Higher Spins in Hyper-Superspace

We extend the results of arXiv:1401.1645 on the generalized conformal Sp(2n)-structure of infinite multiplets of higher spin fields, formulated in spaces with extra tensorial directions (hyperspaces), to the description of OSp(1|2n)-invariant infinite-dimensional higher-spin supermultiplets formulated in terms of scalar superfields on flat hyper-superspaces and on OSp(1|n) supergroup manifolds. We find generalized superconformal transformations relating the superfields and their equations of motion in flat hyper-superspace with those on the OSp(1|n) supermanifold. We then use these transformations to relate the two-, three- and four-point correlation functions of the scalar superfields on flat hyperspace, derived by requiring the OSp(1|2n) invariance of the correlators, to correlation functions on the OSp(1|n) group manifold. As a byproduct, for the simplest particular case of a conventional N=1, D=3 superconformal theory of scalar superfields, we also derive correlation functions of component fields of the scalar supermultiplet including those of auxiliary fields.

The theories on tensorially extended (super)spaces, which we will henceforth refer to as hyper-(super)spaces, offer many interesting and challenging problems regarding higher-spin fields, one of them being the further development and study of generalized (super)conformal theories on these spaces. This motivated our recent work [1] in which, using generalized conformal transformations, we established an explicit relation between the equations of motion of hyperfields on flat hyperspace and on Sp(n) group-manifolds, the latter being tensorial generalizations 1 of AdS spaces. This relation was then employed in order to explicitly derive the Sp(2n)-invariant two-, three-and four-point correlation functions for fields on Sp(n) group manifolds, from the known Sp(2n)-invariant correlation functions on flat hyperspaces, thus, generalizing the results obtained in [5,10,23].
In this paper we further extend the results of [1] to the description of supersymmetric systems of higher spin fields in hyper-superspaces, which were previously studied e.g. in [3, 4, 6-8, 11, 14, 20, 24]. In particular, by means of a generalized superconformal transformation, we establish an explicit relation between the superfield equations of motion [11] on flat hyper-superspace and on an OSp(1|n) supergroup manifold. Furthermore, the explicit solution of the generalized superconformal Ward identities allows us to derive the OSp(1|2n)-invariant two-, three-and four-point superfield correlation functions on flat hyper-superspace and, consequently, using the generalized superconformal transformations, we obtain the corresponding correlation functions on the OSp(1|n) group manifolds. Our results, therefore, generalize the superfield description and computation of superfield correlators in conventional superconformal field theories, considered e.g. in [25][26][27][28], to superconformal higher-spin theories. A byproduct of our analysis is the derivation of correlation functions involving the component fields of the scalar supermultiplet, including the auxiliary fields, for the simple special case of a three-dimensional N = 1 superconformal theory of scalar superfields.
As in the case of the N = 1, D = 3 superconformal theory, the fact that 3-and 4-point correlation functions are non-zero for hyperfields of an anomalous conformal weight may indicate the existence of interacting conformal higher-spin fields which involve higher orders of their field strengths.
It should be noted that in the literature  various supersymmetric higher spin systems have been considered in either irreducible or reducible representations of the Poincaré and AdS groups (see e.g. [50,51] for a discussion of reducible higher spin multiplets in the "metric-like" approach). As we will see, the systems of integer and half-integer higher spin fields considered in [3, 4, 6-8, 11, 14, 20] and in this paper form irreducible infinite dimensional supermultiplets of space-time supersymmetry. These supersymmetric higher spin systems are therefore different from finite dimensional higher spin supermultiplets considered in .
The paper is organized as follows. Section 2 begins with a review of some basic known results about hyper-superspaces. We describe in detail the generalized superconformal algebra, the realization of the generalized superconformal group OSp(1|2n) on hyper-superspace and the precise connection between generalized and conventional conformal weights for scalar superfields and their components in various dimensions. Finally, we demonstrate how an infinite dimensional N = 1 supersymmetry multiplet is formed by the component fields of the hyper-superfield in the case of four-dimensional flat space-time.
