Correlation Functions of Sp(2n) Invariant Higher-Spin Systems

We study the general structure of correlation functions in an Sp(2n)-invariant formulation of systems of an infinite number of higher-spin fields. For n=4,8 and 16 these systems comprise the conformal higher-spin fields in space-time dimensions D=4,6 and 10, respectively, while when n=2, one deals with conventional D=3 conformal field theories of scalars and spinors. We show that for n>2 the Sp(2n) symmetry and current conservation makes the 3-point correlators of two (rank-one or rank-two) conserved currents with a scalar operator be that of free theory.This situation is analogous to the one in conventional conformal field theories, where conservation of higher-spin currents implies that the theories are free.


Introduction
The Sp(2n) invariant description of massless bosonic and fermionic higher-spin fields  is an elegant geometrical approach to study higher-spin gauge theories. The main feature of this approach is that the theory is formulated in an extended space, sometimes called hyperspace, which is parametrized by the n × n matrix valued coordinates X αβ = X βα . These n(n+1) fields depend on the both x-and y-coordinates and obey free field equations [5] which are invariant under the transformations of the Sp(2n) group. The analysis of these field equations for different n [5,13] shows that for n = 4, 8 and 16 they generate the field equations and the Bianchi identities for an infinite set of free conformal higher-spin curvatures in space-times of dimension D = 4, 6 and 10, respectively 4 . The D-dimensional linearized higher-spin curvatures R (s) (x) are the components of a series expansion of hyperfields Φ(x, y) in powers of y, schematically Φ(x, y) = ∞ s=0 R (s) (x)y s . The main hyperfields are a scalar b(X) and a fermion f α (X) (X = (x, y)) transforming under the linear representation of GL(n) ⊂ Sp(2n). We will somewhat loosely call f α (X) the spinor field since it contains half-integer spin curvatures in the corresponding D-dimensional space-time and at y = 0 reduces to a spinor field.
The group Sp(2n) is often referred to as a generalised conformal group, since its structure closely reminds the structure of the conformal group and moreover the Sp(2n) group contains a conformal group as a subgroup. This fact not only makes the Sp(2n) formulation relevant to the study of conformal properties of higher-spin fields but is also useful for better understanding the structure of Sp(2n)-invariant systems themselves, as we shall see below.
A natural question to ask is whether Sp(2n)-invariant higher-spin systems admit interactions. An unsuccessful attempt to obtain interacting Sp(2n)-invariant models was undertaken in [12] in the framework of a generalized supergravity in tensorial superspaces.
Recently, it was shown [24] that in D = 4 models higher-spin current interactions necessarily break Sp(8) group down to the four-dimensional conformal group SU (2,2). However, this result, a priori, does not rule out other types of Sp(2n)interactions, e.g. of some order in higher-spin curvatures. Moreover, there might exist Sp(2n)-theories that do not have any Lagrangian/equations of motion description at all, which is the case for certain conventional conformal field theories.
One way of approaching the interaction problem generically is to use an analogy with D-dimensional conformal field theories. One can make a statement whether a generic D-dimensional conformal theory is free or interacting by looking at the structure of its correlation functions. In this way, for instance, in [25] it was shown that under certain assumptions about the content of D = 3 conformal theory, the presence in the theory of a single conserved higher-spin current implies the existence of an infinite set of conserved higher-spin currents whose correlators with the stress tensor are those of a free conformal theory.
