 2 m 1 10 + ζ ζ m ) m b ...a + n ( 1 = D ...a a . | 2 +1 m b ζ | n ) m +1 1 a ...i 2 m − +1 ζ ...a a 2 +1 n 1 m ζ 1 βi a and . ζ m ...i ...a βi ) D ) i b . | 2 1 ) ], although no mention of the even ) 9 n b m a D 2 | ( ) ( n | γi { 2 n 80 a ) b a 1 m | m + n ζ ( ...a 1 m i nζ nζ 2 ) ...a n a ...β 2 n − − ), one can show that 2 ...a ( a ζ 1 . 2 ...β β β ζ } i β a  1 = 1 i α 1 +1 ζ ∇ σ i − ) 1 4.13 ( i γ α D n β  n n = 0 ( + α β 2 D D 2 n a ) αβ }  1 m ζ ) 1 to ( 1 β αβ a ( α +1 − m D ) bc ) ( ...a n δ b 2 1 1 m a a ...i γ m m a 2 { { – 18 – 1 ...a ( ( 2 i i 2 ζ γ γ n ( + 3 )  b a (˜ (˜ 1 2 ]. An immediate consequence of ( , ζ n n n D n + 4 1 = | ( a 72 b β | ) { n ...β =  1 1 = 0 + 2) + 2) m ) D β − ( ) ( 1 i = n n ) n m  +1 ) ( n mn mn ) and coincides with the one presented in [ and m a ( + + n i i ( ...a = β ζ ) 1 a  ...i ) 1 a m m B ζ 2 2 σ { i 1 m i α ) δ ( 8( 8( m K 3.16a i n ) ( ∇ + mζ − + 1 1 β n  ( m 1 = 0 describes a conformal Killing-Yano tensor superfield, introduced in a flat ( = i β ]. i ( α ). This constraint admits non-trivial generalisations )  = 1 AdS supersymmetry in four dimensions [ 2 74 1) D n − 4.1 , m N + n 1 + n ( = 2 m β (  a n ] 2 , ζ 1 ζ [ Having investigated the structure of conformal Killing tensors, we now return to the For a given curved superspace, the set of conformal Killing tensor superfields may The above definition of the conformal Killing tensor superfield can be recast in con- The supersymmetric extension of the even Schouten-Nijenhuis bracket was proposed for theThe first case time 9 10 also satisfies this constraint. in the framework of Schouten-Nijenhuis bracket was made. superspace context in [ Given two solutions master equation ( It is conformally invariant provided is a conformal KillingKilling vector tensor superfields superfield. ( called This the supersymmetric generalises even the Schouten-Nijenhuis bracket. Lie bracket for conformal sors, then and solves the equation be endowed with an additional algebraic structure. Let formal superspace. Aconformal symmetric Killing if traceless it tensor has superfield the superconformal properties higher symmetries of thetwo kinetic derivatives, in operators accordance of withconformal superconformal Killing [ field tensor theories equation with at most two such tensors is also conformal Killing. As will be shown shortly, such tensors generate JHEP03(2021)157 (5.1) (4.16) (4.17) (4.18) (4.19) (4.21) (4.22) (4.23) (4.20)
] β,δ [  ]. Here we will ` i  71 D , γ α δ 69 − , ] , 67 , γ,δ ) [ dimensions ), while their antisymmetric  2 ) ` 2 m d i  βα,γ i n . ` + . + D 4.13 ) ) 1 1 − β α n n m ( δ m ( ). It is clear from the analysis above i α , ...α ...i , = 0 , 1 5 ) = ] 4.1 δ 3 σχ +1 +1 β 3  1 + 1  .  = 0 γ 2 n = 0 m α
n  γα,β ) α i [ 2 2 ) ) ` 3 β 1  φ χ  +1 γ 1 a = 0 1 +1 β,δ  n + 1 − ( n ) ∇ m  m β a ` − ...α i m  – 19 – αβγδ ...α ...i χ 2 ...i γ 3 1 ∇ 2 2 ε βγ,α 1 i α α D  α i  i ( ` ] ( ( γ α χ = β 2 = δ 1 D χ  1 + = i α φ α . α ( ( ) ) [ 1 − = ] χ  2 n ) D m ( ( ) ( ) 2 αβ,γ i α 1 2 2 β,γ [ γ,δ ` χ n m (   ` σ + + = ` i  δ 1 1 i  n D m ( D ( δ α i α β α δ αβ,γ χ δ ` 2 5 1 3 − be conformal Killing spinors ( 2, including the important publications [ are solutions to this constraint then α 3 =  ) ) 2 2 n m ( ). ( d > βγ,δ i α and ` χ i α α 2 4.18  D be a solution to the conformal wave equation in , α 1 and φ  ) ) 1 1 n m ( ( Let We may also construct a hook field from our three spinors Let α i χ in dimensions review known resultssupersymmetric and analysis present in this some section new will guide observations. our study in The the outcomes next two of sections. the non- 5 Higher symmetries ofThere the has conformal been d’Alembertian extensive study of the higher symmetries of the conformal d’Alembertian Further, it satisfies the conformally invariant constraint which satisfies the Young condition If also solves ( and is conformally invariant provided This constraint may be immediately generalised that their totallyproduct symmetric is product dual is to a a right-handed solution spinor to ( which satisfies JHEP03(2021)157 ), (5.6) (5.7) (5.8) (5.9) (5.2) (5.3) (5.4) , if it 5.5a (5.10) (5.5a) (5.5b) can be 2 n of order )) D . . d 2.15 n K − , cd . , φ . 0 b ab . D C K ≥ c b = 0 φ . 2) , a has the transformation properties ∇ f a = 0 φ 2) − ξ φ
ζ 3) − a 2 d . − , n a ( ∇ D k D d − 1 2 1 a ( a 2 d b − 1 2 = 0 , it is natural to introduce the equivalence d ∇ = − 1 2 2( } 2 − = φ d D ... +1 + bc D φ n 1 + a ⇐⇒ ∇ D D cd M a – 20 – ...a ∇ bc 2 , the following first-order operator ) M ∇ a = 0, and of dimension a a k ζ a ] ( cd ) ω ξ 1 ξ a ⇐⇒ n , , 1 2 a , ab ( ζ { 39 a = 0 = , C ζ − ∇ n =0 b φ 1 2 2 = 0 k a = 0 = 0 X 16 (1) ξ e is a conformal dimension-0 operator. The symmetry oper- K D − (0) ζ φ φ φ D = a = D ∼ = D ) D D K n a a 2  1 ( ζ 2 = 0 and 1 cases in more detail. It is easily seen that zeroth-order b are symmetric and traceless. Making use of the condition ( is called a symmetry of the conformal d’Alembertian, ∇ K D ) D n ∇ is a constant, k , ( D satisfies the conformal Killing tensor equation a a ) (0) ζ ζ ∇ n is primary, (  D a ) ζ n ( a ) means that ζ ) is known to be conformally invariant if ), it is possible to show that every symmetry operator ), 5.5b naturally form an associative algebra. 5.1 is the conformally covariant derivative (compare with ( 5.6 a 5.5b 2 ∇ Let us first study the In the algebra of all symmetry operators of Given a conformal Killing vector field Due to ( symmetry operator where the parameters one observes that Utilising ( reduced to the canonical form Condition ( ators of relation A differential operator obeys the following conditions: Equation ( with the commutation relations [ where JHEP03(2021)157 , on , the a 2). In (5.14) (5.11) (5.13) a n ξ ), and (5.15a) (5.12a) (5.15b) (5.12b) − d 5.5b ( , . . . ξ 1 2 a 2 . The rela- T, w in any confor- , ξ a 1 , . Consequently, (1) ξ i ξ 2 . ) leads to a 1. Similar to the ϕ ) and ( D K b w (1) ξ ξ = 1 5.5b b 5.5a n > D = n ∇ ] = 0. n , = T a ϕ [ i ∇ ≤ (1) ξ ,i ϕ ) for ,A d ξ S k 1 D 2 δ is the following: 5.7 D ≤ + (1) ξ 0 + 2) D , d D a n ξ , (1) ξ − a ) = 0 D k ∇ 1) − . T 1 d ) is uniquely determined by each of the n ... ( to be determined by its top component, − ) b (1) ξ ) + 2 d ) d k (1) ξ = 0 n D ( 5.10 ( ζ ab − a a + D φ ζ satisfies the conditions ( D k 1 M K n . Imposing the condition ( k (1) ξ – 21 – 2 b ) ( (1) ξ − ξ n n n D ( (1) ξ a b − D a ) preserves the background geometry, the matter coupled to conformal gravity. We place this theory − D ζ ∇ 2 ∇ i := (4 ⇒ 1 2 ϕ k ) 3) = (with suppressed indices) of dimension n ... ( + 5.12a 1 − b a ) is simply a special case of the conformal isometry T D d ∇ ∇ , k a + wT ξ 5.10 A k ). ( = = = 0 ), we would like = k are given by the solution to the recurrence relation ) T  ) tell us that 2( 5.5b k a ) when acting on any primary scalar field of dimension k (1) ξ ( D 5.10 a ∇ A = D ζ , 5.13 , 1 5.10 ), the other fundamental property of k (1) ξ − k ) and ( A D = 0 is a symmetry of the conformal d’Alembertian. An immediate corollary of A  5.12b T ) and ( 5.5a (1) ξ a ] is invariant under the conformal transformation D K ϕ [ S 5.12b is a symmetry of the corresponding equation of motion, Let us now return to the general symmetry operator ( Our consideration above allows for important generalisations. Consider a dynamical Actually, the operator ( includes the universal enveloping algebra of the conformal algebra of the background (1) ξ where the constants It may be readilymally shown that flat the background. constructed However, operator when is the a background symmetry Weyl of tensor is non-vanishing, the first-order operator ( which is the conformal Killing tensor on a fixed gravitationalspacetime. background and consider Since aaction the conformal operator Killing ( vector field, D is a symmetry of2 the conformal d’Alembertian.spacetime. Therefore, the algebra of symmetries of system described by primary fields therefore the above consideration isoperator that, for any conformal Killing vector fields addition to ( for every primarytions tensor ( field which reduces to ( The second term onconditions the ( right-hand side of ( is a symmetry of the conformal d’Alembertian, JHEP03(2021)157 := ), it (2) (5.23) (5.20) (5.21) (5.22) (5.16) (5.17) (5.18) (5.19c) 5.6 (5.19a) (5.19b) D .  