hep-th/0312322 ITP-UH-34/03 N=8 superconformal mechanics S. Belluccia, E. Ivanovb, S. Krivonosb, O. Lechtenfeldc a INFN-Laboratori Nazionali di Frascati, C.P. 13, 00044 Frascati, Italy 4 [email protected] 0 0 b Bogoliubov Laboratory of Theoretical Physics, JINR, 141980 Dubna, Russia 2 n eivanov, [email protected] a J c Institut fu¨r Theoretische Physik, Universit¨at Hannover, 3 Appelstraße 2, 30167 Hannover, Germany 2 v [email protected] 2 2 3 2 1 3 Abstract 0 / h We construct new models of N=8 superconformal mechanics associated with the t - off-shell N=8,d=1 supermultiplets (3,8,5) and (5,8,3). These two multiplets are p e derived as N=8 Goldstone superfields and correspond to nonlinear realizations of h the N=8,d=1 superconformal group OSp(4⋆ 4) in its supercosets OSp(4⋆|4) and : | U(1)R⊗SO(5) v OSp(4⋆|4) , respectively. The irreducibility constraints for these superfields auto- i SU(2)R⊗SO(4) X matically follow from appropriate superconformal covariant conditions on the Car- r tan superforms. The N=8 superconformal transformations of the superspace coor- a dinates and the Goldstone superfields are explicitly given. Interestingly, each N=8 supermultiplet admits two different off-shell N=4 decompositions, with different N=4 superconformalsubgroupsSU(1,12) and OSp(4⋆ 2) of OSp(4⋆ 4) beingman- | | | ifestassuperconformalsymmetriesofthecorrespondingN=4,d=1superspaces. We present the actions for all such N=4 splittings of the N=8 multiplets considered. 1 Introduction Supersymmetric quantum mechanics (SQM) [1] - [26] 1 has plenty of applications ranging from the phenomenon of spontaneous breaking of supersymmetry [1, 6, 7] to the descrip- tion of the moduli of supersymmetric monopoles and black holes [12, 13, 20, 21]. The latter topic is closely related to the AdS /CFT pattern of the general AdS/CFT corre- 2 1 spondence [28], andit is thesuperconformal versions of SQM[3,4,5,13,14, 15,17,22,25] which are relevant in this context. Most attention has been paid to SQM models with extended N=2n supersymmetries (where n=1,2,...) 2 because the latter are related to supersymmetries in d>1 dimensions via some variant of dimensional reduction. So far, the mostly explored example is SQM with N 4 supersymmetry. However, the ≤ geometries of d=1 sigma models having N 4 supercharges are discussed in [12, 18]. Up ≥ to now, concrete SQM models with N=8,d=1 supersymmetry have been of particular use. In [12] such sigma models were employed to describe the moduli of certain solitonic black holes. In [11], a superconformal N=8,d=1 action was constructed for the low energy effective dynamics of a D0-brane moving in D4-brane and/or orientifold plane backgrounds (see also [29, 26]). In [21], N=8 SQM yielded the low energy description of half-BPS monopoles in N=4 SYM theory. It appears to be desirable to put the construction and study of N=8 (and perhaps N>8) SQM models on a systematic basis by working out the appropriate off-shell super- field techniques. One way to build such models is to perform a direct reduction of four-dimensional superfield theories. For instance, one may start from a general off-shell action contain- ing k copies of the d=4 hypermultiplet, naturally written in analytic harmonic superspace [30, 31], and reduce it to d=1 simply by suppressing the dependence on the spatial coordi- nates. The resulting action will describe the most general N=8 extension of a d=1 sigma model with a 4k-dimensional hyper-K¨ahler target manifold. Each reduced hypermultiplet yields a N=8,d=1 off-shell multiplet (4,8, ) [32] containing four physical bosons, eight ∞ physical fermions and an infinite number of auxiliary fields just like its d=4 prototype. A wider class of N=8 SQM models can be constructed reducing two-dimensional N=(4,4) or heterotic N=(8,0) sigma models. In this way one recovers the off-shell N=8,d=1 multiplets (4,8,4) and (8,8,0) [33]. The relevant superfield actions describe N=8,d=1 