In all the examples of correlators in the Sp(2n)-invariant theories considered so far [11,21,22] generalized conformal weights of the fields in the correlators were not restricted to their canonical values thus, in principle, leaving room for nontrivial interactions. In this paper we will show that for n > 2 already the presence in the Sp(2n)-invariant theories of a conserved generalized stress tensor and/or of a conserved current associated with a rigid internal symmetry (introduced in [6]) makes their correlators with a scalar operator to be those of free theories. To arrive at this result, the correlation functions were constructed solely under the requirement of their Sp(2n) invariance and current conservation properties, without resorting to a specific form of the operators. The three-point functions turn out to have a structure, whose generating functions were found earlier in [26][27][28][29][30] with the use of a different approach.
Therefore, one concludes that for n > 2 the rigid Sp(2n) symmetry, together with the conservation requirements, turn out to be too restrictive for the existence of the nontrivial interactions of higher-spin fields already at the cubic level. This confirms and generalises the result obtained in [24]. In order to allow for nontrivial interactions of these systems, the rigid Sp(2n) invariance should be broken down to an appropriate subgroup.
Let us emphasise that our results apply to the systems with rigid Sp(2n) symmetry. It would be very interesting to construct and study systems which posses a local Sp(2n) invariance, and see if analogous obstructions for interactions apply also in these cases.
The paper is organised as follows. In Section 2 we introduce a set up for the rest of the paper. We review Sp(2n) transformations of the scalar field b(X) and the spinor field f α (X) and their free field equations. We also introduce a generalised current J αβ (X) and a stress tensor T αβγδ (X), and give their Sp(2n) transformations and conservation laws.
In Section 3 we explain the general procedure of constructing the Sp(2n)-invariant correlation functions and derive two-point functions of two currents and two stress tensors using the requirement of Sp(2n) invariance and the current conservation.
In Section 4 we derive three-point functions which include two scalar or spinor fields and one current or stress tensor. The results obtained in this Section are completely analogous to those in ordinary D-dimensional CFTs in the sense that there is no restriction on the generalised conformal dimensions of the scalar and spinor operators.
In Section 5 we derive three-point functions of a scalar operator with two conserved currents or stress tensors. Here the situation turns out to be different from the one in D-dimensional CFTs. Namely, apart from the n = 2, D = 3 case, the values of the conformal dimensions of the scalar operators of the Sp(2n)-invariant systems in these correlation functions are fixed.
In Section 6 we discuss the generic Sp(2n)-invariant structure of three-point correlation functions of higher-rank tensorial fields whose three building blocks are provided by basic two-and three-point correlators of bosonic and fermionic fields and their currents.
In Conclusion we discuss the obtained results and their implications. Some lengthy calculations are given in the Appendices. The basic object in the Sp(2n)-invariant description of integer higher-spin fields is a hyperfield b(X µν ) (see [21] for a recent review). The hyperspace coordinates X µν = X νµ and the field b(X) transform under the Sp(2n) transformations as follows where the parameter a µν = a νµ corresponds to the translations, the parameter k µν = k νµ corresponds to the generalised conformal boosts and the parameter g µ ν is that of GL(n) transformations. The latter can be split into l µ ν = g µ ν − 1 n δ ν µ g ρ ρ which parametrizes the sl(n) subalgebra of gl(n) and its trace g µ µ which corresponds to the dilatations. Together these transformations generate an Sp(2n) group which contains D = n 2 + 2 dimensional conformal group as its subgroup 5 . The parameters a µν , l µ ν and k µν contain conventional translations, Lorentz transformations and conformal boosts respectively, whereas the parameter of dilatation is proportional to the trace g µ µ .
The constant ∆ which is present in the equation (2.2) is a generalized conformal weight or the Sp(2n) weight of the scalar hyperfield. It is related to the conventional conformal weight of the scalar fields in the corresponding space-time dimensions D = n 2 + 2 as follows [22] This is because the D-dimensional dilatation parameter g D is g D = 2 n g µ µ The free field b(X) satisfies the field equations These equations are invariant under the Sp(2n) transformations (2.2) provided that ∆ = 1 2 , which is therefore a canonical dimension of b(X) [4].