b 2 ξ , b , bd ∇ ζ ab a ζ ∇ b a 2 abcd . ∇ ξ and the corresponding a C c , a 2 bd + ∇ , ξ ζ ∇ b 2 i ξ . 2 . b = 0 2) ,ξ 3 abcd ). Then their product 1 + 2) ∇ and φ ξ − 2) C a i (0) h − d = 0 c = 0 2 a 1 d d ∇ D ξ − ( ,ξ i 5.10 ): ∇ a bd 1 1 2 d d ξ ξ ζ (0) h ( 2 2) , ξ  3 + 1)( 5.7 − d ) is tedious, thus here we will present 1 D + 1) 2 d − − d ξ abcd − 2 bd d h d d C a 4( ζ − d ( c c 4( 5.17 b 2 2 d + ξ ∇ ∇ + 1) b − − a − ] 2) d defined via ( ∇ 2 3 d ∇ abcd a 1 bd ,ξ 4( 2 − ξ 1 ζ ab – 22 – − C (1) ξ ξ a c (1) [ ζ d d ⇐⇒ ∇ − − b ( D ∇ ∇ D d φ , ∇ 2 = 1 2 = 2 case. a 1 + d (2) ζ abcd and − ξ n Z + is a symmetry only on a limited class of backgrounds, b d + 2 C a D d bd 1 = 0 ) (1) ξ ∇ ζ d − 2 ∇ (2) ζ n b c 2 − ( ζ φ D ξ D + b = ∇ 2 ∼ D , ξ b (2) ζ i − ∼ a φ 2 , a ∇ 2 D Z abcd ∇ } a ξ (2) , ξ b 2 2 b a (2) C 1 1 + ξ ∇ ξ D ξ a ∇ D b h { b ab 1 1 2 (2) ζ a ξ ξ ∇ ζ is a symmetry operator by construction, we obtain D ∇ = = = = 2 = ξ (1) i a ab 2  (2) ζ D ζ 2 (2) 2 the operator , ξ D 1 1 ξ (1) , ξ only results in a symmetry when the right-hand side vanishes. We thus expect D ξ 1 ≥ h D ξ  is a second-order symmetry operator. Modulo the equivalence relation ( (2) ζ n = D 2 (1) ξ (2) D A natural extension of the analysis above is to determine if there exists a symmetry The direct computation necessary to verify ( We assert that the second-order operator D 1 (1) ξ such as conformally flat ones. operator of the form Hence, that for which has the immediate consequence Now a direct computation leads to As where first-order symmetry operators D may be expressed as a sum of operators of the form ( a simpler proof. Consider two conformal Killing vectors this claim it suffices to analyse the only results in a symmetry in backgrounds satisfying existence of a higher symmetry implies non-trivial restrictions on its structure. To prove JHEP03(2021)157 1 D (6.1) (6.2) (6.3) (6.4) (6.5) (6.6) must be Z . αβ ]. It is now time to extend W . 75 ). This fails since ] that the higher symmetries } b 2 ξ , 72 = 0 a i { 1 i q . . ξ q i
D i 2 q 6= q is a primary superfield of dimension 2 D αβ ab = 2 i . = 2 ζ . q i − ∇ q i 1 4i q D = 0 D − D = 0 . It is easily verified that no such term exists. ) D j ) ab j = q , – 23 – ζ q , i j D ( α i q ( α j β ∇ = 0 ⇐⇒ ∇ = 0 ∇ i i i α q q A ∇ 2 D K D A K ∼ = 1 superspace were analysed in [ 1 N D is a superconformal dimension-0 operator. = 4, d D ) yields the useful corollary It may be shown that our arguments immediately generalise to the higher 6.1 ), we will also require 11 is a symmetry operator (of the hypermultiplet) if ]. On the mass shell, the hypermultiplet is described by an isospinor superfield 6.2 ]. D 15 are equivalent if 82 , 2 81 D Here we will study the higher symmetries of this model. We will say that a differential The off-shell formulation for a hypermultiplet coupled to conformal supergravity is Our analysis in this section has lead to several non-trivial results regarding the struc- This is in keeping with the fact that the background geometry generally restricts Killing tensors, see, satisfying the equation 11 i Owing to ( which means that e.g., [ It is useful to introducedant an structures equivalence can relation on beand the discarded. space of Specifically, symmetries we so say that redun- that two symmetry operators operator The constraint is conformally invariant provided Additionally, ( operator in curved this analysis to the hypermultiplet. given in [ q ematical physics. Moreplored. recently their Specifically, supersymmetric in generalisationsof flat have so-called also superspace ‘super-Laplacians’ been (superspace itLaplacian ex- differential was as operators shown their containing highest-dimensional in the component)conformal [ spacetime are Killing in tensor one-to-one superfields. correspondence with Further, the higher symmetries of the Wess-Zumino symmetries of superconformal operators.should lead In to particular, non-trivial the restrictions existence on of the super-Weyl such tensor 6 symmetries Higher symmetries ofThe the study of hypermultiplet symmetries of relativistic wave equations has a long-standing history in math- primary, of dimension 0 and first order in ture of higher symmetryfirst-order operators. case, The their most existenceWeyl important implies tensor. observation previously is unknown that, constraints beyond on the the spacetime when the conformal Killing tensor is irreducible ( JHEP03(2021)157 (6.7) (6.9) is (6.13) (6.11) (6.12) (6.8a) (6.8b) (6.10c) (6.10a) (6.10b) ) , it may n a ( ξ is gamma- D i β ) k ( , , including a , which is neces- ) . ) ij ζ n n ( J i β ( , 1 ,ij a . D k k ∇ A ζ . k a ∇ b ...A ...A = 0 ∇ m m 1) is completely determined ) ... A A − j ) k n q ... n ( +1 +1 ( A i a 1 ] , the operator m m , a ) yields the unique first-order ζ D ξ , ∇ a n b ) [ ) j ∇ ...A ...A σ ,ij i n in terms of αβ 1 1 k ∇ β 3.12 ) ) b A A , b n ...a ij to the canonical form k ˜ γ ζ ζ , . . . ξ (1) ξ + 2 2 0 γ ( ...A i 1) a a 2 ) (˜ a ζ ( 1 . -order symmetry operator D b +1 +1 n ζ − ζ jk A ( , ξ ∇ n i β + m m th 1) 1 J ζ ij ( a 1 ] A A D n ... − − a =0 1 ξ J ∇ ξ ε ε k n X n [ ζ k − =0 β ··· ( 2 m m a k X n a (1) ξ + 3) α jk A A ). Further, if + ) ζ + ε ε ∇ ) n D + k i α ) 1 and + 1) a is symmetric in its isospinor indices. 1) 1) 1 . a 1 n + Λ ( (1) ξ ∇ ) ( ζ – 24 – ... 4.11 A n b ∇ − − n ij βi ( 1 ( ζ D B D ) γ a ∇ + 2)( ( 1) n k i ), ( ( D ∇ ... − = = ( = ( ∇ + 2) a n i := 1 B n ... ζ ) ij ( a k ) ξ n n + 4 ) 8( n a k ,ij n h ( k 4.10 ∇ ( k ζ i ( A n 4( − ) ( α ...A a D k D ∇ ζ ( ∇ ...A +1 k = = = a 1 , the most general m ζ ) allows us to bring ) +1 − n =0 = ij A βi n k X n ...A m ) ( 1 n 1) m =0 j 1) a A 6.5 = 0 , and k X + A q − ζ − m i ζ n n ...A i α ( ( = (1) ξ 1 bβ a a ...A ∇ ) admits a decomposition as a sum of symmetry operators determined n =0 A ζ ) 1) ζ D 1 k X ζ i n − A ( α ( k ζ = ( ) yields numerous constraints on the parameters of 6.12 ∇ D a ) , we will denote it by ζ n ) ( 6.4 n ( αβ D a ) ). Therefore, the algebra of such symmetries contains the universal enveloping b ζ γ 6.4 If the supergravity background admits a conformal Killing vector superfield Equation ( Given a positive integer satisfies ( algebra of the conformalprevious algebra section, ( of the backgroundby their superspace. top As component was discussed in the Thus, given conformal Killing vector superfields in terms of be shown that thesymmetry corresponding operator conformal isometry ( Hence, we obtain expressionssarily for a conformal Killing tensor ( traceless, ( Here all parameters are symmetric and traceless in their vector indices, The equivalence relation ( where the coefficients may be chosen to be graded-symmetric in their superspace indices JHEP03(2021)157 (6.14) (6.16) (6.15f) (6.17c) (6.15c) (6.15e) (6.17a) (6.15a) (6.17b) (6.17d) (6.15b) (6.15d) ab j ab j q ζ ζ q ) ) j j β αj , αj bβ D , ∇ , D ab i . A routine de- ( . N ζ ab αi αi i ab ζ ( α ζ (2) ζ ˆ ij bα b ζ = 0. i ζ ab 1 2 j D 1 2 C ζ D N ) bβ j a δ αβ ) C − αβ − αβ δ D ) ∇ i ) . When acting on the i W a αβ β q a ∇ ab q ) j ( + γ a γ γ ζ a (2) ζ a (˜ j ( j ∇ i(˜ γ ) ∇ ab D D i i γ (˜ δ ∇ a a ζ 10 i 3 ˆ ) ζ ) 80 ζ i D ∇ j 10 β β + β j ( + γδ + − ∇ ) + D j j D ab b α i q i ( ab q i ( γ ζ γ a ab ) a ζ (˜ j ∇ D ) bα ζ ∇ j β D i αβ j bβ j γδ γδ N ) is ) D ) ) bβ C i ai ai a b b i ( αβ ( ˆ α ζ ζ γ N γ γ 6.6 ) (˜ (˜ (˜ bα D a + + αβ C γ ) αβ αβ 1 j j αβ ) ) i(˜ a 800 q ) q αβ a a γ , ) a 5 γ γ αj αj 16 a γ i(˜ – 25 – + (˜ (˜ ) and ( , ab (˜ γ D ∇ ζ (˜ − ab 11 50 i a a ab 1 1 , , 6.4 10 ij b ζ i ζ 800 800 D 80 b ∇ ab b − ab C ab ζ − ζ ∇ ζ + + 2 5 ) ∇ − ) aαi ab aαi j ) i β i β j ˆ ab j ζ ζ ζ β ab ab − ab ζ ∇ , ∇ 1 2 ∇ 1 2 ζ ζ ∇ , . D ζ k βj b b b i a b i ab ab ˜ αβ ( γ i αβ − D D ˜ γ − i , ζ ) D ∇ ) ζ D i b ( ˆ ) i ζq i a b bα i ij ( a β ) i b β ζq j ˜ γ q j ab q γ γ C ∇ ) b D C b (˜ ∇ ∇ (˜ i D + ζ j D + D b b D αβ j i i i a ∇ 3 3 D j αβ αβ ˜ γ αβ ) a 20 80 20 16 80 q i a ˜ ∇ γ i ) ) , q ) ( a j i ( j a a b D i 3 4 ( k a − − − − − γ ∇ i ˆ γ γ γ ab ζ D ζ ab (˜ (˜ D C i(˜ (˜ i ζ ab ======ζ + b ζ + i i i i i 5 3 20 25 80 80 16 16 80 . a D ζ ij = αi ζ = aij 3 4 ab ζ − + − + + − − in backgrounds with vanishing super-Weyl tensor, aαi ζ i ζ ζ i ζ ) q q = = = = = n ( ζ (2) ζ a (2) ζ ij D αi ˆ ζ aij ˆ D ζ aαi ˆ ζ D ˆ ζ ˆ ζ For completeness, we also present the SU(2) superspace form of Here we will restrict our attention to the evaluation of where have employed