In Section 3 we provide a description of the geometric structure of OSp(1|n) manifolds. These manifolds exhibit the property of generalized superconformal flatness (or GL-flatness) observed earlier in [7,8], which is similar to the superconformal flatness property of certain conventional AdS superspaces and superspheres [52][53][54][55][56]. We then consider the relation between the OSp(1|2n)-invariant field equations for scalar superfields on flat hyper-superspace and those on the OSp(1|n) group manifold derived in [11]. We show that, similarly to the non-supersymmetric case [1], the supersymmetric field equations on flat hyper-superspace and on OSp(1|n) group manifolds are related to each other via a generalized superconformal transformation of the scalar hyper-superfield and its derivatives.
In Section 4, as a preparation for the computation of correlation functions on flat hypersuperspace and on OSp(1|n) supergroup manifolds, we consider the simplest example of an OSp(1|4)-invariant superconformal theory of a conventional N = 1, D = 3 massless scalar superfield. Even though higher-spin fields are absent in this case, it is a simple setup in which one can illustrate the salient features of our approach. To this end, we present the OSp(1|4)invariant two-, three-and four-point correlation functions of scalar superfields, as well as the correlators of the component fields of the scalar supermultiplet, including those of auxiliary fields.
Finally, in Section 5 we use the requirement of OSp(1|2n) invariance to derive the expressions for two-, three-and four-point correlation functions of the scalar hyper-superfields. Again, in a complete analogy with the non-supersymmetric systems [1], the correlation functions on flat hyper-superspaces and OSp(1|n) supergroup manifolds are related via generalized superconformal Weyl rescaling. Thus, our basic result is that the GL-flatness is a key property of Sp(n) and OSp(1|n) manifolds that renders them amenable to the same type of analysis as for the case of flat hyper (super) spaces.
We conclude with a discussion on open problems and perspectives for further development of the hyperspace formulation of higher-spin fields.
2 Scalar superfields in flat hyper-superspace, equations of motion and correlators
The four coordinates x m parametrize the conventional flat space-time which is extended to flat hyperspace by adding six extra dimensions, parametrized by y mn = −y nm . This bosonic hyperspace is then further extended to the hyper-superspace by adding four Grassmann-odd directions parametrized by θ µ , which transform in the spinor representation of the D = 4 Lorentz group SO (1, 3).
The supersymmetry variation of the coordinates leaves invariant the Volkov-Akulov-type one-form 3) The round brackets denote symmetrization of indices with the standard normalization The supersymmetry transformations form a generalized super-translation algebra with P µν generating translations along X µν . Namely, δX µν = ia ρλ P ρλ · X µν = a µν , with a µν being constant parameters. The realization of P µν and Q µ as differential operators is given by where, by definition, Furthermore, in the case n = 4, D = 4, the partial derivative associated with (2.1) takes the form (2.8) The algebra (2.5) is invariant under rigid GL(n) transformations generated by which act on P µν and Q µ as 11) and close into the gl(n) algebra The algebra (2.5), (2.11) and (2.12) is the hyperspace counterpart of the conventional super-Poincaré algebra enlarged by dilatations. That this is so can be most easily seen by considering e.g. n = 2 (i.e. µ = 1, 2), in which case this algebra is recognized as the D = 3 super-Poincaré algebra with L µ ν − 1 2 δ ν µ L ρ ρ = M m (γ m ) µ ν generating the SL(2, R) ∼ SO(1, 2) Lorentz rotations (note that m = 0, 1, 2) and D = 1 2 L ρ ρ being the dilatation generator. Note that the factor 1 2 in the definition of the dilatation generator is required in order to have the canonical scaling of the momentum generator P µν with weight 1 and the supercharge Q µ with weight 1 2 , as follows from eq. (2.11).
This algebra may be further extended to the OSp(1|2n) algebra, generating generalized superconformal transformations of the flat hyper-superspace, by adding the additional set of supersymmetry generators together with the generalized conformal boosts 14) The generators S µ and K µν form a superalgebra similar to (2.5) while the non-zero (anti)commutators of S µ and K µν with Q µ , P µν and L µ ν read

Generalized superconformal algebra OSp(1|2n)
We now collect together all the non-zero (anti)commutation relations among the generators of the OSp(1|2n) algebra Let us note that in the case n = 4, in which the physical space-time is four-dimensional (see eq. (2.1)) the generalized superconformal group OSp(1|8) contains the D = 4 conformal symmetry group SO(2, 4) ∼ SU (2, 2) as a subgroup, but not the superconformal group SU (2, 2|1). The reason being that, although OSp(1|8) and SU (2, 2|1) contain the same number of (eight) generators, the anticommutators of the former close on the generators of the whole Sp (8), while those of the latter only close on an U (2, 2) subgroup of Sp (8), and the same supersymmetry generators cannot satisfy the different anti-commutation relations simultaneously. In fact, the minimal OSp-supergroup containing SU (2, 2|1) as a subgroup is OSp(2|8).