Spinor field f α (X)
The half-integer spin fields are packed in a Grassmann-odd "spinor" hyperfield f α (X) transforming under the linear representation of GL(n). Under Sp(2n) it transforms 5 The relation D = n 2 + 2 between the space-time dimension D and n is valid only for n = 2, 4, 8 and 16. This is related to the fact that in these dimensions the massless momentum condition P m P m = 0 has the general twistor-like solution P m =λγ m λ, where commuting spinors λ α are transformed under a fundamental representation of the D-dimensional conformal group whose subgroup is Sp(n). as follows f α (X) satisfies the free equations of motion where [αβ] denotes anti-symmetrization of indices, while (αβ) will indicate symmetrization. These equations are Sp(2n) invariant provided that the field f α (X) has the generalized conformal dimension Let us assume that the fields b A (x) and f A α (X) belong to a vector representation of the O(N) group (A = 1, . . . , N) and define a generalised current (2.8) The current is symmetric with respect to its indices (α, β) and anti-symmetric in A, B. Using (2.1), (2.2) and (2.5), one can show that the Sp(2n) transformations of the current are As it can be seen form (2.10) (and (2.11)), the canonical conformal weight of the current is where the subscript of ∆ 1 labels the "spin" s = 1 of J αβ (see also (2.7)). In general, the "spin" of a symmetric tensor T α 1 ...αr of rank r is defined as s = r 2 . This is a natural extension of the notion of spin of fields in D = 3, 4 described by spin-tensors.
One can show that the current (2.8) satisfies the generalized conservation conditions introduced in [6] provided that the fields b(X) and f α (X) satisfy the free field equations (2.4) and (2.5), respectively. On the right in (2.13) we indicated the Young symmetry in the indices (µναβ) of the left hand side which is annihilated by the conservation condition. Note that, in the general case, even if the current J αβ is not composed of the matter fields, but satisfies the conservation law (2.13), the Sp(2n) invariance of the latter requires that its generalized conformal dimension is always canonical (2.12), which is in accord with the usual CFTs.

Stress tensor T αβγδ (X)
Similarly one can define a free generalised stress (or energy-momentum) tensor [6] T αβγδ =T (αβ,γδ) , (2.14) withT or for the fermionic field f αT αβ,γδ = f γ ∂ αβ f δ . One can check that the totally symmetric tensor T αβγδ transforms covariantly under the Sp(2n) transformations (2.2) as follows From the form of the above transformations we see that the canonical conformal dimension of T αβ,γδ is One can also check that the stress tensor (2.14) satisfies generalized conservation conditions [6] provided that the fields b(X) and f α (X) satisfy the free field equations (2.4) and (2.5), respectively. Again, we indicated on the right hand side of (2.19) the relevant symmetry of ∂T that is set to zero by the Sp(2n) conservation condition.
As in the case of the conserved current J αβ (X), the Sp(2n) invariance of the conservation law (2.19) always requires that the conformal dimension of T αβµν (X) is canonical (2.18).
By analogy with J αβ and T αβγδ one can introduce higher-spin conserved currents T α 1 ...α 2s (2s = 1, 2, 3, . . .) [6] which transform under Sp(2n) as follows The Sp(2n) conservation condition [6] sets to zero the most anti-symmetric component of ∂T : We will see that when applied to correlation functions these conservation conditions restrict the structure of the former to those of free theories. Heuristically, this happens because the Sp(2n) conservation leads to an over-determined system of equations, i.e. the number of equations generally exceeds the number of the components of T , as it can be seen from the Young diagram above. On the contrary, in the usual CFTs the conservation condition has fewer components than the current J m 1 ...ms . For s = 1, 2 the conservation is not too restrictive, while for s > 2 it requires an advanced machinery of Ward identities [25] or higher-spin algebras [32], see also [33][34][35], to see that the presence of at least one higher-spin current implies the presence of full infinite-dimensional higher-spin symmetry of the model and makes it free. Now we are in a position to consider various Sp(2n) invariant two-, three-, and four-point functions. As we mentioned in the Introduction, when computing the correlation functions we will, a priori, assume that the fields b(x) and f α (X) may have arbitrary Sp(2n) weights ∆ 0 = ∆ and ∆1 2 = ∆ + 1 n , respectively, with anomalous (spin-independent) dimensions ∆ and then see how the Sp(2n) invariance and conservation laws restrict their values.