sigma models withstrong torsionful hyper-K¨ahler or octonionic-K¨ahler bosonic target geometries, respectively [12]. Although any d=1 super Poincar´e algebra can be obtained from a higher-dimensional one via dimensional reduction, this is generally not true for d=1 superconformal algebras [34, 35] and off-shell d=1 multiplets. For instance, no d=4 analog exists for the N=4,d=1 multiplet with off-shell content (1,4,3) [5] or (3,4,1) [9, 10, 20] (while the multiplet (2,4,2) is actually a reduction of the chiral d=4 multiplet). Moreover, there exist off- shell d=1 supermultiplets containing no auxiliary fields at all, something impossible for d 3 supersymmetry. Examples are the multiplets (4,4,0) [16, 19] and (8,8,0) [33] of ≥ N=4 and N=8 supersymmetries ind=1. This zoo of d=1 supermultiplets greatlyenlarges the class of admissible target geometries for supersymmetric d=1 sigma models and the relatedSQMmodels. Besidesthoseobtainablebydirectreductionfromhigherdimensions, 1See [27] for a more exhaustive list of references. 2By N in SQM models we always understand the number of real d=1 supercharges. 1 there exist special geometries particular to d=1 models and associated with specific d=1 supermultiplets. In fact, dimensional reduction is not too useful for obtaining superconformally in- variant d=1 superfield actions which are important for the study of the AdS /CFT 2 1 correspondence. One of the reasons is that the integration measures of superspaces hav- ing the same Grassmann-odd but different Grassmann-even dimensions possess different dilatation weights and hence different superconformal transformation properties. As a re- sult, some superconformal invariant superfield Lagrangians in d=1 differ in structure from their d=4 counterparts. Another reason is the already mentioned property that most d=1 superconformal groups do not descend from superconformal groups in higher dimensions. The most general N=4,d=1 superconformal group is the exceptional one-parameter (α) family of supergroups D(2,1;α) [35] which only at the special values α=0 and α= 1 (and − at values equivalent to these two) reduces to the supergroup SU(1,1 2) obtainable from | superconformal groups SU(2,2 N) in d=4. | Taking into account these circumstances, it is advantageous to have a convenient su- perfield approach to d=1 models which does not resort to dimensional reduction and is self-contained in d=1. Such a framework exists and is based on superfield nonlinear re- alizations of d=1 superconformal groups. It was pioneered in [5] and recently advanced in [22, 36]. Its basic merits are, firstly, that in most cases it automatically yields the irreducibility conditions for d=1 superfields and, secondly, that it directly specifies the superconformal transformation properties of these superfields. The physical bosons and fermions, together with the d=1 superspace coordinates, prove to be coset parameters associated with the appropriate generators of the superconformal group. Thus, the dif- ferences in the field content of various supermultiplets are attributed to different choices of the coset supermanifold inside the given superconformal group. Using the nonlinear realizations approach, in [36] all known off-shell multiplets of N=4,d=1 Poincar´e supersymmetry were recovered and a few novel ones were found, in- cluding examples of non-trivial off-shell superfield actions. With the present paper we begin a study of N=8,d=1 supermultiplets along the same line. We will demonstrate that the (5,8,3) multiplet of ref. [11] comes out as a Goldstone one, parametrizing a specific coset of the supergroup OSp(4⋆ 4) such that four physical bosons parametrize | the coset SO(5)/SO(4) while the fifth one is the dilaton. The appropriate irreducibility constraints in N=8,d=1 superspace immediately follow as a consequence of covariant in- verse Higgs [37] constraints on the relevant Cartan forms. As an example of a