Scalar superfields and their OSp(1|2n)-invariant equations of motion
Let us now consider a superfield Φ(X, θ) transforming as a scalar under the super-translations given in eq.
To construct equations of motion for Φ(X, θ) which are invariant under (2.18) and comprise the equations of motion of an infinite tower of integer and half-integer higher spin fields with respect to conventional space-time, we introduce the spinorial covariant derivatives which (anti)commute with Q µ and P µν . The Φ-superfield equations then take the form [11] where the brackets denote the anti-symmetrization of indices with unit overall strength similarly to (2.4). As was shown in [11], these superfield equations imply that all components of Φ(X, θ) except for the first and the second one in the θ µ -expansion of Φ(X, θ) should vanish (i.e. A µ 1 ...ν k = 0 for k > 1) while the scalar and spinor fields b(X) and f µ (X) satisfy the equations first derived in [4] ( For n=4, 8 and 16 these equations encode the Bianchi identity and equations of motion for the curvatures of infinite towers of conformally invariant, massless higher-spin fields in 4-, 6-and 10-dimensional flat space-time, respectively (see [4,12]). The superfield equations (2.20) are invariant under the generalized superconformal OSp(1|2n) symmetry, provided that Φ(X, θ) transforms as a scalar superfield with the 'canonical' generalized scaling weight 1 2 where the factor 1 2 in the second line is the generalized conformal weight and ǫ µ , ξ µ , a µν , k µν and g µ ν are the rigid parameters of the OSp(1|2n) transformations.
Scalar superfields with anomalous generalized conformal dimension ∆ transform under OSp(1|2n) as It is instructive to demonstrate how the generalized conformal dimension ∆, which is defined to be the same for all values of n in OSp(1|2n), is related to the conventional conformal weight of scalar superfields in various space-time dimensions. As we have already mentioned in Section 2.1, the dilatation operator should be identified with D = 1 2 L µ µ . Therefore, considering a GL(n) the part of the transformation corresponding to the dilatation reads whereg = 2 n g µ µ is the genuine dilatation parameter. From (2.25) it then follows that the conventional conformal weight ∆ D of the scalar superfield is related to the generalized one ∆ via In the n = 2 case corresponding to the N = 1, D = 3 scalar superfield theory the two conformal dimensions coincide, whereas in the case n = 4 describing conformal higher spin fields in D = 4 one finds ∆ 4 = 2∆. Relation (2.27) indeed provides the correct conformal dimensions of scalar superfields (and consequently of their components) in the corresponding space-time dimensions. For instance, when ∆ = 1 2 , in D = 3 one finds 1 2 as the canonical conformal dimension of the scalar superfield, while in the cases D = 4 and D = 6, n = 8 it is found to be equal to one and two, respectively. For convenience, we shall henceforth associate the scaling properties of the fields to the universal D-and n-independent generalized conformal weight ∆.
The D = 4 higher-spin field curvatures are contained in b(X) and f µ (X) as the components of the series expansion in the powers of the tensorial coordinates y mn of the flat hyperspace (2.1) Remember that in (2.29), C ρµ = −C µρ is the charge conjugation matrix used to raise spinor indices, φ(x) and ψ ρ (x) are a D = 4 scalar and a spinor field, respectively, F m 1 n 1 (x) is the Maxwell field strength, R m 1 n 1 ,m 2 n 2 (x) is the curvature tensor of linearized gravity, R ρ m 1 n 1 (x) is the Rarita-Schwinger field strength and other terms in the series stand for generalized Riemann curvatures of spin-s fields 2 that also contain contributions of derivatives of the fields of lower spin denoted by dots, as in the case of the Rarita-Schwinger and gravity fields (see [12] for further details).
The fact that the higher spin fields should form an infinite-dimensional representation of N = 1, D = 4 supersymmetry is prompted by the observation that the spectrum of bosonic fields contains a single real scalar field φ(x), which alone cannot have a fermionic superpartner, while each field with s > 0 has two helicities ±s. Indeed, from (2.28) we obtain an infinite entangled chain of supersymmetry transformations for the D = 4 fields and so on.
3 Scalar superfields on OSp(1|n) group manifolds and their equations of motion 3.1 Geometric structure of the OSp(1|n) group manifolds The geometric structure of the OSp(1|n) group manifolds in the form we shall review below and use extensively in this paper for the description of higher spin fields in the associated AdS spaces has been discussed in [3,7,8,11,24]. The OSp(1|n) superalgebra is formed by n anti-commuting supercharges Q α and n(n+1) where C αβ = −C βα is the Sp(n) invariant symplectic metric and ξ is a parameter of inverse dimension of length related to the AdS radius via r = 2/ξ (see also [1]). The OSp(1|n) algebra (3.1) is recognized as a subalgebra of (2.17) with the identifications where S α = S β C βα , L αβ = L α γ C γβ and K αβ = K γδ C γα C δβ . The OSp(1|n) manifold is parametrized by the coordinates (X µν , θ µ ) and its geometry is described by the Cartan forms where O(X, θ) is an OSp(1|n) supergroup element. The Cartan forms satisfy the Maurer-Cartan equations associated with the OSp(1|n) superalgebra (3.1) with the external differential acting from the right. The Maurer-Cartan equations (3.4) are then solved by the following forms where Θ is related to θ through while the covariant derivative contains the Cartan form of the Sp(n) group manifold and Note also the relations 11) and the fact that the inverse matrix of (3.10) is given by The form of the bosonic Cartan form (3.5) prompts us that the latter is related to the superinvariant form (2.3) in flat hyper superspace via the GL(n) tansformation with matrix element (3.10). This property was revealed in [7] and called GL-flatness of the OSp(1|n) supermanifold. It will allow us to generalize the results of [1] and relate the scalar superfield Φ(X, θ) and its field equation (2.20) in flat superspace to a scalar superfield and its equation of motion on the supergroup manifold OSp(1|n).