The correlation function is invariant under the Sp(2n) transformations (2.1) if for any values of the Sp(2n) parameters the following equation holds
where the variations δX µν i are given in (2.1). The equation (3.2) plays an important role in the further analysis, since it determines a general condition for a given multipoint function to be Sp(2n)-invariant.  (1) and ∆ (2) . The condition that this two-point function is Sp(2n)-invariant under the transformations (2.2) i.e., satisfies (3.2) implies that it is nonzero if ∆ (1) = ∆ (2) = ∆, and has the following form [11] b where C bb is an arbitrary constant, X αβ Analogously, for the two fermionic fields f A α (X 1 ) and f B β (X 2 ) one gets [11] f The correlator of b(X 1 ) with f α (X 2 ) is zero, as are all the multi-point functions with an odd number of f α (X), since their index structure is not even GL(n) invariant.
For further use let us also give the form of the Sp(2n) variations (2.1) of the matrix ( Again the variations δX µν 1 and δX µν 2 are given in (2.1). From eq. (3.6) it follows that the matrix (X −1 12 ) αβ , regarded as a bi-local tensor, is Sp(2n)-invariant, i.e.
−(δX µν With the help of relation (3.7) one can immediately check Sp(2n) invariance of correlation functions away from the singularity point. Notice that the conformal boosts act effectively on the first index of (X −1 12 ) αβ with the matrix X 1 and on the second index with the matrix X 2 (or vice verse). This ensures the same Sp(2n)invariant properties of the left-and right-hand side of (3.5) in accordance with the generic formula (3.2). The matrix (X −1 12 ) αβ , together with its determinant, is thus one of the elementary building blocks of all the Sp(2n)-invariant correlation functions.

Two current correlator JJ
The two-point function of the conserved currents J AB αβ can be derived by writing the most general expression compatible with its index symmetries, requiring its invariance under O(N) and Sp(2n), and imposing the current conservation condition. The Sp(2n) invariance condition (3.2) reduces a general expression to the one constructed in terms of the matrix X −1 12 and its determinant as follows where we defined and
The constant C JJ is arbitrary, while the power of the determinant is fixed to be −1, i.e. its absolute value is equal to the canonical value ∆ = 1 of the spin independent part of the conformal weight of J αβ . One can easily check that (3.8) obeys the current conservation law (2.13) 6 .
The Sp(2n) variations of (3.9) are determined by those of X −1 12 , given in (3.7), and have the following form It is important for the proof of the Sp(2n) invariance of (3.8) that in the right hand side of (3.12) the first pair of the indices of (P 12 ) αβ,γδ gets rotated with the matrix k ασ X σδ 1 and the second pair gets rotated with k µσ X σδ 2 . The bi-local tensor (3.9) is a building block of correlation functions of higher even-rank tensors such as the stress tensor.

Stress tensor correlator T T
Following the same reasoning as above, one gets the Sp(2n)-invariant two-point function of the two stress tensors where the total symmetrization of the both sets of indices (αβγδ) and (µνρσ) is assumed. The Sp(2n) invariance (3.2) and the conservation properties of (3.13) dictated by (2.19) can be checked with the use of the form of the variations (3.11), (3.12) and of det |X 12 |.

.1 Scalars and fermions
A three-point function for three fields b(X) with weights ∆ (1) , ∆ (2) and ∆ (3) has the following form [11] b( where (4.2) Correspondingly, the correlator of two fermions and a scalar is [11] Let us mention that in a similar way one can find four-point functions which include four scalars, four fermions and two scalars and two fermions. As in the ordinary CFTs the Sp(2n) symmetry fixes the four-point correlators up to an arbitrary function of cross-ratios [21].