different N=8 mechanics, we construct a new model associated with the N=8,d=1 supermulti- plet (3,8,5). This supermultiplet parametrizes another coset of OSp(4⋆ 4) such that | SO(5) OSp(4⋆ 4) belongs to the stability subgroup while one out of three physical ⊂ | bosons is the coset parameter associated with the dilatation generator and the remaining two parametrize the R-symmetry coset SU(2) /U(1) . This model is a direct N=8 ex- R R tension of some particular case of the N=4,d=1 superconformal mechanics considered in [22] and the corresponding bosonic sectors are in fact identical. For both N=8 supermul- tiplets considered, we construct invariant actions in N=4,d=1 superspace. We find an interesting peculiarity of the N=4,d=1 superfield representation of the model associated with a given N=8 multiplet. Depending on the N = 4 subgroup of the N = 8 super Poincar´e group, with respect to which we decompose the given N = 8 superfield, we obtain different splittings of the latter into irreducible N=4 off-shell supermultiplets and, 2 consequently, different N=4 off-shellactions, which however produce the same component actions. There exist two distinct N=4 splittings of the considered multiplets, namely (5,8,3) = (3,4,1) (2,4,2) or (5,8,3) = (1,4,3) (4,4,0) , (1.1) ⊕ ⊕ (3,8,5) = (1,4,3) (2,4,2) or (3,8,5) = (3,4,1) (0,4,4) . (1.2) ⊕ ⊕ The first splitting in (1.1) is just what has been employed in [11]. The second splitting is new. Forboth of them we write down N=8 superconformal actions instandard N=4,d=1 superspace. The latter is also suited for setting up the off-shell superconformal action cor- responding to the first option in (1.2). As for the second splitting in (1.2), the equivalent off-shell superconformal action can be written only by employing N=4,d=1 harmonic superspace [23], since the kinetic term of the multiplet (0,4,4) is naturally defined just in this superspace. Thepaper isorganizedasfollows. InSection2we giveaN=8superfield formulationof the multiplet (3,8,5), based on a nonlinear realization of the superconformal supergroup OSp(4⋆ 4). In Sections 3 and 4 we present two alternative N=4 superfield formulations | of this multiplet and the relevant off-shell superconformal actions. Section 5 is devoted to treating the multiplet (5,8,3) along the same lines. A summary of our results and an outlook are the contents of the concluding Section 6. 2 N=8, d=1 superspace and tensor multiplet 2.1 N=8, d=1 superspace as a reduction of N=2, d=4 Itwillbemoreconvenient forustostartfromtheN = 8,d = 1superfielddescriptionofthe off-shell multiplet (3,8,5), since it is tightly related to the N = 4,d = 1 model studied in [22]. We shall first recover it within the dimensional reduction from N = 2,d = 4 superspace. The maximal automorphism group of N = 8,d = 1 super Poincar´e algebra (without central charges) is SO(8) and so eight real Grassmann coordinates of N = 8,d = 1 superspace can be arranged into one of three 8-dimensional real irreps of SO(8). For our purpose in this paper we shall split these 8 coordinates in another way, namely into two real quartets on which three commuting automorphism SU(2) groups will be realized. A convenient point of departure is N = 2,d = 4 superspace (xm,θα,θ¯iα˙) whose auto- i morphism group is SL(2,C) U(2) . The corresponding covariant derivatives form the R × following algebra: 3 i, j = 2iǫij∂ , (2.3) Dα Dα˙ αα˙ n o where ( j). The N = 2,d = 4 tensor multiplet is described by a real isotriplet jα˙ α D ≡ − D superfield (ik) subjected to the off-shell constraints V (i jk) = (i jk) = 0. (2.4) DαV Dα˙V These constraints can be reduced to d = 1 in two different ways, yielding multiplets of two different d = 1 supersymmetries. 3We use the following convention for the skew-symmetric tensor ǫ: ǫijǫjk =δik , ǫ12 =ǫ21 =1. 