the definitions gauging leads to In particular, we findKilling that tensor all parameters are expressed solely in terms of the conformal The unique solution compatible with ( operators hypermultiplet it takes the form if the superspace is conformally-flat. Therefore, it is of interest to construct the symmetry JHEP03(2021)157 (7.3) (7.1) (7.2) (6.18) (6.19) ab (6.17e) , which ζ i ab (2) ζ subject to bβ ζ ) D C j αi i α D F ab D b ζ ˜ γ i αβ ( βi ) a D D , i γ ij a (˜ bα ] is C i i C 10 = 0 , as was shown for the ) 84 ) . αβ n + 73 ( ) 800 βj a a ab ζ , γ F + ζ = 0 3 2 i i(˜ . αβ α ab i bβ ζ αi 91 F ∇ j 400 ←→ N ) i i α F i δ α ( is uniquely determined in terms of , 3 2 + α ∇ i D D ) q γ n . X ab j ( = ( ζ αβ , ζ 5 2 ) (2) ζ D D a i αi βδ ) D + γ = 0 β F (˜ σ = 0 N ) α ], which is a locally supersymmetric extension D D j i ) β 50 αγ α q i ( F 16 γj = 2 N – 26 – + (2) ζ D i F i βij . We expect that, in conformally-flat superspaces, q γδ ( γ D γδ , ) ab α i ) ab b ( α ζ (2) ζ ∇ Y b . i ζ γ ) belong to a broader family of symmetry operators 2 β ) α γ D (˜ D δ (˜ 5 δ = 0 σ − 6.7 1 4 D αβ δ αβ ) . αi β ] ) ij i ( a − a F γ ab D ) X γ β (˜ j 83 (˜ ζ a , ) is a superconformal dimension-0 operator S βj αγ 5 βi 12 ∇ 1 F N a i 78 D 800 ( α , i − 6.16 ∇ γδ ) ∇ is a primary superfield of dimension + bα 16 b ab ζ γ := N ab αi (˜ ζ ij bij F αβ b αβ ) G C ) D a a ij a a γ γ C i(˜ D (˜ i 3 10 47 80 20 10 81 + − − − . The vector multiplet can be realised in terms of the field strength = F ˆ ζ 2 As a result, we have shown the existence of the higher symmetry operator F The equation of motion for the superconformal vector multiplet [ the Bianchi identities [ The field strength 7.1 Superconformal vector multiplet Consider a vector multiplet coupledthe to higher-derivative conformal action constructed supergravity. in Its [ dynamicsof is descried by The higher symmetry operatorsacting ( on tensor superfieldsoperators by of adding arbitrary Lorentz dependentthe index terms vector structure. multiplet. via an Here analysis of we the will higher generalise symmetries these of this is true forits symmetries top of component, allnon-supersymmetric the orders; case conformal in every Killing section tensor superfield 7 Higher symmetries of the vector multiplet is completely determined in terms of It may be verified that ( and yields a symmetry on conformally-flat superspace backgrounds JHEP03(2021)157 (7.7) (7.9) (7.4) (7.5) (7.6) , and ) is a T subject 3.12 αi which are and . F αi S β k X F k α ∇ i and 4 defined by ( − ) may be generated by βij = α (1) ξ 7.3 . Y D  is also a solution. Given a γ k (7.8) F = 0 αi k γ α . is a primary superfield with the = 0 F i ) and ( ∇ ). A superconformal dimension-0 ) . F β α D αi δ i α βj 7.1 n 7.3 1 4 = 0 . The compensator appears in the F D (1) ξ ) is not super-Weyl invariant, since F ) i . ( D α − , for arbitrary superfields αj , 7.8 3.3 β αi k T F ... ) F ], and the action is super-Weyl invariant. (7.10) ) and ( Φ S k α 2 σF i = 0 (1) ξ a ( α 15 3 2 ) ∇ 7.1 ⇒ D D ∇ D :  γj ( α = 1 – 27 – i i 4 (1) ξ + Φ = 1 F − F i ) αi ( γ − D ). In this subsection we make use of the super-Weyl T αj F , the first-order operator D a a σ β α := := F 7.8 ξ δ i δ ∇ = 0 = ) ( α β 1 4 n S ) α ( D αj − Φ D := ) F 1 4 ) involves the torsion superfields i ( α T βj a D F 7.3 ,F ). i ( α ) ∇ ←→ D γj S 2.23 F i ( γ ∇ i 4 := be a solution of the equations ( is called a symmetry of these equations if ). ij αi D X F 7.10 It is important to note that the equation ( Let Choosing the super-Weyl gauge reducing the equation of motiongauge to ( ( superspace action forThe the corresponding vector equation multiplet of [ motion for the vector multiplet is the super-Weyl invariance hasIn been the fixed superconformal bythe setting imposing to tensor an Poincar´esupergravity, compensator the appropriate Φ vector gauge introduced multiplet condition. in couples section to super-Weyl transformation When the vector multiplet is placed on-shell, it obeys the additional equtation to the constraints These constraints are super-Weyl invariant provided considering products of the first-order symmetries, 7.2 Supersymmetric MaxwellIn theory SU(2) superspace, the off-shell vector multiplet is described by a superfield defined according to ( operator conformal Killing vector superfield symmetry. Higher-order symmetries of the equations ( introduced the following descendants of The equation of motion ( where we have defined JHEP03(2021)157 ). D ) for 7.11 (7.14) (7.11) (7.13) (7.18) (7.15) (7.16) (7.17) (7.12) , 6.9 ij via ( J reduces to bc n αi M b k . F a 1) ] in the framework of αi D − n 80 F (
a ... ζ 1 bc a  i M . D ] bc D ξ [ b , M ˜ γ bc ,bcij , it may be shown that their b i = 0 k ) ) a k ) 1) n . K D ( ( D − a β α 1 2 b 2 +1 n ζ δ ζ n = 1, its action on ( . 2 1 4 a + ... i β = 0 n − =0 ζ ...a , 1 ) D k 2 X n α − jk a i j = 2, is analogous to that of ( a k and = 0 J 1 2 F a ζ αβ D ] γi ∇ , ) n 1 ) b b ξ D D b a β [ ,bc D m + ˜ ( γ 2) ) ( i γ α i 0) Killing tensor superfields in six dimen- α i α (˜ k a ( jk 1 − ) , bc ( ... b D ζ n D a ∇ ). When K cd 1 1) + M ζ a γ βγ − i α 1 , m + ( ) D n 4.8 ( = (1 – 28 – b − =0 ( i ⇒ D D a B k γ X a n + 3) ] β 1 bcd ), say for (˜ ) 2 ζ − D N = 0 1 2 n k N k i α  ( ) , ζ b + 1) B a a 1 ξ ∇ + 1) 7.12 ζ βj n D ζ [ ( 1 − ij b F = 0 = n + 2)( + 2) i n − =0 J = ( i ... D k X n b D k a + 2) n n i i 1 a ( 1) α ) define αi a + n 8( 8( nζ D − D F D k n ] 4( i − − + ( a ξ [ a 7.17 ... D ζ K = = = ), is also Killing 1 b α,bc δ a β 1) D ij ... βi D = α − -order symmetry operator for the vector multiplet is 1 1) 4.12 1) ij k a 1) ) ( th − αi − ) and ( a k − n D n n ( ( ζ ) F ( n a a ( k a 1 4.8 ζ ( a ζ ζ (1) ξ − a =0 1 ζ k X n ζ − =0 D k X n 1 2 is a conformal Killing tensor ( n =0 ) k X + + k ( Given two Killing tensor superfields a = ζ ) 12 n Equations ( As the procedure to compute ( We now turn to the analysis of the higher symmetries of this theory. The operator ( The concept of a Killing tensor superfield was introduced for the first time in [ = 1 AdS supersymmetry in four dimensions. D 12 bracket, defined by ( N as well as the Killing condition for tensor superfields sions. Our analysis reveals the following restrictions on its parameters: the hypermultiplet, we willnon-trivial not pursure information such regarding analysis the here. structure Instead, of we this will operator extract for some general where is said to be a symmetry of the (on-shell) vector multiplet if The most general JHEP03(2021)157 0) , = 0. (8.5) (8.6) (8.1) (8.2) T (8.4c) (8.3c) (8.7c) (8.4a) (8.7a) (8.3a) (8.3b) (8.4b) (8.7b) (8.7d) = (1 = 0, and ) together N 8.4 ) is imposed) abc such that its bar- . W ) implies 3.32 T } ) j β j . ) C.1 i ) and ( a D ) , , C , . i α ) i 8.3 l ( abc . k ap ε N {D = 0 bc , where we discuss how to = 0 C + k , kl ( M + 2 C k a p ) ) abc l d C a N = 0, = 2 abc d C abc i α C ˜ 6 , D i , ij W N ( D kl e a ( e j ] − C ] β k ε C bc ij bc ξ ( + 2 } γ , bkl j β N W , β αβ ) ) , C ) ) abc D γδ e e ) , a a bc [ [ N W i α bcd = 0 d d γ ) reduces to dab γ ( = 0 ,C γδ ( N N 1 2 N {D + ( abc kl W abc bcd 3.22 kl – 29 – + N α C β + 2 + 2 + 2 d J δ a , , W N e e ˜ jk 4 1 i α kl C a a D 3 2 b [ [ , , a d dab d D − C C = = 0 = 0 = W W W 3 βj d d = 0 = = 0 kl ξ 6( 6( 2( , βγ − C C a γ ij ij , − − − N a β C e e cjk ) abc abc D = 0 C C αγ = = = C ab N d d W = 0 ] ] γ bil W := kl bc bc abc kl abc abc a C a a N = ( W W C N ], in such backgrounds it may be shown that ( ˜ W D d C d αβ d γ d k αβ ) ˜ αβ i α 57 ξ D ) D D ) D a D = 0, and this condition is supersymmetric, then the entire superfield is zero, de abc de | ) can be compactly rewritten as D ) leads to severe restrictions on the backgroud superspace geometry. a γ to [ a [ [ ( T γ γ ( 8.3 ( 8.1 13 = 0 imply the following differential equations abc N } j β ] for a more detailed discussion. D Note that ( Equation ( Here we restrict ourselves to the case of eight supercharges, i.e. maximally , 57 For any background admitting eight supercharges, if there is a tensor superfield i α 13 projection vanishes, See [ A lengthy, though straightforward, analysiswith of the the superspace consistency Bianchi identities of leads ( to the following algebraic constraints where we have defined together with the algebraic conditions {D Additionally, the Killing spinor equation ( In particular, the integrability conditions supersymmetric backgrounds, and derivesimilar constraints analysis on the superspace geometry. By a The existence of (conformal) Killing vectortions and on tensor the superfields places superspace non-trivialby geometry. restric- such conditions. So In far thisis we section further have the not elaborated case examined of on.obtain Killing the component More vectors constraints (where results results imposed eq. from are ( superspace. given in appendix 8 Maximally supersymmetric backgrounds JHEP03(2021)157 ], A cd D (8.8) (8.9) , (8.12) M (8.13c) (8.13a) (8.10a) (8.11a) will, in kl (8.13b) (8.10b) (8.11b)  , 0) back- J ij , , C abc . = 0 a W e = 0 C = (1 ] ). In conjunc- ] = 0 cd ] and [ αβ N ] ) A 8.7 def W and . . e D , which are covari- def acd W , cd ab γ [ abc N abc cd abc = 0 M N [ N W M abc  [ ] ) and ( W d bcd + 2i( , N , , e ]