4.2
Three-point functions Jbb , T bb , Jf f and T f f One more tensor structure appears in the Sp(2n)-invariant three-point correlators which involve two scalars of dimensions ∆ (1) and ∆ (2) , and one conserved current or stress-tensor of the spin-independent conformal dimension ∆ where constants k i again satisfy the conditions (4.2). In the above expressions where a, b, c = 1, 2, 3 and a = b = c.
The quantities (Q c ab ) αβ are manifestly invariant under the translation of the coordinates and transform in the following way under the GL(n) rotations and the generalized boosts For the Sp(2n) invariance (3.2) of the correlators (4.4) and (4.5) it is important to notice that the contraction of the indices of Q 3 12 on the right hand side of the boost transformations (4.8) only involves the coordinate X 3 .
Using (A.1) and (A.8) one can check that (4.4) and (4.5) satisfy the current and stress-tensor conservation laws if and k 3 is arbitrary. This means that the generalized dimensions of the fields b(X 1 ) and b(X 2 ) are equal to each other but otherwise unrestricted (i.e. can be anomalous). Correspondingly, for the correlators of two fermions of dimension ∆1 2 = ∆ + 1 n with the current J and the stress tensor T we have Looking at the form of the two-and three-point correlation functions constructed above we come to the conclusion that except for some degenerate cases 7 the most general multi-point function can be written as a sum over all possible polynomials of a required rank in three structures p ab = X −1 ab (3.10), P ab (3.9) 8 and Q c ab (4.6) times a pre-factor which in the case of four-point and higher order correlators is a function of Sp(2n)-invariant cross-ratios: This statement is completely analogous to the one for the usual CFTs (see [37] for a proof). We will discuss the general structure of the three-point correlators in more detail in Section 6.

Three-point functions with fixed conformal dimension
We will now consider Sp(2n)-invariant three-point functions in which the requirement of the current and stress-tensor conservation completely fixes the conformal dimension of the scalar operator to be 1, i.e. twice that of the canonical dimension of the free scalar hyperfield. At this point the properties of correlators in Sp(2n)-invariant systems (for n > 2) become different from those of conventional D-dimensional CFTs where in the analogous correlators a restriction on the dimension of the scalar operator does not occur. The obtained restriction suggests that the Sp(2n)-invariant systems under consideration are free, as we will discuss in more detail below.

JJO three-point functions
The simplest three-point function of this kind is JJO , where O(X) is a scalar operator of dimension ∆ which, in general, can be a composite of the elementary fields b(X). From the requirement of Sp(2n) invariance one finds that the correlator has the following form Degenerate cases correspond to the situations in which additional invariants can be built with the help of ǫ-symbols, as it happens in 3d [36]. 8 P ab is actually a square of p ab = X −1 ab , but it is convenient to regard it as an independent structure in order to separate contributions from bosons and fermions (see Section 6 for more details).
where A and B are some yet undetermined constants. Now let us impose the current conservation condition (2.13) on the three-point function (5.1). Requiring, for example, the conservation of the current J µν (X 1 ) one gets (see the Appendix B for details) and similarly for the conservation of the current J αβ (X 3 ). We thus conclude that for the three-point function (5.1) to be non-zero the dimension of the scalar operator O must be equal to one. There at least two interpretations of this result. and we will soon find more examples of the same kind. Firstly, one can start from a free theory (2.4) with a fundamental field b(X) and see if any interactions are possible. Here we do not assume that interactions admit any realization in terms of either Lagrangian or equations of motion. Then the above constraint implies that JJ[b k ] = 0 for k = 1, 3, 4, ... in the interacting theory, where [b k ] is a quasi-primary field built of k fields b(X). The only non-zero correlator corresponds to k = 2 and does not allow for any anomalous dimension. The property of allowing for anomalous dimensions is the most important property of any CFT. Therefore, we interpret our result as the fact that there are no nontrivial interactions possible for the field b(X). Secondly, one can start with any Sp(2n) invariant CFT and assuming that it has a scalar operator O(X) and a conserved current This situation is different from the one in D ≥ 3 CFTs (see e.g. [37][38][39]) where an analogous three-point function does not impose any condition on the dimensions of the scalar fields. Using the same ansatz as above with the understanding that each pair of the indices (αβ) should be replaced by one vector index of the Lorentz group SO(1, D − 1) one gets the condition which relates the three parameters but does not impose any restriction on the conformal weight ∆. The reason for this difference is that in the Sp(2n)-invariant systems in the hyperspace with extra coordinates the conservation conditions are more restrictive than in ordinary CFTs. The situation does not change even if we consider weaker conservation conditions, e.g. only an Sp(n)-invariant part of (2.13) and (2.19) obtained by contracting the latter with the symplectic metrics C αν C βµ . The exception is the case of n = 2, D = 3 in which the Sp(4) ∼ SO(2, 3) symmetry simply coincides with the three-dimensional conformal symmetry. In this case the condition (5.2) does not arise due to extra 2 × 2 matrix identities which are present in D = 3 as it is shown in Appendix B.
We can also consider a correlator which contains two currents and an antisymmetric tensor operator O [αβ] . This operator can be constructed e.g. by taking the product of two fermionic fields f α (X 2 )f β (X 2 ). Such a correlator has the following structure Again the requirement of current conservation fixes the conformal dimension of O [γ 1 γ 2 ] to be ∆ = 1.

