Lagrangian description of the partially massless higher spin N=1 supermultiplets in $AdS_4$ space

In the recent paper [1] the classification of non-unitary representations of the three dimensional superconformal group has been constructed. From AdS/CFT-correspondence they have to correspond to N=1 supermultiplets containing partially massless fields in $AdS_4$. Moreover, the simplest example of such supermultiplets which contains a partially massless spin-2 was explicitly constructed. In this paper we extend this result and develop explicit Lagrangian construction of general N=1 supermultiplets containing partially massless fields with arbitrary superspin. We use the frame-like gauge invariant description of partially massless higher spin bosonic and fermionic fields. For the two types of the supermultiplets (with integer and half-integer superspins) each one containing two partially massless bosonic and two partially massless fermionic fields we derive the supertransformations leaving the sum of four their free Lagrangians invariant such that the $AdS_4$ superalgebra is closed on-shell.


Introduction
In the recent paper [1] the classification of non-unitary representations of the three dimensional superconformal group has been constructed. From AdS/CF T -correspondence they have to correspond to N = 1 supermultiplets containing partially massless fields in AdS 4 . Moreover, the simplest example of such supermultiplets which contains a partially massless spin-2, massless spin-1, massless spin-3/2 and massive spin-3/2 was explicitly constructed. The dynamical description of the arbitrary supermultiplets was not studied. In this paper we fill this gap and construct explicit Lagrangian realization of all N = 1 supermultiplets containing partially massless fields with arbitrary integer and half-integer superspins.
The partially massless fields [2][3][4][5][6] (non-unitary in AdS) of integer s or half-integer s+1/2 spins are labelled by depth t ∈ {0, 1, 2, ..., (s − 1), s}. Two boundary values t = 0 and t = s correspond to massless and massive cases respectively. For other values of t we have pure partially massless field which propagates 2(t + 1) degrees of freedom. As it was shown in [1] the general partially massless N = 1 supermultiplets are described by the diagrams For the description of the individual partially massless higher spin bosonic and fermionic fields we use the frame-like gauge invariant description similar to the massive case [7][8][9]. In such formalism partially massless spin s(s + 1/2) of depth t is described by a set of massless fields with spins s, s − 1, ..., s − t combined together into one system. To combine partially massless fields into supermultiplets (1.1) we follow the strategy of our recent paper [9] where massive higher spin supermultiplets were constructed. For the Lagrangian we just take the sum of four free Lagrangians for the two partially massless bosonic and two partially massless fermionic fields entering the supermultiplet. Then for each pair of bosonic and fermionic fields (we call it superblock in what follows) we find the supertransformations leaving the sum of their two Lagrangians invariant. After that we combine all four possible superblocks and adjust their parameters so that the algebra of the supertransformations be closed on-shell. The paper is organized as follows. In section 2 we give non-unitary frame-like gauge invariant formulation for free partially massless arbitrary integer and half-inter spins. In section 3 we consider superblocks containing one partially massless bosonic and one partially massless fermionic fields and find corresponding supertransformations. In section 4 we combine the constructed partially massless superblocks into the partially massless supermultiplets. Notations and conventions. We use the same frame-like multispinor formalism as in [9], where all the Lorentz objects have local totally symmetric dotted and undotted spinorial indices. AdS 4 space is described by pair one-forms: background frame e αα that enters explicitly in all constructions and background Lorentz connections ω αβ , ωαβ that are hidden in the one-form covariant derivative. Basis elements of 1, 2, 3, 4-form spaces are: They are defined as follows: The covariant derivative satisfies the following normalization conditions:

Partially massless higher spin fields
In this section we provide frame-like gauge invariant formulation for (non-unitary) partially massless fields with arbitrary integer and half-inter spins in AdS 4 space.