⇐⇒ 2.7 b W , kl   and l δ A J ] ⇐⇒ ] W acd (5) (5) c c c D D  ⇐⇒ [ bcd = 0 j ω ω e N  ) a l e . N [ k l ] = 0 ε , (4) (4) b b δ ), two branches of solutions exist, ) , = 0 k N cd e + 2 δ ω ω ( ] j i kl γ e ] N ε ) cd a a (3) = 0 (3) e [ [ C 6= ]). These one-forms may be normalised 8.7c acd cd b bc 5, are orthogonal one-forms, that is, [ ω ω W ). i + 16 , abc a γ e N ( 86 a W bcd e ≤ + − N )( ef ab C W ] [ i N ab b 8.11 [ ] ] αβ A αβ (2) (2) c abc c W ) ) ≤ ), due to the (anti-)self-duality conditions on = 0. W N ) and ( cd ω D ω , a N 0 ⇐⇒ ef abc γ – 30 – ] M (1) (1) ( b abc b , γ d ,  8.7d ) ( ⇐⇒ ω + 2 8.7b ω as in ( ij = 0 ) define the most general superalgebras with eight i e N ⇐⇒ ( a ε ] (see also [ W = 0 3 a 8i a a C β (0) c (0) [ a ) [ ω = [ 2i [ e e abc j = 0 ] 85 ω ω δ ( = 0 8.13 + = 0 C − cd  W  ] ω e kl ] ] abc ( e ) d b kl and α β ] i  W + 2 δ C 1 2 ( a cd W C e c cd a [ A α [ ω b = = + a [ ef W δ − N -symmetry commutation relations, [ ] a D e a C 4 b e l b b R [ k N [ D abc abc N a αβ − a , C ) ] N W ef b αβ a W N d ] ) d ), and obey the algebraic constraints ( γ C a C ⇐⇒ ( δ ] = 0 γ N b γ ij ( ), provided that 8.8 c a a ) [ [ ε ⇐⇒ ⇐⇒ C ij δ c C a = 0; and (ii) ab = 0 [ ε [ 6i 8 A γ δ a 8.7d  2i  ( 8 de D = 0 = 0 C ] −  + +  , is equivalent to any of the following relations b de de ] ] W = abc b b ] = ] = de } N b N k γ W a j β [ D de D de D N , a , a [ , [ a a and i α N In accordance with our analysis, for every maximally supersymmetric By a routine calculation it may be shown that Note that the algebraic constraint ( D D W [ [ abc {D The algebra is determinedantly by constant, the eq. four ( tensors tion with the Lorentzthe graded and commutation relations ( ground the algebra ofrelations: covariant derivatives is given by the following graded commutation This result was originally derived into [ constitute an orthonormalgeneral, basis, involve overall and factors then the expressions for is a solution to ( W while the following relations hold identically It should be emphasiseddefined that, by: due (i) to ( as well as the conditions JHEP03(2021)157 ] 87 ) is = 0, ), or ]. As a (8.15c) (8.14c) (8.15a) (8.14a) (8.15b) (8.14b) , which abc 8.13c 1 8.13 16 , , , 5 W ij R , = cde J l δ ≡ . W D abc abc 6 o cd ) takes a particu- N cde k N l M δ M N αβ δ )  γ 8.13 ab e ) η abc C bc 8 3 e γ γ ( . C − )( ] ] 3 8i d b δ abc = 0; the bosonic body of this bcd efg c a − is replaced with the torsionful [ [ 0 and a pp-wave spacetime if N a δ W W a = 0. The graded commutation cd 4 , C cd > D b ] = 0. When this is not the case, M + 2 a efg − = 0, though the reverse is not true d abc 2 ] ij η W d N , the commutation relation ( c N C abc b [ C 4 ] abc C a a abcd d ] C W b − η W C C ( c C [ Rη = 0 and c a a αβ − [ [ bcd = 1 η ) when δ 10 l N abc k 8 16 15 spacetime or a pp-wave, see later. acd R  + cd 6= 0, which necessarily have N C abc γ a  – 31 – 3  + a N + δ × γ a S N ] δ d C 5 d ae a ] ˜ × 16 D d R = 0 implies + 2i( C ) 3 c 3 d [ − a W b , ˜ e c ab abc η D − c [ , which decompose as the sum of two orthogonal simple C ) that there are two disconnected branches of solutions, b b N cde 0, AdS W c − αβ = 0 and ) C N b abc C W ] δ a = 0 implies that the superspace is conformally flat [ < a d + 2 γ 8.7c γ ab W 16 ) ( C cde d R η 2 abc c ij ab − [ ab N C a ε = 0. γ W ( η be and W a ] 2i 16  + 16  d C for − n b ) and ( 1 2 − abc W C 5 c e ). One obtains -symmetry curvature vanishes if = − a c N ). The most obvious solution is Minkowski space, S [ C R ] = 4( ] = a 8.7b c C } b 8.6 k γ j β × ) it is clear that ˜ = 0, the corresponding geometries are described by the covariantly con- C N D 16 abcd 6= 0 or D 8.15 D , , R a is constant, the possible backgrounds are locally equivalent to the following , − R a a a i ). If either of the corresponding simple forms are null, the background is a α ˜ C C ˜ a D ]. D [ = = 80 = = 16 8.14c [ C 0) Poincar´esupergravity. These superalgebras were derived two years ago [ {D 87 a , R 8.11 ab C abcd R When Solutions belonging to the branch We now employ the above analysis to identify all possible maximally supersymmetric It should be remarked that the algebra of covariant derivatives ( In accordance with the discussion in appendix C = = 0 [ = (1 2 2 three cases, C stant three-forms forms ( corresponds to theit choice follows from ( defined by are characterised by the existenceC of a parallel, nowhere vanishing vector field. Thus, since relations take a remarkably simplesuperspace form is if a conformally flat AdS spacetimes, which areequivalently the ( bosonic bodies of the superspaces with geometry ( We see that the larly simple form if theone torsion-free defined covariant by derivative ( Note that the condition result of ( in general. read off the Ricci tensor, the scalar curvature and the Weyl tensor: N using sophisticated algebraic techniques.techniques Here allow we one have to demonstrated derive that these the superalgebras superspace via aequivalent simple to calculation. that of the spacetime covariant derivatives. Therefore we can immediately supercharges, which are associated with the maximally supersymmetric backgrounds of JHEP03(2021)157 . . q Ξ S Ξ G σ × = Ξ (8.21) (8.16) (8.17) (8.18) (8.19) (8.20) p do not w Ξ )). This 3 = S Ξ 8.11 ) reduces to σ δ and , invariant, in ( 3.35 , specifically: the 3 Ξ ij β w G 3.3 0) supergravity in six . , 3 S 14 × 0. = (1 , 1 ). In particular, the possible , ≡ 2 . N R = 0 i j , 8.11 δ
abcd k − . Then it is possible to identify ) C = 0 j ji = G ij G i j ( ) hold for all maximally supersymmetric l k G . ε =  . G ; and (iii) . k 3 kl + i ij 8.20 = 0 leads to the constraint J l R = 0 ] ) G G j = 0 ξ kl = 0 [ × jk G G – 32 – 3 kl i ij a ( G abc } k i ) and ( j α K β C N ε ), the tensor multiplet constraint ( D D ⇐⇒ + , 8.16 i α αβ B must leave the linear compensator ) 7.10 is a descendant of the compensators Ξ which is a singlet D {D B ; (ii) AdS abc B Ξ 3 ξ = 1 ξ γ ( S 2  = Φ as has been done above, we can instead choose G × abc ] for an interesting discussion of superconformal flatness of AdS Ξ is annihilated by the spinor covariant derivatives, 3 ). In the case of a maximally supersymmetric background, this N 90 ij ), where ]. When this is not the case, it follows that locally the spacetime 6= 0, their radii must be equal (proportional to G 3.34b 88 3.31 abc W ]: (i) AdS ]. It is also an example of a superspace which is not superconformally flat, 87 89 = 0 and 6= 0, though its bosonic body is conformally flat, Instead of identifying Let us analyse the case of the compensators introduced in section So far we have not specified any conformal compensators Ξ. We have worked in the In general, for backgrounds belonging to (i) the radii of the AdS abc The reader can consult [ N abc 14 Next, we impose the super-Weyl gauge superspaces based on coset constructions. which is solved by We have shown that the conditions ( backgrounds of Poincar´esupergravity with the tensor and linear compensators. Now, the integrability condition in accordance with ( equation implies that tensor multiplet Φ andwith the Φ. linear In multiplet the super-Weyl gauge ( Every Killing supervector field under the structure groupit and is has nowhere the vanishing; properties: andAdditional (i) (iii) constraints it it on is has supergravity ancompensators a backgrounds algebraic is non-zero often function made. super-Weyl occur weight of once Ξ; a (ii) specific choice of if background is one ofdimensions the [ well-known solutions toW minimal super-Weyl gauge ( decomposes into the productferred of by two the three-dimensional structure symmetric ofsolutions spaces. the are This three-form [ can fluxes be given in in- ( necessarily coincide — in particular, (ii) and (iii) are degenerate cases of (i). Additionally, pp-wave spacetime [ JHEP03(2021)157 ) obey (8.22) (8.23) (8.27) (8.25) (8.26) 8.24a ). The ij (8.24a) (8.24b) ) tell us G . Both of must leave 8.18 G 8.21 k β B ξ = )Υ ij Ξ G αβ , ∇ ), and thus the integra- 0) Poincar´esupergrav- ( ). kl , jk G 8.18 ij G 8.16 3 ) takes the form ( G ) becomes = (1 1 G δl 2 , . Υ N = Φ; and (ii) 8.25 8.24b − γk ) = 0 Ξ Υ = 0 ij ). Specifically, the integrability condi-  βj . G ,  Φ G Υ α , j Φ . G ) reduces to ( 2.5a  X )  = 0 j β