3 Tostartwith, ithasbeenrecentlyrealized[36]thatmanyN = 4,d = 1superconformal multiplets have their N = 2,d = 4 ancestors (though the standard dimensional reduction to N = 4,d = 1 superspace proceeds from N = 1,d = 4 superspace, see e.g. [9]). The relation between such d = 4 and d = 1 supermultiplets is provided by a special reduction procedure, whose key feature is the suppression of space-time indices in d = 4 spinor derivatives. The superfield constraints describing the N = 4,d = 1 supermultiplets are obtained from the N = 2,d = 4 ones by discarding the spinorial SL(2,C) indices and keeping only the R-symmetry indices. For instance, the constraints (2.4) after such a reduction become D(iVjk) = D(iVjk) = 0. (2.5) Here, the ‘reduced’ N = 4,d = 1 spinor derivatives Di,Dj are subject to Di,Dj = 2iǫij∂ (2.6) t n o and ∂ ∂ . The N = 4,d = 1 superfield Vij obeying the constraints (2.5) contains four αα˙ t → bosonic (three physical and one auxiliary) and four fermionic off-shell components, i.e. it defines the N = 4,d = 1 multiplet (3,4,1). It has been employed in [9] for constructing a general off-shell sigma model corresponding to the N = 4 SQM model of [2] and [8]. The same supermultiplet was independently considered in [10, 20] and then has been used in [22] to construct a new version of N = 4 superconformal mechanics. It has been also treated in the framework of N = 4,d = 1 harmonic superspace [23]. Other N = 2,d = 4 supermultiplets can be also reduced in this way to yield their N = 4,d = 1 superspace analogs [36]. It should be emphasized that, though formally reduced constraints look similar to their N = 2,d = 4 ancestors, the irreducible component field contents of the relevant multiplets can radically differ from those in d = 4 due to the different structure of the algebra of the covariant derivatives. For example, (2.4) gives rise to a notoph type differential constraint for a vector component field, while (2.5) does not impose any restriction on the t-dependence of the corresponding component fields. As an extreme expression of such a relaxation of constraints, some N = 2,d = 4 supermultiplets which are on-shell in the standard d = 4 superspace in consequence of their superfield constraints, have off-shell N = 4,d = 1 counterparts. This refers e.g. to the N = 2,d = 1 hypermultiplet without central charge [38] (leaving aside the formulations in N = 2,d = 4 harmonic superspace [30, 31]). Another, more direct way to reduce some constrained N = 2,d = 4 superfield to d = 1 is to keep as well the SL(2,C) indices of spinor derivatives, thus preserving the total number of spinor coordinates and supersymmetries and yielding a N = 8,d = 1 supermultiplet. However, such a reduction clearly breaks SL(2,C) down to its SU(2) subgroup iα, jβ = 2iǫijǫαβ∂ . (2.7) t D D n o A closer inspection of the relation (2.7) shows that, besides the U(2) automorphisms 4 R and the manifest SU(2) realized on the indices α,β and inherited from SL(2,C), it also possesses the hidden automorphisms δ i = Λβ i, δ i = Λ¯β i, Λα = 0, (2.8) Dα αDβ Dα − αDβ α 4Hereafter,wereservetheterm‘R-symmetry’forthoseautomorphismsofd=1Poincar´esuperalgebra which originate from R-symmetries in d=4. 4 which emerge as a by-product of the reduction. Together with the manifest SU(2) and overall phase U(1) transformations, (2.8) can be shown to form USp(4) SO(5). The R transformationswithΛ¯α = Λα closeonthemanifest SU(2)and,togetherw∼iththelatter, β − β constitute the subgroup Spin(4) = SU(2) SU(2) USp(4). The manifest SU(2) forms × ⊂ a diagonal in this product. Passing to the new basis 1 i Dia = ia + ia , iα = iα iα ,(Dia) = D , ( iα) = , ia iα √2 D D ∇ √2 D −D − ∇ −∇ (cid:0) (cid:1) (cid:0) (cid:1) (2.9) one can split (2.7) into two copies of N = 4,d = 1 anticommutation relations, such that the covariant derivatives from these sets anticommute with each other Dia,Djb = 2iǫijǫab∂ , iα, jβ = 2iǫijǫαβ∂ , Dia, jα = 0. (2.10) t t ∇ ∇ ∇ These(cid:8)two mutu(cid:9)ally anticommut(cid:8)ing algebr(cid:9)as pick up the lef(cid:8)t and righ(cid:9)t SU(2) factors