T JO and T T O correlators
Other examples of correlation functions in which the conformal weights of the operators are completely fixed by the conservation laws are T JO , and T T O . The Sp(2n) invariance restricts the correlation function T JO to have the following form where the total symmetrization of the indices denoted by the same letter is implied. The conservation of the stress-tensor ∂ 1 (T α(4) ) and the current ∂ 2 (J β(2) ) require Therefore we have found again that the structure of the three-point function and the conformal dimension of the scalar field are fixed by the conservation laws. The same happens with the Sp(2n)-invariant correlator T T O which has the form ) ββ (P 12 ) αα,ββ ] +C (P 12 ) αα,ββ (P 12 ) αα,ββ .

Generic structure of the three-point correlation functions of symmetric tensor operators
The form of the Sp(2n)-invariant correlator of three currents J i αβ , where i is an internal group index (e.g. i stands for AB in the O(N) case) manifests a general structure of the three-point functions which include three tensor structures which we called p ab , P ab and Q c ab (given in (3.10), (3.9) and (4.6), respectively) where f ikj are structure constants of the internal group. As in the conventional CFTs, the first two structures entering (6.1) with the parameters A and B are associated with the currents constructed with the bosonic fields, while the third structure is associated with the form of the fermionic currents (2.8).
The current conservation leaves the parameter C arbitrary. It fixes the parameters k a and relates the parameters of the bosonic structures which is consistent with the fact that the canonical spin-independent part of conformal dimension of the current is ∆ = 1. The above correlator has the same structure as in the ordinary CFTs, expect for the 3D case where an extra odd structure exists [36].
We are now in a position to discuss the general structure of the three-point correlators of conserved currents which are symmetric tensors of a rank r = 2s with s being an integer 'spin'. To this end it is convenient to hide the tensor indices away by contracting them with auxiliary variables λ α a , where a refers to the point of operator insertion: no summation over a . (6.5) For instance the correlator of two scalar operators O of the same dimension ∆ with a conserved current of an integer spin s obeying (2.24) is ) s (6.6) Imposing the current conservation condition leads to the same result as for the currents with s = 1, 2, i.e. k 1 = k 2 = 1 However, if we consider a three-point function of a scalar operator and two conserved currents J s = J α 1 ...α 2s λ α 1 · · · λ α 2s of ranks 2s 1 and 2s 2 with s ≥ 1 we will again find that, up to an overall factor, all the free parameters in the correlator are fixed. For example, is the unique solution of the Sp(2n) conservation conditions (2.24).
Summarizing, we have found that the restrictions imposed by the Sp(2n) conservation laws on the correlation functions fix their structure up to an overall factor and the correlators are those of free Sp(2n) invariant CFT. We studied thoroughly the correlators which include generalised currents and stress-tensor. We can make further progress by computing correlators involving higher-rank tensors. However the study of examples with higher-spin currents do not lead to any new conclusions comparing to the the study of the correlation functions involving a rigid symmetry current J 1 and the stress tensor T 2 . In particular, the correlators with higher-spin currents also look like those in a free Sp(2n) invariant CFT. This suggests that all other correlators with higher-spin currents follow the same pattern. The generating function of correlators in free theories were already obtained [26][27][28][29][30]. For example, a generating function of the three-point functions of currents built out of free scalars b(X) is JJJ = cos(p 12 ) cos(p 13 ) cos(p 23 ) exp 1 2 [Q 1 23 + Q 2 13 + Q 3 12 ] (det |X 12 | det |X 23 | det |X 13 |) 1/2 . (6.8) It contains operators J s , s = 0, 1, 2, ... and the correlator J s 1 J s 2 J s 3 is obtained as the coefficient in front of (λ 1 ) 2s 1 (λ 2 ) 2s 2 (λ 3 ) 2s 3 . The generating function obtained from the currents built out of free fermions f α (X) is The generating function of multi-point correlators can be found in [28][29][30].
The above expressions deal with the bosonic symmetric tensor currents of even rank. The generating function which produces 3-point correlators involving two fermionic currents of odd ranks is similar, see e.g. [25].