αβγδ ) ε 3.35 ∇ Φ = 0 ij i αβ ) Φ = 0 jk 5 ( α j β + 10i . To make contact with our previous results, we C B – 33 – G G i k ∇ i ∇ i β ] ( α D i β ( α 16 Υ B ∇ Φ = const Υ + 4i , ξ ∇ ) and the super-Weyl gauge condition ( αβ α + ) k [ ) holds. Every Killing supervector field j β ). The superspace torsion is determined by the super- ∇ W βj D = 0 i 3.36 ( i 4 ( α Υ 8.20 4 αi G D 8.20 αβ := W = 0 must hold. The latter contains nontrivial information, in + F i β αβ kl ij ). In the case of a maximally supersymmetric background, this F Υ G G } 3 ) and ( αβ j β ]. They have the form 1 G and ∇ D 2 3.34b 84 , 1 8.16 G ij + i α which is covariantly constant. Such a superspaces are the only maximally G = {D ), while the linear multiplet constraint ( αj abc αi ]. The superfield equations of motion for ∇ W 2 3 3.35 is the field strength of a composite vector multiplet W 91 , αi is annihilated by the spinor covariant derivatives, eq. ( := i α ij 84 W , G The equations of motion for have Poincar´esupergravity the simplest form in conformal We have discussed the two possible choices: (i) 47 now degauge to SU(2) superspace.turns Upon into degauging, the ( tensor multipletsecond constraint ( equation of motion for Poincar´esupergravity in ( with Υ ity were derived in [ where For more details,to [ including the Poincar´e supergravity action, we refer the reader Weyl tensor supersymmetric solutions of Poincar´esupergravity. Let us discuss this point in more detail. superspace. In this setting, thethe tensor constraints compensator Φ and the linear compensator As a result, the tensor multiplet constraint ( them lead to theby same the maximally conditions supersymmetric ( backgrounds, which are characterised in accordance with ( equation implies that Φ is annihilated by the spinor covariant derivatives, and therefore that bility condition accordance with the anti-commutation relationtion ( tells us that thethe condition tensor ( compensator Φ invariant, Then the analyticity constraint ( JHEP03(2021)157 5 0) supergravity , ), or equivalently generate all (non- . 8 | ) ]. 6 = (1 n 8.13 ( a 89 M N ζ – are covariantly constant; 87 ij 0) supergravity in six dimen- ) is satisfied. The first equa- , G 8.27 = (1 N , which generate extended conformal su- α  – 34 – ). ) hold. Now eq. ( 7.17 8.20 superfields was shown. The former parametrise the confor- = 0, is also satisfied, since all maximally supersymmetric ) n ( ) and ( αi a ζ W ). 8.16 ), 4.12 8.25 and tensor a ξ ), which contains three distinct branches. Further, their corresponding space- 8.15 Our approach to the highermay be symmetries immediately of generalised to the the conformal studyIn of d’Alembertian particular, more in complex it conformal section field would theories. namics be in interesting four to dimensions. extend this analysis to Maxwell electrody- The maximally supersymmetric backgrounds of sions were classified. Our( analysis leads totime the backgrounds superalgebra are ( derived, reproducing the results of [ the explicit form of everyit higher symmetry was operator proven on that curvedtrivial) backgrounds. the symmetries For conformal of (ii), their Killing kinetic tensorthat operators. its superfields higher Finally, in symmetries the arealso case parametrised of introduced by Killing (iii), in tensor we this superfields, deduced work which ( were the conformal Killing tensors ofto a the fixed bracket superspace ( form a superalgebraWe with studied respect the highernamely: symmetries of (i) three the on-shellconformal conformal models vector multiplet. in scalar In curved field; our backgrounds, analysis (ii) of the (i) hypermultiplet; we have, and for the (iii) first the time, non- derived The conformal Killing spinorpersymmetries, superfields were introduced. Invector addition, their relation tomal the isometries conformal of Killing superspace, whiletries the of latter the are kinetic associated operators with of the on-shell higher multiplets. symme- Additionally, it was proven that backgrounds within the SU(2)finitesimal case and they conformal were superspacebackground. shown to formulations. Further, form we detailed a In how(conformal) closed the these algebra Killing in- may on spinor be any equation utiliseduplifted fixed to to supergravity at trivially a the read unique component (conformal) off Killing the level. vector superfield Its on solutions may be We have described the structure of (conformal) isometries of Interesting open problems include the following: • • • • • 9 Conclusion To conclude, we summariseareas the for future main work. results Our of main this outcomes are paper as and follows: outline some interesting persymmetric backgrounds: (i)and both (ii) compensators the conditions Φtion ( and of motion inbackgrounds ( have no covariant background spinor superfields. The analysis given above tells us that the following properties hold for all maximally su- JHEP03(2021)157 is and (A.1) (A.2) (A.3) a abcdef ε , T a 8 matrices Γ Γ − × = 1 − is defined by . ], with a few minor modifi- C ∗ 1 a 15 − Γ = 1 supergravity backgrounds. . C . 0 ∗ N and by Fonden. L¨angmanska The Γ ∗ Γ ∗ 1), the Levi-Civita tensor C 15 , ,C 1 = 5, = = Γ , a d abcdef 1 T Γ ε , − C 1 , = 1 . = = ] , s f † 1 f ,B ) Γ e e a – 35 – = 2 and 1 − r Γ e − (Γ d e N B q Γ ∗ d c ) e Γ a p b = 1, and the Levi-Civita tensor with world indices is c = 4, = diag( Γ obey the relations , e (Γ a d n [ 1 B ab b C Γ e η ab = η ¨ m UBITAK does not mean that the content of the publication is approved 012345 2 ,C a ε a , it would be interesting to study the higher symmetries of the e Γ − 1 − 6 = = = ¨ UBITAK (Project No:120C067) } is antisymmetric. The chirality matrix Γ abcdef ¨ C UBITAK. b † ε 1 Γ − , C a := C 012345 Γ a ε { mnpqrs ε metry operator for thein hypermultiplet terms and of vector its top multiplet component. is It uniquelyAs would determined an be extension interesting to ofmultiplet prove in our this section analysis explicitly. ofmassive hypermultiplet the on higher symmetries of the (massless) hyper- We believe that, as was shown for the conformal d’Alembertian, every higher sym- We exclusively use four component spinors in the body of the paper, but it is useful The Minkowski metric is However the entire responsibility of the publication belongs to the owners of the publication. The • • 15 As a consequence of the above conditions, one can show that financial support received fromin T a scientific sense by T In particular, Γ to relate these tothe eight charge component conjugation spinor matrix conventions. The Dirac 8 Our 6D notation and conventionscations. are All similar relevant to details to are those summarized of here. [ normalised by given by Capacity Building Package of the University of Queensland. A Conventions A.1 Spinors in six dimensions in part byof the U.L. Australian has Research been(CoCirculation2) Council, partially of project supported T by No.work the of 2236 DP200101944. Co-Funded ESNR Brain isAustralian The Circulation supported Government research Scheme2 Research by the Trainingby the Program. Hackett Australian Postgraduate Research Scholarship The Council UWA, work (ARC) under of Future the Fellowship GT-M FT180100353, is and supported by the Acknowledgments We are grateful tothank Paul Joseph Howe Novak for useful for comments collaboration on at the manuscript. early The stages work of of SMK this is project. supported We JHEP03(2021)157 (A.8) (A.9) (A.4) (A.5) (A.6) (A.7) (A.12) (A.11) (A.10a) (A.10b) . ! ˙ and a four- β . α 0 α ∗ B ψ ˙ Ψ β . using the conven- ∗ α B 0 2 4 ) ∗ i (4). It is natural to , , B Γ ∗ . .