of Spin(4) as their automorphism symmetries. Correspondingly, the N = 8,d = 1 super- space is parametrized by the coordinates (t,θ ,ϑ ), subjected to the reality conditions ia kα (θ ) = θia, (ϑ ) = ϑiα. (2.11) ia iα Such a representation of the algebra of N = 8,d = 1 spinor derivatives manifests three mutually commuting automorphism SU(2) symmetries which are realized, respectively, on the doublet indices i,a and α. The transformations from the coset USp(4)/Spin(4) ∼ SO(5)/SO(4) rotate Di and i through each other (and the same for θ and ϑ ). a ∇α ia iα In this basis, the N = 8,d = 1 reduced version of N = 2,d = 4 tensor multiplet (2.4) is defined by the constraints D(iVjk) = (iVjk) = 0 . (2.12) a ∇α This off-shell N = 8,d = 1 multiplet can be shown to comprise eight bosonic (three physical and five auxiliary) and eight fermionic components, i.e. it is (3,8,5). We shall see that (2.12) implies a d = 1 version of the d = 4 notoph field strength constraint whose effect in d = 1 is to constrain some superfield component to be constant [5]. In the next Section we shall show that the N = 8,d = 1 tensor multiplet Vij defined by (2.12) can support, besides the manifest N = 8,d = 1 Poincar´e supersymmetry, also a realization of the N = 8,d = 1 superconformal algebra osp(4⋆ 4). While considered as | a carrier of the latter, it can be called ‘N = 8,d = 1 improved tensor multiplet’. In fact, like its N = 4,d = 1 counterpart (3,4,1), it can be derived from a nonlinear realization of the supergroup OSp(4⋆ 4) in the appropriate coset supermanifold, without any reference | to the dimensional reduction from d = 4. 2.2 Superconformal properties of the N=8, d=1 tensor multi- plet The simplest way to find the transformation properties of the N = 8,d = 1 tensor multiplet Vij and prove the covariance of the basic constraints (2.12) with respect to the superconformal N = 8 superalgebra osp(4⋆ 4) is to use the coset realization technique. | All steps in this construction are very similar to those employed in [22]. So we quote here the main results without detailed explanations. 5 We use the standard definition of the superalgebra osp(4⋆ 4) [35]. It contains the | following sixteen spinor generators: QiaA, QiαA, (QiaA) = ǫ ǫ QjbA, (i,a,α,A = 1,2), (2.13) 1 2 ij ab and sixteen bosonic generators: TAB, Tij, Tab, Tαβ, Uaα . (2.14) 0 1 2 The indices A,i,a and α refer to fundamental representations of the mutually commuting sl(2,R) TAB and three su(2) Tij,Tab,Tαβ algebras. The four generators Uaα belong to the co∼set0SO(5)/SO(4)with S∼O(4) ge1nera2ted by Tab and Tαβ. The bosonic generators 1 2 form the full bosonic subalgebra sl(2,R) su(2) so(5) of osp(4⋆ 4). R ⊕ ⊕ | The commutator of any T-generator with Q has the same form, and it is sufficient to write it for some particular sort of indices, e.g. for a,b (other indices of QiaA being 1,2 suppressed): i Tab,Qc = ǫacQb +ǫbcQa . (2.15) −2 The commutators with the c(cid:2)oset SO(cid:3)(5)/SO(cid:0)(4) generators(cid:1)Uaα have the following form Uaα,QibA = iǫabQiαA, Uaα,QiβA = iǫαβQiaA . (2.16) 1 − 2 2 − 1 h i (cid:2) (cid:3) At last, the anticommutators of the fermionic generators read QiaA,QjbB = 2 ǫijǫabTAB 2ǫijǫABTab +ǫabǫABTij , 1 1 − 0 − 1 n o (cid:0) (cid:1) QiαA,QjβB = 2 ǫijǫαβTAB 2ǫijǫABTαβ +ǫαβǫABTij , 2 2 − 0 − 2 n o (cid:16) (cid:17) QiaA,QjαB = 2ǫijǫABUaα . (2.17) 1 2 n o From (2.17) it follows that the generators QiaA and QiαA, together with the corresponding 1 1 bosonic generators, span two osp(4⋆ 2) subalgebras in osp(4⋆ 4). For what follows it is | | convenient to pass to another notation, P T22, K T11, D T12, V T22, V T11, V T12, ≡ 0 ≡ 0 ≡ − 0 ≡ ≡ 3 ≡ Qia Qia2, iα Qiα2, Sia Qia1, iα Qiα1. (2.18) ≡ − 1 Q ≡ − 2 ≡ 1 S ≡ 2 One can check that P and Qia, iα constitute a N = 8,d = 1 Poincar´e superalgebra. Q The generators D,K and Sia, iα stand for the d = 1 dilatations, special conformal S transformations and conformal supersymmetry, respectively. The