Conclusion
We have studied restrictions imposed by the generalized conformal group Sp(2n) and the conservation laws on various correlation functions involving conserved currents, stress-tensor and higher-spin currents. The general structure of the correlators is similar to the one in the usual conformal field theories. It is build of three conformallyinvariant tensor structures, which were found by working out the simplest two-and three-point functions.
The difference between Sp(2n)-and SO(2, D)-invariant correlation functions arises when conserved tensor currents are involved. If we assume that the theory has a conserved stress-tensor T and, possibly, a conserved current J, then computation shows that while the simplest correlators T OO and JOO allow operators O to have an anomalous dimension, their correlators with two conserved tensors, T T O , JJO and T JO , fix the conformal dimension of O to be twice that of a free field. This is not the case in the SO(2, D) CFTs in which similar restrictions arise only if they contain in addition to the stress tensor a conserved higher-spin current [31][32][33][34][35].
The fact that the above restriction on the conformal weight of O implies that the theory is free can also be understood as follows. If a free theory contains, e.g. elementary scalar fields b(X) of the canonical dimension 1/2 then we can take O = bb. In the Sp(2n)-invariant free theory we expect that T T O = 0, which is already true for a free field. If a free Sp(2n)-invariant model could be deformed by some interactions with a coupling constant g, quantum corrections would make fields to acquire anomalous dimensions ∆(g). If so, then as we have seen, the conservation of T requires that the correlator T T O ∆(g) = 0. Such a theory would not have a smooth free limit g → 0, since at g = 0 the correlator is non-zero.
Therefore, for n > 2 the rigid Sp(2n) conservation condition turns out to be much more restrictive than the SO(2, D) one. Only in the n = 2, D = 3 case the Sp(4) ∼ SO(2, 3) becomes the usual three-dimensional conformal symmetry and the conservation condition reduces to the usual current conservation which does not restrict conformal dimension of operators in the correlators with spin-one and spintwo currents.
Assuming the presence in the Sp(2n)-invariant theory, containing a stress-tensor, of at least one higher-spin current one can easily repeat the proof given in [31][32][33][34][35] and conclude that there should be infinitely many (symmetric) higher-spin currents associated with a unique higher-spin algebra that is generated by the corresponding charges. In this sense our work shows that in the Sp(2n) setup with n > 2 spin-one and spin-two currents already behave like higher-spin currents, forcing the theory to be a free one. If we do not assume the existence of the conserved stress-tensor, like in gravity theories, then the above reasoning does not apply. However, the possibility of introducing 'hypergravity' interactions of the Sp(2n)-invariant systems is an open problem itself.
Another option that may still lead to interacting Sp(2n)-theories without contradicting our results is the existence of certain contact terms in the Sp(2n) correlators, i.e. the terms that have δ-like singularity when points collide.
Therefore, the main conclusion is that the generalized conformal field theories with Sp(2n) symmetry only admit a free field realization, with the few loopholes mentioned above. Still the Sp(2n)-invariant formulation can be useful for the study of free CFTs. In particular, one can derive the generating functions for all the correlation functions [26][27][28][29][30] and work out the operator algebra [29]. In order to allow for nontrivial interactions the Sp(2n) symmetry should be broken, as happens for the current interactions [24].
In this paper we have mainly studied the correlation functions of scalar operators with conserved currents that are totally-symmetric tensors of even rank from the Sp(2n) point of view. As we have mentioned, in the theory with fermions f α one can find operators with more complicated types of symmetry. It should be possible to generalize the classification of the Sp(2n)-invariant correlators to the case of fields with mixed-symmetry as well as to consider the hyperfields which are p-forms in hyperspace (see [35] for a discussion of p-forms in SO(2, D) CFTs).
Another interesting application of the generalized conformal Sp(2n) symmetry is the study of conformal higher-spin fields on Sp(n) group manifolds (see [7,8,10,21,22] for details), which is a generalisation of conformal higher-spin theories on AdS D backgrounds (see for example [40][41][42]). For instance, the infinite sets of bosonic and fermionic symmetric higher-spin fields in AdS 4 are packed into a scalar and a spinor field which propagate on a 10-dimensional group manifold Sp(4) and enjoy Sp(8) invariance. As was shown in [21,22] the correlation functions of the fields on Sp(4) can be obtained from the flat hyperspace ones by performing a certain GL(4) transformation and rescaling of the latter. So the results of this paper are directly generalized to Sp(2n)-invariant systems on the Sp(n) group manifolds.
One more application of the free Sp(2n)-invariant systems is to compute partition functions of free higher-spin theories along the lines of [43,44] (and references therein), an advantage being that Sp(2n)-fields encode the infinite multiplets of higher-spin fields and therefore should evaluate the sum over the spins automatically.