Scalar superfield on OSp(1|n) and its OSp(1|2n) invariant equation of motion
The scalar superfield equation on OSp(1|n) takes the form [11] where the Grassmann-odd covariant derivatives ∇ α and their bosonic counterparts ∇ αβ satisfy the OSp(1|n) superalgebra similar to (3.1), namely A somewhat tedious but straightforward algebra then shows that the superfield Φ OSp (X, θ) satisfying (3.13) is related to the superfield Φ(X, θ) satisfying the flat superspace equation (2.20) by the super-Weyl transformation while the OSp(1|n) covariant derivatives are obtained from the flat superspace ones by the following GL ('generalized superconformal') transformations Substituting (2.21) into (3.17) and using the definition (3.7), together with the fact that on the mass shell all higher components in (2.21) vanish, we find where the first two terms are the fields propagating on the Sp(n) group manifold, and O(Θ 2 , b(x)) stands for higher order terms in Θ 2 which only depend on b(X). The fields (3.20) satisfy the equations of motion discussed in detail in [1]. Note that in (3.21) and (3.22) the covariant derivatives are restricted to the bosonic group manifold Sp(n), i.e. ∇ αβ = G −1 µ α (X) G −1 ν β (X) ∂ µν . Since the flat superspace field equation is invariant under the generalized superconformal OSp(1|2n) transformations (2.24), the above relation leads us to conclude that also the OSp(1|n) superspace equations (3.13) are invariant under the OSp(1|2n) transformations, under which the superfield Φ OSp (X, θ) varies as Here and Using the relations and The other generators of the OSp(1|2n) are and Taking into account the commutation relations (3.32) we see that the operators Q µ , S µ , P µν , L µ ν , K µν obey the same OSp(1|2n) algebra (2.17) as the operators Q µ , S µ , P µν , L µ ν and K µν .