(Ψ − su ··· = 1 B = , ! i , = , ∗ , = 1 β , γ α Γ ∗ α γ a 0 α 1 ∗ δ δ γ δ = 1 2 − Ψ ) ε 1 ab ab β β ,B = ˜ η η − = χ 0 α ∗ = 2 2 ∗ (4). They obey ( δ ∗
C ˙ ) ∗ 2 β ) 0 − − a 1 i

γδ α Γ su γ ) , α ( B = = a = (Ψ − γ 1 ! (˜ = − = βγ βγ β are antisymmetric and related by ) ) ˙ α δ , ε α c C a a 0 ¯ , δ β 0 χ j Ψ γ γ αβ Γ (˜ ( − B Ψ ) γδ ˙ ,C γ a ) − ij β αβ αβ a α γ ε ) ) 0 α ! γ = , b b – 36 – B δ ( ⇒ = γ γ i ∗ αβ

= ) i = ) Ψ β 0 a and (˜ Ψ + ( + (˜ = αβγδ ψ γ αβ ε ( ∗ so that ) ˙ αβ β βγ βγ 1 2 a Γ ) ) ) α α γ ⇒ b b a αβ (˜ C χ , = ) B γ γ γ = 0 (˜ (˜ ( a j T ) γ = ), αβ c Ψ , , ( αβ αβ ) ) ) α ∗ ij i a

a a
¯ ε ! ψ γ A.3 of a Dirac spinor is conventionally defined by γ γ α (˜ α γδ ( (˜ = are valued in the two inequivalent fundamental representations of c = ) χ ψ =: (Ψ = Ψ a i a α c 0

Γ γ Ψ ) , charge conjugation is an involution only for objects with an even χ ( Γ i 4 matrices ( 1 † ˙ δ − Ψ × (Ψ β Ψ = and B ≡ = ˙ 1). We further take γ α , α ¯ Ψ is the canonical antisymmetric symbol of ψ B B (5 ∗ matrix to define bar conjugation on a four component spinor via so B = B αβγδ = ∼ ε αβ ) We employ a Weyl basis for the gamma matrices so that an eight-component Dirac a (4) γ ∗ ( A dotted index denotesuse the the complex conjugate representation in and as a consequence of ( where The Pauli-type 4 The spinors su spinor Ψ decomposescomponent into right-handed spinor a four-component left-handed Weyl spinor index. Conventionally this is denoted for a spinor of eithertions chirality. We raise and lower SU(2) indices Because number of spinor indices, soOne it can is instead not have a possible symplectic to Majorana have Majorana condition when spinors the in spinors six possess dimensions. an SU(2) The charge conjugate Ψ JHEP03(2021)157 are βα (A.20) (A.18) (A.19) (A.13) (A.14) 1) and (A.16c) (A.17c) V (A.16a) (A.17a) (A.15a) (A.16b) (A.16d) (A.17b) (A.17d) (A.15b) , − (5 . = so ), has Weyl ba F αβ via A.5 V , − β T = ) , α , F β ab ef , γ α def ˜ γ ( f F . ˜ γ , ˜ γ − α . ) . , β ]  αβ , ) ( c = δ abc -supersymmetries are both V ˜ abcdef γ γ ] ab T b b ε S δ abcdef abcdef γ γ γ αβ ε β 1 2 ( = 0 ε ) a a abcdef α [ [ 1 1 2 of a 3! ) and obeying ( − − ε δ , are symmetric. Further antisym- γ γ . γδ γ ) (˜ i α = = = = = i α η 1 4 abc , := ˜ := ˜ A.7 abc + 2 χ ) abc ab δ γ ,  γ ) β abc = ab ( F δ (˜ = β abcd abc γ ˜ ] γ ˜ γ abcde a δ T ˜ ˜ δ γ , γ β . We also note that, as a consequence of abcdef ˜ γ αβ γ δ V αi δ α a and ˜  ) ˜ γ α α γ χ γ ( ψ , δ α δ [ = ) abc and antisymmetric matrices − abc 8 δ a ⇐⇒ ) γ γ a ˜ γ −  = , b ( def – 37 – , V ⇐⇒ γ α = 4 = = 48 = ( T α i def , ψ δ − ψ γδ γδ γδ γ γ = 0 a b ) ) ) ) ef are related to traceless matrices , ˜ γ a V , = α γ ab f a abcdef γ abc abc α ab γ (˜ γ ε γ αβ γ γ αi ( ( and the parameters ) F abcdef (˜ (˜ 1 β 3! a ψ , decomposed as in ( αβ 1 2 ε i ) α α i γ abcdef , αβ αβ abcdef ) 1 a ,F θ 3! ) ) ε ] ε (4). Vectors abcdef γ c := ab ) and similarly for ˜ ∗ 1 2 ab ( − ε − γ ] γ abc abc := ( b b F ( su γ γ ˜ ˜ γ γ = = = = β ( ( a a α [ [ αβ A.11 ) γ γ V abc are traceless, whereas ab abcd γ abcde γ γ := := ab abcdef γ ( γ γ 1 4 ab using ( abc γ − γ and ˜ αβ ) := a ab , β γ γ α 1 F − = ( = Making use of the completeness relations We define the antisymmetric products of two or three Pauli-type matrices as αβ ) B a ∗ γ Self-dual and anti-self-dual rank-three antisymmetric tensors matrix representations of related by Antisymmetric rank-two tensors it is straightforward to establish natural isomorphisms between tensors of metric products obey Note that components that obey The Grassmann coordinates symplectic Majorana-Weyl using this definition. ( B with the natural extensionsymplectic to Majorana any spinor tensor Ψ carrying an odd number of spinor indices. A with the obvious extension to any object with multiple spinor indices. For example, JHEP03(2021)157 ), a P (A.22c) (A.22e) (A.23a) (A.24a) (A.25a) (A.21a) (A.22a) (A.23b) (A.24b) (A.25b) (A.21b) (A.22b) (A.22d) ), where ij J , , i α . α i Q , ). The fermionic S 1 2 α i 1 2 αβ αβ , ,S T T − a ] = βi i α αβ αβ Q K ) ) ] = ) naturally arise from an , ,Q αβ α i a abc abc ) 16 = ( , D 4.1 γ γ | a [ K ,S (˜ ( a 6 γ A 1 8 1 8 − D P i(˜ , [ . M K , c − ] = = j ) b i , ] = j ] = α ) ) J , a a S j α − M α β ] = i (+) ( k ) and SU(2) generators ( , abc abc a ( k ) δ [ a Q ,P j T T ,K δ d D i , ( η D , J D ,P k [ i [ ab 2 + 8 ( ε α i l ] = 0 ] = via α ε S are M − a α [ k β , d , ] = 0 ] = + ] B αβ ] ⇐⇒ ⇐⇒ ] b ,S a b ,K M k α l + 2 b P , ) T j i α ij i j M P K δ ,P – 38 – D S J βi ,Q a a a J 4 [ [ [ [ [ i α i c c c ab S ij ( and and η η η η Q k − J [ ), dilatation ( ε [ αβ A a ) D αβ j i a K K δ T γ βα ] = 2 ] = ] = 2 ] = 2 ] = 2 0) superconformal algebra contains the translation ( βα , α β b c c , i( c . Their algebra is T T kl , cd δ 2 − c ,P K ,P , , ,K = = ,J a , P k δ ,M ab = 2 = (1 ab ij αβ δ k K ) Q ] = αβ ab [ abc abc J = 1 } S M c δ ) [ M i α [ j β N γ T T [ c γ γ M δ ) (˜ [ γ ) obeys the algebra i, j ,Q ( αβ αβ superconformal algebra ,Q ij ab ) ) a ab ε ij α α i i γ γ ε ( K 0) ]. By employing this construction, we will explicitly prove our earlier S S 2i ( abc abc [ , 2 1 { 2i γ γ 1 2 0) superspace reduction. − (˜ ( , 92 − − 5 and ) and the special conformal generator to , , 1 1 = 3! 3! i α (1 4 = (1 = ] = 77 } obeys the algebra , ] = γ ), special conformal ( k } β j 3 ,Q k γ := := 0) Minkowski superspace in six dimensions. Such analyses were previously N i α j β , ab a , −→ ,S 2 ,S Q P , ,Q M αβ αβ ab α i ,Q 1 0) T ab i α T S , , M = ( { [ = (2 Q M [ { A Its superconformal generalisation is obtained by extending the translation generator = 0 N P = (2 The aim of thisof appendix is to studyconducted the in structure [ of conformalclaim Killing that supervector the fields proposedN conformal Killing spinor superfields ( B The conformal Killing supervector fields of Finally, the (anti-)commutators of while the generator with all other commutators vanishing. to generator The bosonic sector of the Lorentz ( a, b A.2 The are related to symmetric matrices JHEP03(2021)157 (B.5) (B.6) (B.7) (B.8) (B.9) (B.1) (B.2) (B.3) (B.4) (4). It is USp 0) Minkowski , . Its covariant 4 , . = (1 ··· , N = 0 ,  = 1 I α b , I D , ∂ ] a , a ξ ∂ [ ∂ .  , is parametrised by the coor-  σ 2 βI α K 16 , 2 1 θ | ξ 4 and 6 , , K α . − . αβ 0) R-symmetry group , M , ) = 1 , D J a α J a β i I α ˆ I ξ J ··· βI = 0 , γ D δ D , ξ ] D b  ) i( βI 1 4 ξ ξ α I = (2 I β α [ J αβ a ξ D ξ − J ) = 1 ∂ − I I α a N β ,D + α αβ I ) to the form α J γ , i, α a ) α D ∂ a ξ ) a ( ∂ ∂θ . ∂ I α  + Λ γ ! 2i( 5, B.5 a ab − (˜ a – 39 – , j ˆ I D β ξ γ − = i ξ ˆ 0 i (  = ε , a 12 D I α = ] 1 4 1 4 = ∂  ··· ξ ij − ¯ , 0 ξ I [ αβ α 1 6 a such that it takes the form − − ε 1 β ∂ ξ = , = α I α

IJ ,D IJ ω α I ξ, D ξ D ] = ] = ] = a  ξ = ξ ξ ξ − = 0 [ [ [ ∂ 2iΩ β J ∂x σ a = IJ I − α 0) Minkowski superspace,  Ω Λ ω = , I α = a } ∂ J β = (2 ξ, D  ), where ,D N α I I α is an invariant tensor of the D , θ a { JI x Ω − = ( = A z IJ We now briefly consider the problem of performing a reduction to We will say that the real supervector field We recall that It is clear thatmations, the respectively. above parameters generate Lorentz, R-symmetry and scalingsuperspace. transfor- Without loss of generality, we will assume that this coincides with the section By a routine computation, we may bring ( where we have made use of the definitions which yields is conformal Killing if it satisfies This constraint implies the fundamental equation Here Ω convenient to choose a basis for Ω dinates derivatives take the form and satisfy the algebra JHEP03(2021)157 , , 6 8 | 6 M M (C.6) (C.1) (C.2) (C.3) (C.4) (C.5) (B.10) (B.11) . | obey kl a ab D ) connections. In R ). As a result, we kl = a 4.1 φ . kl ( m ab R ∂ . 0) supersymmetric back- = 0 R , m . (4). By making use of the | kl a e J , | =0 abc | = (1 α i USp ˆ kl N := θ cd , | A i α ) a N ab j ˆ Φ D α R = 0 ξ parametrise a curved spacetime ) and SU(2) − i ˆ , ( , α α i = bc 0) conformal Killing supervector field, satisfies m bc γ j ˆ a , D m x , e  , α i =0 cd ˆ ω 1 4 M i ψ γ θ  kl

| ab J ) D bc = (1 = 0 R β := α kl A δ | a j ˆ x, θ i ˆ N 1 4 Ω φ ( abc ⇐⇒ – 40 – 1 2 0). It then follows that, upon such a reduction, . The bar-projection of the superspace covariant U − = , , 8 W | − α i β j ˆ i bc 6 α kl  := a θ J D i , λ α | M M M D kl ∂ bc (2) R-symmetry within U D | =0 a ab = α i ˆ ) = ( ω M | SU θ ). The coordinates α | , i ˆ a 1 2 A α i ˆ − R D , θ E ξ are conformal Killing spinor superfields ( − ) will be assumed. The covariant derivatives α x, θ i = 0 cd α 2 ( a θ = ), one may show that =  | e | U C.4 M α i ˆ kl A  = a = ( B.6 cd D generate non-manifest extended superconformal symmetries in C a and ) decomposes into an ) = α ab I i α ], the bar-projection of a superfield is defined as usual: j ˆ θ z α 1 D i ˆ R (  D λ B.5 1 2 U 94 , ), one can completely gauge away the gravitini such that the projection of − and 93 , C.1 ] = α i ˆ b  20 generates the hidden D defined by j ˆ , i ˆ a λ 16 | D 6 [ Due to ( M is a spacetime covariant derivativewhat with follows, Lorentz the ( gauge ( the vector covariant derivatives takes the simple form where derivatives is defined similarly by: Following [ for any superfield the bosonic body of the superspace In this appendix, we introducegrounds a starting from formalism a to superspace studybackgrounds, perspective. meaning 6D that Our the analysis following will conditions be hold restricted to bosonic which implies that have proven our original claim. C Bosonic backgrounds where fundamental equation ( every solution to ( a spinor superfield and an additional triplet of scalar superfields defined by The spinor of JHEP03(2021)157 . 8 | ) we 6 (C.7) (C.8) (C.9) (C.17) (C.11) (C.12) (C.13) (C.14) (C.15) (C.16) (C.10) 3.5 M . ij a c ] v [ . ω a . v + a | k ) D j a abc c 1 6 ] N v [ ). By bar-projecting ( = k = i | ( ] ξ C.8 k . transformations are generated by [ . , , abc ] σ . R . v + 2 [ c b , = 0 = 0 6= 0 ) is that every conformal Killing vector ω v ij b | , n v | | ] ] := c c | ] α . ab i α ] ij i ξ v v ξ [ ) reduces to η [ v D C.1 [ [ ab abc ij b ω ab a R := = 0 η W ] + , ξ K is a conformal Killing vector field k C.14 ) v 1 6 α i , ω [ b ⇒ is even if = a ] + – 41 – v ] = b v ab = = 0 a v v ] := A ( k ij [ | 6 a ) abc ξ ,  v [ a b [ a ij D c v = ξ k ij D b , it may be shown that every conformal Killing vector a b a ( k = 0 ] = 0 = v D 3 ) that = , w := b ξ D | a D [ | | a v ] a σ D ) equation ( ξ kl ξ v [ a + C.10 b ab C v 3.20 K = ij ab kl a ] := R encode complete information about the parent conformal Killing c is a Killing vector field v [ a α i  ab v ] = k v [ ij and k ) can be uniquely decomposed as a sum of even and odd ones. We will say a may be lifted to a unique even conformal Killing vector superfield on a D 3.4 v 6 M be an even conformal Killing supervector field ( A It should be remarked that in the case of Poincar´esupergravity, we must supplement We also note that, at the component level, SU(2) An important feature of such backgrounds ( ξ which implies that Further, by making use of ( the constraints above with field on which satisfies the differential equation In particular, it follows from ( By employing the results of section obtain where we have defined vector superfield. This appendixspacetime is level. devoted to the study of symmetriesC.1 they induce at the Conformal KillingLet vectors or odd if The fields superfield ( that the conformal Killing supervector For convenience, we also make the definitions JHEP03(2021)157 gen- (C.28) (C.22) (C.23) (C.24) (C.25) (C.26) (C.27) (C.18) (C.19) (C.20) (C.21) α i  ): ,
C.19 δ k . jk  b ) gives δ ] c βk β , δj η ˆ  , ) D 3.22 γ βγ βδ ) βk α abc ) [ . The spinor a ˆ η b n δ α i γ γ ), from which one can 2 3 (˜ ξ βγ , i ) 2 + 2 − 2i( a bc C.9 γ ( (˜ − = is identically zero. abc M is a solution to ( ) + i 2 ) γ k w γ k A βk  α i (  abc ξ  η ≡ δ k abc a n αβ .  n
ˆ D . ˆ , β k D βγ δβ is the 6-vector jk )  c ) . + 2 -supersymmetry transformations are b β j a a + 2 α i . V c  S γ c  ˆ abc α i | D (˜ δj abc ]  γ  i abc D 2 = 0 ξ ( w [ ij αβ w ( i ab a βδ ) 6 ε σ ( ) β k = η a V i α b  ⇐⇒ a αβ γ 6 1 − β k − γ γ spinor ) ( D  β V a i 3 γ kl ) – 42 – = γ 2i( β J := ) bc − ) βδjk as the bar projection of := b kl − γ c i α bc δ k V = ( ( 2 a α i η  a := δj γ c 1 2 b  ( a βk V  )( η V ˆ + D D 3 αk 4i + ˆ a η abc βδ commuting βγ jk ) + n D ) b b δ k a c γ  γ ( := (˜ + 2 βj βδ  i 2 a γβ γ ) D ˆ abc β a D i = ) 3 γ w (˜ ( γ k ab −  1 6 1 2 γ a ) implies = − ˆ − D = ( γ j = βk  γ k η 3.23 j γ k  k  a a a c D ˆ -supersymmetry transformation, while D ) becomes − Q γ k  a C.19 Associated with a non-zero The conformal Killing spinor equation then takes the particularly simple form With the previous assumptions at hand, bar-projecting equation ( D which proves to be a conformal Killing vector field when Furthermore, it is a null vector In particular, we see that the gamma-traceless part of and where we have defined or, equivalently, Moreover, eq. ( and ( parametrised by Our analysis in thisleast subsection one will conformal be supersymmetry.possessing restricted an Such to odd spacetimes thoseidentify are conformal a backgrounds associated which conformal Killing with admit Killingerates supervector a at spinor a field superspace C.2 Conformal Killing spinors JHEP03(2021)157 , ij ij e (C.36) (C.31) (C.32) (C.33) (C.34) (C.35) (C.29) e jk c (C.30c) (C.30a) cjk (C.30b) c a d ] c d c ] bc i α bc bil η ) becomes w c n ) αβ αβ αβ k ) ) + 8i ) C.19 ap de
de c abc a j [ a ) j [ γ l ( γ , γ a p c 4i( i . These results exem- ( αβ k c − + 6i( abc ε  il + 6i( ) = 0
ε jk N j + ) a abc abc i abc c k  12i a  ) w n N l e c e ] ] αβ i i a α and − ( αβ c bc αβ , bc k i D D ( n w ε ( D , . D il j , jk ) ) abc cjk , ε ε ij b a a c ij + [ [ c ε , ) ε jk W abc i 2i i abc k b de de a , δ αl ρj , ) w c n l .  abc = 0 η V n n i α a i α jk a ρj | − n | − | − . c η i α η a βρ  i ) η 32i ( D jk + 2 + 2 C b abc j + δρ = 0 abc a + ) = 0 + 2 ) ε γ a a − d [ [ b d ρ k ( C = 0 ] W N ] ] ⇒ ] ]  γ abc ) de de bc l β j j bc abc β β βγ b αβ bjk δρ ) W w w n w ) – 43 – w D V c D D = 0 = a ( ( ˆ i ( β β α , a i D , , γ γ ( = 6( a α α i α bcd i i η α α D αβ αβ ) ) δk αβ V γ γ ( ρ k D ) )  D a a ) D D ( a [ [ ] = 0 =  [ α [ [ d d b = (˜ d d ) ξ ˆ γ γ 1 1 2 2 γ [ D δρ γ k ( ( bcd γ γ − ( ab ) are superconformal. The bar-projection of the above  σ ( ( , δ ˆ ij ij il D γ a − − i i N β β  ε ε ε η η ˆ β l ). The conformal Killing spinor equation ( 3 D   8i C.1 3 3  β j β j +6i −  +6i − +4i − +8i(  ) = 0 3.32 jk = = 4i a | | 0 = C σ σ i α ) k γ δk D ( D D δ j k γ ( β D D i α i α D D γδ βγ ) ) a abc is a Killing vector field γ γ (˜ a (˜ V i In the case of the Poincar´esupergravities, equations given above must be supplemented By construction, the following identities hold 32 This implies that thus which implies and by the additional condition which is a consequence of ( Restrictions on higher mass-dimensioninvariance component of parameters higher-order may spinor be derivativesplify of obtained how results from in the components can be efficiently obtained from a superspace setting. which implies the conditionsconditions ( imply the following constraints JHEP03(2021)157 i α B 2.1 α i (D.4) (D.1) (D.2) (D.3) E (D.5f) (D.5c) (D.5e) (D.5a) (D.5g) (D.5b) (D.5d) one can ]. + a 16 ] 95 , B 16 a 37 , , E 29 . αβ ) = c αβ γ . ) . (˜ B c ij βi γ kl ( J F J c , ij ), see e.g. [ F kl ∧ β A c α ∧ Φ ) AB 2.11 E βi ab i ]. − , . R γ E ( − i α 40 i bc − αβ j β – ) Λ ji M − F a cd α i 37 i Φ bc γ ∧ j E (˜ M A ∧ 2 Φ α j cd βk Ω E − provide new contributions to the torsion, ∧ αj F 1 2 it is clear that they are SU(2) superspace . , AB a α j E B 5, see [ ) ∧ − R j α Λ F + 2 R A k α a − F ] ≤ A 1 2 F = 0 b F − , E i ∧ E B F 2 D A − i α – 44 – i B ( − ∧ ∧ = SU(2) − F B C B, α ≤ ∧ a α i [ B ∧ B × D = E ∧ α i E 8 E C α ∧ i F a K 1) 1 2 4 B a 1 2 E , − E AB F − AB + k − ) + (Λ) b T F α j + 2 − c β a K β − a a b Φ + Ω δ α b Ω F Ω = Ω A ∧ ∧ Ω ∧ i ∧ ∧ } ∇ ( ∧ ∧ ac β k a i b B i β b Φ E E E := D F F , + Ω − + A + − A + + 2 a α D i ab ij . a i α covariant derivatives are given by D [ A F F E E B ∇ . = dΩ = dΦ = d = d = d = d B ) = d a a i α α i ij ab ) ) ) D ) T ( T S degauged J K ( ( ( = 0. These exactly