A Properties of P and Q tensor structures
The derivatives of the matrix valued coordinates and corresponding determinants have the form ∂ ∂X µν X αβ = These relations can be used to derive useful properties of the tensors (P ab ) αβ,µν and (Q c ab ) αβ , in particular

B.2 Three Dimensions
In this subsection we shall explicitly show that in the case n = 2, D = 3 win which the generalised conformal group Sp(4) coincides with the three-dimensional conformal group, the current conservation condition in the three point function with two currents and one scalar does not impose any restriction on the scaling dimension of the later, thus reproducing a known result from D = 3 CFT (see for example [36]).
In D = 3 the coordinates X αβ = x m γ αβ m (m = 0, 1, 2) are 2×2 symmetric matrices parametrising the D = 3 space-time. Taking the generalized conformal dimensions of the current to be equal to 1 we obtain from (4.2) that k 2 = 1 − k 1 . The conventional conservation law is obtained by contracting expressions (B.9)-(B.13) with ǫ αν ǫ βµ , where the convention for the spinorial metric is ǫμνǫνρ = −δμ ρ . Doing so one gets For the expression (B.14) to be zero for an arbitrary k 1 one should prove that To this end note that in vew of the definitions (B.1) the following relation holds Note also that for the 2 × 2 symmetric matrices X µν X 2 = 2 det X = 2 det X −1 .

Now using that
we can rewrite [X(X − Y )X] nm as follows We thus find that in (B.15) a = 2. Hence, from (B.14) it follows that A(1 − k 1 ) − Bk 1 = 0, which means that in this case there is no restriction on the parameter k 1 and hence on the conformal dimension of the scalar field.