full structure of the superalgebra osp(4⋆ 4) is given in Appendix. | Now we shall construct a nonlinear realization of the superconformal group OSp(4⋆ 4) | in the coset superspace with an element parametrized as g = eitPeθiaQia+ϑiαQiαeψiaSia+ξiαSiαeizKeiuDeiφV+iφ¯V . (2.19) The coordinates t,θ ,ϑ parametrize the N = 8,d = 1 superspace. All other supercoset ia iα parameters are Goldstone N = 8 superfields. The stability subgroup contains a sub- group U(1) of the group SU(2) realized on the doublet indices i, hence the Goldstone R R 6 ¯ superfields φ,φ parametrize the coset SU(2) /U(1) . The group SO(5) is placed in the R R stability subgroup. It linearly rotates thefermionic Goldstone superfields ψ and ξ through each other, equally as the N = 8,d = 1 Grassmann coordinates θ and ϑ’s. To summarize, in the present case we are dealing with the supercoset OSp(4⋆|4) . U(1)R⊗SO(5) The semi-covariant (fullycovariant only under N = 8 Poincar´e supersymmetry) spinor derivatives are defined by ∂ ∂ Dia = +iθia∂ , iα = +iϑiα∂ . (2.20) t t ∂θ ∇ ∂ϑ ia iα Their anticommutators, by construction, coincide with (2.10). A natural way to find conformally covariant irreducibility conditions on the coset superfields is to impose the inverse Higgs constraints [37] on the left-covariant Cartan one-form Ω valued in the superalgebra osp(4⋆ 4). This form is defined by the standard | relation g−1dg = Ω . (2.21) In analogy with [22, 36], we impose the following constraints: ω = 0 , ω = ω¯ = 0 (2.22) D V V | | where denotes the spinor projection. These constraints are manifestly covariant under | the whole supergroup. They allow one to express the Goldstone spinor superfields and the superfield z via the spinor and t-derivatives, respectively, of the remaining bosonic ¯ Goldstone superfields u,φ,φ iDiau = 2ψia , i iαu = 2ξiα , u˙ = 2z , − ∇ − iD1aΛ = 2Λ ψ1a +Λψ2a , iD2aΛ = 2 ψ1a +Λψ2a , − i 1αΛ = 2α(cid:0)ξ1α +Λξ2α(cid:1), i 2αΛ = 2Λ(cid:0) ξ1α +Λξ2α(cid:1) , (2.23) ∇ ∇ − where (cid:0) (cid:1) (cid:0) (cid:1) ¯ ¯ tan φφ tan φφ ¯ Λ = φ , Λ = φ . (2.24) ¯ ¯ pφφ pφφ ¯ Simultaneously, eqs. (2.23) implypsome irreducibilitpy constraints for u,φ,φ. After intro- ducing a new N = 8 vector superfield Vij such that Vij = Vji and Vik = ǫii′ǫkk′Vi′k′, via Λ Λ i 1 ΛΛ V11 = i√2eu , V22 = i√2eu , V12 = eu − , − 1+ΛΛ 1+ΛΛ √2 1+ΛΛ V2 VikV = e2u , (2.25) ik ≡ and eliminating the spinor superfields from (2.23), the differential constraints on the remaining Goldstonesuperfields canbebrought inthemanifestly SU(2) -symmetric form R D(iVjk) = 0 , (iVjk) = 0 , (2.26) a ∇α which coincides with (2.12). For further use, we present one important consequence of (2.26) ∂ DaD Vij + α Vij = 0 α Vij = 6m DaD Vij, m = const. (2.27) t i ja ∇i ∇jα ⇒ ∇i∇jα − i ja (cid:0) (cid:1) 7 The specific normalization of the arbitrary real constant m is chosen for convenience. Besidesensuringthecovarianceofthebasicconstraints(2.26)withrespecttoOSp(4⋆ 4) | and clarifying their geometric meaning, the coset approach provides the easiest way to find the transformation properties of all coordinates and superfields. Indeed, all transfor- mations are generated by acting on the coset element (2.19) from the left by the elements of osp(4⋆ 4). Since all bosonic transformations appear in the anticommutator of the con- | formal supersymmetry and Poincar´e supersymmetry, it is sufficient to know how Vij is transformed under these supersymmetries. The N = 8,d = 1 Poincar´e supersymmetry is realized on superspace coordinates in the standard way δt = i η θia +η ϑiα , δθ = η , δϑ = η (2.28) ia iα ia ia iα iα − and Vik is a scalar with re(cid:0)spect to these(cid:1)transformations. On the other hand, N = 8 superconformal transformations are non-trivially realized both on the coordinates and coset superfields δt = it ǫiaθ +εiαϑ + ǫiθja +εiϑjα θ θb +ϑ ϑβ , − ia iα a α ib j iβ j (cid:16) (cid:17) δθ = tǫ(cid:0) iǫjθ θb +2(cid:1)iǫjθ(cid:0)bθ iǫjϑ ϑα(cid:1)+2iεαϑ θj , ia ia − a jb i b i ja − a jα i j iα a δϑ = tε iεjϑ ϑβ +2iεjϑβϑ iεjθ θa +2iǫjθaϑ , iα iα − α jβ i β i jα − α ja i a i jα δu = 2i ǫiaθ +εiαϑ , δΛ = a+ibΛ+a¯Λ2 , (2.29) ia iα − where (cid:0) (cid:1) a = 2i(ǫaθ +εαϑ ), a¯ = 2i(ǫaθ +εαϑ ), 2 2a 2 2α 1 1a 1 1α b = 2(ǫaθ +ǫaθ +εαϑ +εαϑ ) . (2.30) − 1 2a 2 1a 1 2α 2 1α The conformalsupersymmetry transformationofVij has themanifestly Spin(4) SU(2) R × covariant form δVij = 2i (θkaǫ +ϑkαε )Vij +(ǫa(iθ +ǫaθ(i +εα(iϑ +εαϑ(i)Vj)k . (2.31) ka kα ka k a kα k α Thus we kn(cid:2)ow the transformation properties of the N = 8 ‘tensor’ m(cid:3)ultiplet Vij under the N = 8 superconformal group OSp(4⋆ 4). Before turning to the construction | of superconformal invariant actions, in the next Section we will study how this N = 8 multiplet is described in N = 4,d = 1 superspace. 3 N=8, d=1 tensor multiplet in N=4 superspace While being natural and very useful for deriving irreducibility constraints for the su- perfields and establishing their transformation properties, the N = 8,d = 1 superfield approach is not too suitable for constructing invariant actions. The basic difficulty is, of course, the large dimension of the integration measure, which makes it impossible to write the action in terms of this basic superfields without introducing the prepotential and/or passing to some invariant superspaces of lower Grassmann dimension, e.g. chiral 8 or harmonic analytic superspaces. Another possibility is to formulate the N = 8 tensor multiplet in N = 4,d = 1 superspace. In such a formulation half of N = 8 supersymme- triesis hiddenandonlyN = 4 supersymmetry ismanifest. Nevertheless, it allows arather straightforward construction of N = 8 supersymmetric and superconformal actions. Just this approach was used in [11]. In the next sections we shall reproduce the action of [11] and construct new N = 4 superfield actions with second hidden N = 4 supersymmetry, both for vector and tensor N = 8 multiplets. In the present section we consider two N = 4 superfield formulations of the N = 8 tensor multiplet. 3.1 (3,8,5) = (3,4,1) (0,4,4) ⊕ In order to describe the N = 8 tensor multiplet in terms of N = 4 superfields we should choose the appropriate N = 4 superspace. The first (evident) possibility is to consider the N = 4 superspace with coordinates (t,θ ). (3.1) ia In this superspace the N = 4 conformal supergroup OSp(4⋆ 2) P,K,D,Tij,Tab,Qia,Sia | ∼ 1 is naturally realized, while the rest (cid:8)of the osp(4⋆ 4) generato(cid:9)rs mixes two irreducible | N = 4 superfields comprising the (3,8,5) N = 8 supermultiplet in question. Expanding the N = 8 superfields Vij in ϑ , one finds that the constraints (2.26) leave in Vij the iα following four bosonic and four fermionic N = 4 projections: vij = Vij , ξi Vij , A α Vij (3.2) α ≡ ∇jα ≡ ∇i∇jα (cid:12) (cid:12) (cid:12) where means restriction to ϑ(cid:12)iα = 0. Each N(cid:12)= 4 superfield is s(cid:12)ubjected, in virtue of | (2.26), to an additional constraint D(ivjk) = 0, D(iξj) = 0, a a α A = 6m DaD vij, m = const, (3.3) − i aj where we used (2.27). Thus, we conclude that our N = 8 tensor multiplet Vij, when rewritten in terms of N = 4 superfields, amounts to a direct sum of the N = 4 ‘tensor’ multiplet vij with the (3,4,1) off-shell content and a fermionic analog of the N = 4 hypermultiplet ξi with the α (0,4,4) off-shell content,5 plus a constant m of dimension of mass (i.e. (cm−1)). The conservation-law type condition (2.27) yielding this constant is in fact a d = 1 analog of the ‘notoph’ condition ∂ Am(x) = 0 of the N = 2,d = 4 tensor multiplet; the appearance m of a similar constant in the case of the N = 4,d = 1 supermultiplet (1,4,3), which is defined by the d = 1 reduction of the N = 1,d = 4 tensor multiplet constraints, was earlier observed in [5]. 5This supermultiplet hasbeen introducedin [33]; its off-shellsuperfieldactionwasgivenin[23]inthe framework of N =4,d=1 harmonic superspace. 9