Correlation functions in N = 1, D = 3 superconformal models
Before considering correlation functions for superfields in hyper superspaces, it is instructive to discuss in detail analogous structures arising in the superconformal theory of a real scalar superfield in a conventional N = 1, D = 3 superspace. The reason being that this model is the simplest example (with n = 1) of the OSp(1|2n) invariant systems considered above. The physical content of this system is a real scalar and a D = 3 Majorana spinor field whereas the massless higher spin fields are absent. The superconformally invariant two-and three-point correlation functions of the N = 1, D = 3 model have been constructed in [26] with the use of a slightly different notation. Below we shall discuss properties of the two-and three-point functions for the D = 3 scalar superfield and its components using a formalism which straightforwardly generalizes to higher-dimensional hyperspaces.
Let us use the spinor-tensor representation for the description of the three-dimensional spacetime coordinates where α, β = 1, 2 are D = 3 spinorial indices and m = 0, 1, 2 is the vectorial one. Since (4.1) provides a representation of the symmetric 2 × 2 matrices x αβ , no extra coordinates, like y mn , are present and, hence, no higher spin fields. The inverse matrix of (4.1), takes the simple form We may now consider a real scalar superfield in D = 3 and imply the supersymmetry transformations of the component fields where we have made use of the identity Moreover, under conformal supersymmetry, Φ(x, θ) transforms as where ∆ is the conformal weight of the superfield. The superconformal transformations of the component fields are given by (4.14) The conformal weights of φ, f α and F are ∆, ∆ + 1 2 and ∆ + 1, respectively. It should be noted that the field equation (4.5) is superconformally invariant if the superfield Φ(x, θ) has the canonical conformal weight ∆ = 1 2 .

Two-point functions
The form of correlation functions in superconformal theories is drastically restricted by the requirement of their superconformal invariance. The two-point correlation function of the superfield Φ(x, θ) with conformal weight ∆ is obtained by first solving the superconformal Ward identities which involve Q-and S-supersymmetry transformations. The invariance under bosonic translations, rotations, conformal boosts and dilations then follows as a consequence of the properties of the superconformal algebra. The Qand S-supersymmetry Ward identities are and The solution to these equations takes the form where c 2 is an arbitrary normalization constant and is invariant under Q-supersymmetry. As usual, for the two-point function to be non-vanishing, the conformal weights of the two superfields should be equal.
Expanding the expression on the right hand side of (4.16) in powers of θ, we obtain (4.18) Using the identities and one may rewrite the expression (4.18) as .
Let us note that when the superfield Φ(x, θ) has the canonical conformal dimension ∆ = 1 2 , due to the identity the last term in (4.18) is proportional to the δ-function if one moves to the Euclidean signature. Then one has for the two point function for the auxiliary field Note that the correlation functions of the auxiliary field F with the physical fields and with itself (for x m 1 = x m 2 ) vanish. On the other hand, if the conformal weight of the superfield (4.4) is anomalous, i.e. ∆ = 1 2 , the correlators of the auxiliary field with the physical ones still vanish (in agreement with the fact that their conformal weights are different), but the F F correlator is This situation may correspond to an interacting quantum N = 1 superconformal field theory [57], where the auxiliary field is non-zero, and fields acquire anomalous dimensions due to quantum corrections.

Three-point functions
We now consider three-point functions involving three real scalar superfields carrying scaling dimensions ∆ i (i=1,2,3). Solving the supeconformal Ward identities for Q-and S-supersymmetry transformations we find where Using the expansion (4.21), one obtains the three-point functions of the component fields of , whose labels of scaling dimension we skip for simplicity The remaining three-point functions containing an odd number of fermions, as well as the correlator F φφ , vanish. Note that, dimensional arguments would allow for a non-zero F φφ correlator, but supersymmetry forces it to vanish. The correlator F (x 1 )F (x 2 )F (x 3 ) is zero as well, since it is proportional to (γ m γ n γ p )x m 12 x n 23 x p 31 = 2iǫ mnp x m 12 x n 23 x p 31 = 0. Moreover, from the above expressions we see that superconformal symmetry does not fix the values of the scaling dimensions ∆ i (4.30) entering the right hand side of (4.29). This indicates that quantum operators may acquire anomalous dimensions and the quantum N = 1, D = 3 superconformal theory of scalar superfields can be non-trivial, in agreement e.g. with the results of [57].
If the value of ∆ were restricted by superconformal symmetry to its canonical value and no anomalous dimensions were allowed (for all the operators which are not protected by supersymmetry) one would conclude that the conformal fixed point is that of the free theory. This is the case, for instance, for the N = 1, D = 4 Wess-Zumino model in which the chirality of N = 1 matter multiplets and their three-point functions restricts the scaling dimensions of the chiral scalar supermultiplets to be canonical. This implies that in the conformal fixed point the coupling constant is zero, i.e. the theory is free [58,59].