reproduce the super-Weyl transformations ( M R ( R R R B R We use different symbols for the degauged derivatives and the SU(2) ones of section The The first step in this procedure is to eliminate the dilatation connection. Under an There exists a class of combined local dilatations and special conformal transformations preserving the 16 gauge Lorentz, and SU(2) connectionsgive are expressions exactly for thosecounterparts. the of This new conformal can be torsion superspace, doneand and by it curvature using curvature is the two-forms expression in tensors easy of terms the in to of conformal terms the superspace torsion vielbein of and their connection superfields conformal [ The degauged special conformal connections and by extension to the other curvatures. since, as we will see, they satisfy slightly different torsion constraints. Since the vielbein, Since their structure group iscovariant SO(5 derivatives. They satisfy the algebra As a result, theextracted from special conformal connection becomes auxiliary and must be manually transforms as Thus, in exchange for agauge loss away of unconstrained special conformal gauge freedom, As is well known, SU(2) superspaceThe exists process as of a moving gauge-fixed from versionit the of latter here conformal to extending superspace. the previous former analysis is known in as 3 ‘degauging’ and weinfinitesimal outline special conformal gauge transformation the one-form D From conformal to SU(2) superspace JHEP03(2021)157 ), (D.9) (D.6) (D.7) (D.8) D.5c (D.14) (D.10) (D.11) (D.12) (D.13) kl J transforming . kl 0 a . U A abc 0 Y U αβ ) } αβ a i α ) γ ∇ ( . ), constrains the special abc , . ij γ C kl ε ( J . αβ 2.19 1 6 . K , ) [ kl a c ij + 2i C , B = γ C αβ ε J j β (˜ Y A cd . F + ε abc βi βα ij abc M ε F Y N + 1) bc Y 0 − ∧ 6= 0 ) the dilatation curvature, eq. ( − αβ = αβ M + i ( ) U c c ) b = } ij γ E ab αβ abc , both of these torsions are required to j β D.2 BA ( abc T αβ F γ ∇ W abc + i ( 2 bcd , A 2.1 + α i ,Y C i 3 4 Y 8i , − a ji ij K T − [ ε + D – 45 – AB 0, one finds the torsion tensors are related by βα C 6= 0 ,Y = = cd F i α = ) A ij ≡ α i α i k F + 4i a a M satisfy − β j β T A a j = 2 D C + F ( αβ = B D αβ β ) 0 . In the gauge ( − , a A Y U ji B AB = 4 αβ a bcd } ) = E T ) A γ j β αβ a , ij D j β F ( T a γ ( = i α A ∇ i α ( ij , A b , and = F D R ij B = i α ij ε ), one can extract the structure of the torsion constraints in the A a = i ij 2 2i T αβ {∇ a i . α + − D.6 A A D = = 2.1 0 αβ ) U } a j } β γ j β ) it is necessary to make the following identifications D D , , = ( i α i α 2.5a D ij D { { αβ A At this point it is possible to derive the degauged algebra of covariant derivatives. An To elaborate further, we must analyse the additional superfields introduced by the By investigating ( , except that A and To match ( as a tensor inexample, some to determine representation the of anti-commutator the of remainder spinor derivatives of we the consider superconformal algebra. For The resulting algebra is efficient way to do this is to consider a weight-zero primary superfield The vanishing of the dilatationconformal curvature connection at as dimension-1, see ( where the superfields special conformal connections is given by In the SU(2) superspacevanish. geometry As we of will section see,covariant derivative. these Then, conditions the resulting can geometry begeometry exactly reproduces satisfied of the by section SU(2) redefining superspace the degauged vector degauged geometry. We findD that these are all the same as for the covariant derivatives For example, in the gauge JHEP03(2021)157 j β = B (D.19) (D.20) (D.15) (D.21c) (D.17c) (D.18a) (D.21a) (D.16a) (D.17a) (D.21b) (D.17d) (D.18b) (D.17b) ··· and i α B + a ε i α j δ F ), one obtains , 1) B D = 0 γβ ε B δ − ) ε U ( β a A 1) ) } k δ 1) γ a D.16a − ( cd , B, ( − , ∇ γ c p ( kj F ) ) ( , δ l ε i B, γδ δj C ε γ ) ) F k C acd . c ), with ) implies ( ρ − K [ cd γ N γδ γδ j β (˜ A, ) ) βi γ C a . c ( a F − = 0 and ( j β , δk , γ γ k X F γ D.5c ρ p j F ( δ k γ ( D.18a A δk B, 8 i δ l B 5i δk j β . γα δ ( D F − δ ) A = 0 k i α X δ , k γ A, a δ ) A ε 0 + γ ) a ε γ F γδ ) kl + 2 S A, U j β j β ( ) + 8 + − ( βγ ε ] } a F cd γδ F d B ) ki B ) j β F γ ε N 2 R j abc ε ε c ), one obtains ( i A, + i A α γ ∇ + i γ i ( ε − = F 1) a γ , (˜ k γ c − c acd C D , [ B j β c C 1) B − i 2.22 D, k γ δ D, a ( δ ( + i( k αβ W − , a k p K F β F F ( [ ) ] ε c F l W c D − + i D C + 4 ε γδ i β A, αβk , a − k ] – 46 – ) k N αβ δ = 2 ( ρ γ d δk F c ) AB F D β AB D a γ γ X B, j β γδ B, B, T δ (˜ 1 T ) , γ + = 0 a 16 ) δkij F β ( F F ) c k δ αβ 0 δk ) ) ab C c + ρ p + [ [ A ij γ ) D + γδ U ( A, δ γ c k γ ab jk ε A A, γδ ] ( ) ) ) δ l ( 4 δ j β γ ) γ c F . j β F abc -curvature equation, eq. ( ) ( a a 8 B B 4 3 δ B γ R k γ γ β k (˜ − δ S γ γ ∇ F F F δρ + 2i − ) , ] at dimension-1. One finds − C c B, A bj kl A A a j [ [ i cd β k γ δ α F ε cd = ( i ( W kl i i C = ( = = i( i α D D ∇ γ D k . Since at mass dimension-3/2, ( ρ a D ( F , − − AB − j β , + ij j β j γ k ) AB a β D a c γ k a abc abc ) = [ k F β B = 2 = 2 D F D M J δ C 0 N W D ( ( c k k γ 2 AB AB k γ γ k U + k γ αβγδ ] j ε T T R − R D D j k γ β D a AB AB ij j β ) ) D C ε = = = = F , S i K 8 i α c ( a γ k ( kl cd D + ] = D R R [ j AB β AB AB AB T D T R R , 0 = a ), one can obtain D [ D.5 At dimension-3/2 only the By examining the expressions for the conformal superspace torsion and curvatures of Finally, we turn to Next, we compute [ derivatives of dimension-1 torsions is Its solution implies the differential constraints which indicates that the decomposition into irreducible and nontrivial tensors of the spinor gives nontrivial constraints. In particular, by using as well as the following conditions on the special conformal connections eq. ( by employing the dilatation curvatures of eq. ( which implies JHEP03(2021)157  γj C (D.26) (D.27) (D.28) (D.29) (D.22) (D.23) (D.24) (D.25) + 4 γj X . 3 4i  j ) k l δ − ] ε . b   j  β βγ k − C a ) ( ] k , N d δ aβ ] . γ k δ 1 2 b ( a c β k [ N p a F − − C ) δ δ l  j W , k γ ε ( β ) i γδ + k 4 a . ( ρ ) δk cd aβ ] a C a −  [ b  γ F 1 2 N γ j ( F γj j = p j k  C δ + aγ γδ δ 4 β k ρ ) β γ β + i(˜ a + δ δj δ [ − − C 2 C γ  γj .  j β k + 8 βδ − k 2i(˜ X ) ( ] aγ  X a γ kl d ρ k 3 γ ,X α 2i − γ j β N C j β δ C ( a 4 i  – 47 – γ k F  ) 3 4 αβ 1 2 c γ [ a ab J − − β βγ δ γk ( − ) ) − T ρ k 4 k ( k R cd W cd γ δj X − = a i γ 4 βγ γ X = 5 X α γ k 12 cd − + ( j β βδ 3 ), which permits any use, distribution and reproduction in ab X kl  + ( a ) 10i = j β γ  )  a T j a ρ  γ α ) β ( αβ γδj M ] β R ) and identifying βγ ( ρ d ) ab ) γk 5i γ 24 a ab X R ρ X γ γ − − C ( δ +( D.14 = ) j  = CC-BY 4.0 β = ( cd curvature derives from ] cd j β d γ = j β γ k a ( R This article is distributed under the terms of the Creative Commons a N F kl ab βγ  R j β ) then implies ) T c a [ a a δ γ R D.19 +2 ) upon using ( = 2i( 2.1 cd It is straightforward to continue the degauging procedure and obtain results at dimen- To conclude the analysis at dimension-3/2 we derive the corresponding torsion and j β a R sions higher than 3/2. We will not pursueOpen such Access. an analysis here. Attribution License ( any medium, provided the original author(s) and source are credited. One can thention prove ( that these results coincide with the dimension-3/2 results of sec- which implies Finally, the SU(2) which becomes The dimension-3/2 Lorentz curvature can be computed by using which leads to curvatures. For the dimension-3/2 torsion it holds that Equation ( JHEP03(2021)157 279 Phys. = 2 100 , Nucl. ]. N , = 2 ]. (2001) 667] Lect. Notes Springer N ]. , , = 2 SPIRE SPIRE IN N (1978) 149 598 Nucl. Phys. B IN ][ , SPIRE ][ Phys. Lett. B IN supergravity 135 supergravity-matter , [ Gauge and matter fields ]. Dimensions = 2 = 2 (2012) 119] 6 ]. ]. , N N Action in Six-dimensions and SPIRE supergravity from Chiral superfields in ]. 5 01 , 2 IN Erratum ibid. Actions, Conformal Invariance Higher derivative extension of action in six-dimensions [ R [ (1982) 309 2 2 = 6 SPIRE = 4 SPIRE R R IN SPIRE D IN Nucl. Phys. 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