Correlation functions in OSp(1|2n)-invariant models
Following the example of the N = 1, D = 3 superconformally invariant model of the previous section, we now proceed to compute correlation functions on hyper superspace for generic OSp(1|2n) invariant models. Again, it is sufficient to require the invariance of the correlation functions under Q-and S-supersymmetry transformations. The invariance under the generalized translations, rotations and conformal transformations will then be guaranteed by the form of the OSp(1|2n) superalgebra. As we will see, the form of the super-correlators will be exactly the same as in the D = 3 case with only difference that the superinvariant intervals (4.17) are now n × n matrices.

Two-point functions
Let us denote the two-point correlation function by The invariance under Q-supersymmetry requires is the interval between two points in hyper-superspace which is invariant under the rigid supersymmetry transformations (2.2). We next impose invariance of the correlator under the S-supersymmetry transformation which is solved by The two point function (5.6) reproduces the correlators of the component bosonic and fermionic hyperfields b(X) and f µ (X) after the expansion of the former in powers of the Grassmann coordinates θ 2 , should vanish. This is indeed the case, as a consequence of the field equations.
To see this, let us recall that in the separated points the two-point function of the bosonic hyperfield of weight 1 2 satisfies the free field equation. Therefore for X 1 αβ = X 2 αβ one has 3 Similarly, for X 1 αβ = X 2 αβ the fermionic two-point function satisfies the free field equation for the fermionic hyperfield. Written in terms of the superfields, these equations are encoded in the superfield equation Expanding the two-point function (det|Z 12 |) − 1 2 in powers of the Grassmann theta-variables one may see that terms in the expansion starting from (θ 2 ) 2 vanish due to the free field equation (5.7). From equations (5.6), (5.9) and from the explicit form of the superfield (2.21), one may immediately reproduce the correlation functions for the component fields [10] b(X 1 )b(X 2 ) = c 2 (det|X 12 Notice also that, contrary to the non-supersymmetric case, where the two-point functions for bosonic and fermionic hyperfields contain an independent normalization constant each, in the supersymmetric case the number of independent constants is reduced to one. The two-point functions on the OSp(1|n) manifold may now be obtained from (5.6) via the re-scaling (3.17), which relates the superfields in flat superspace and on the OSp(1|n) group manifold Φ OSp (X 1 , θ 1 )Φ OSp (X 2 , θ 2 ) = (5.11) (det G(X 1 )) − 1 2 P (Θ 2 1 )(det G(X 2 )) − 1 2 P (Θ 2 2 ) Φ(X 1 , θ 1 )Φ(X 2 , θ 2 ) .

Conclusion
A detailed study of the OSp(1|2n)-invariant generalized superconformal theories is still an interesting open problem, which is important for better understanding the properties of conformally invariant higher spin field theories (see e.g. [60][61][62][63][64][65][66][67][68] for recent progress in studying conformal higher spin fields). Our results are a further step in this direction. Following the program outlined in [1], we have extended the results on the structure of Sp(2n)-invariant field equations to supersymmetric higher spin systems. We constructed generalized superconformal transformations relating the field equation on flat hyper-superspace and on OSp(1|n) supergroup manifolds, which correspond to a generalization of supersymmetric AdS spaces. We computed the two-, three-and four-point functions of real hyper-superfields both on flat and on OSp(1|n) supergroup manifolds and, as a simple illustration of our approach, applied this technique to the example of N = 1, D = 3 superconformal theory of scalar superfields. It is important to further investigate possible interactions (which might be associated with non-trivial three-and four-point correlation functions) in this type of models. A detailed study of the higher-spin content of field equations on higher-dimensional curved hyper superspaces is yet another interesting issue. We hope to address these problems in future publications.