Partially massless bosons
The gauge invariant formulation for the partially massless fields can be easily obtained from the general massive case just by adjusting the value of mass parameter. But unitarity requires that the sign of the cosmological term be positive so that naturally the partially massless fields live in de Sitter space. In this work we use the gauge invariant formulation for the partially massless fields in AdS 4 space where half a number of components have wrong signs of the kinetic terms. Such a description is explicitly non-unitary but the Lagrangian is hermitian and all coefficients are real. In such approach a partially massless integer spin-s field of depth t = (s − l − 1) is formulated in terms of massless fields with spins (l + 1) ≤ k ≤ s. Each massless bosonic fields with spin k ≥ 2 is described by the physical one-form f α(k−1)α(k−1) and the auxiliary one-forms Ω α(k)α(k−2) , Ω α(k−2)α(k) . They are two-component multispinors symmetric on its local dotted and undotted spinorial indices separately. These fields satisfy the following reality condition In these notations the gauge invariant Lagrangian for the partially massless bosonic field can be written as follows: This Lagrangian is invariant under the following gauge transformations: In what follows we assume that all parameters a k are positive. The parameter σ in Lagrangian (2.2) determines the common sign of the Lagrangian that will be important for the construction of the supermultiplets.
In the gauge invariant formalism we use, for each field (physical or auxiliary) there exist a corresponding gauge invariant object ("torsion" or "curvature"). Their form is completely determined by the structure of the gauge transformations (2.3) 1 : In this work we use a formalism analogous to the so-called 1 and 1/2 order formalism, very well known in supergravity. Namely, we do not introduce any supertransformations for the auxiliary fields, instead all calculations are done using the "zero torsion conditions": As for the supertransformations for the physical fields, the variation of the Lagrangian can be compactly written using the gauge invariant curvatures given above:

Partially massless fermions
To construct a gauge invariant Lagrangian for the fermionic fields one only needs physical fields. So to describe partially massless spin s+1/2 field of depth t = (s − l − 1) we introduce a set of one-forms Φ α(k)α(k−1) , Φ α(k−1)α(k) , l + 1 ≤ k ≤ s which are symmetric on their dotted and undotted spinorial indices separately and satisfying reality condition The Lagrangian for the partially massless fields in AdS 4 has the form As in the bosonic case half a number of components have wrong signs of the kinetic terms. Such a description is explicitly non-unitary but the Lagrangian is hermitian and all coefficients are real. In what follows we assume that the parameters c k are positive while τ (even/odd) in Lagrangian (2.8) parametrizes the common sign of the Lagrangian.
This Lagrangian is invariant under the following gauge transformation: In the fermionic case for each field we also have a corresponding gauge invariant object (as in the bosonic case we omit any extra fields): Using these curvatures, the variation of the Lagrangian (2.8) under the supertransformations can be compactly written as follows: (2.12)

Partially massless superblocks
As it has been shown in [1], there are two types of the partially massless supermultiplets which (similarly to the massive case) contains two bosonic and two fermionic partially massless fields with the properly adjusted depths (see below): In this work we follow the same strategy we used for the construction of massive supermultiplets [9]. At first, we consider two possible pairs of the bosonic and fermionic fields (superblocks), namely (s, s + 1/2) and (s − 1/2, s), and find the supertransformations which leave the sum of their free Lagrangians invariant. Then we consider the whole system of four fields and choose the parameters in such a way that the algebra of the supertransformations is closed.

Ansatz for the supertransformations
We choose the following ansatz for the supertransformations for a pair of the partially massless bosonic and fermionic fields: where all coefficients are complex. Note that this ansatz is valid for both types of the superblocks which differ only by the boundary conditions. In the gauge invariant formulation that we use it is easy to see that not only spins but the depths of the superpartners must be related. Indeed, let us consider partially massless bosonic field with maximal helicity s and minimal one l. Then there are only four possible superpartners, namely, partially massless fermions with maximal helicities s ± 1/2 and minimal ones l ± 1/2: Now let us consider a sum of the bosonic (2.7) and fermionic (2.12) variations under the supertransformations (3.1). Using the torsion zero conditions (2.6), it can be written in the Schematically, the structure of the variations has the form "curvature × field". The fact, that both the Lagrangians and their variations are defined only up to a total derivative, leads to a number of non-trivial identities on such terms. The general form of these identities were given in Appendix A of [9] and they are applicable to the case at hands, the difference being in the explicit expressions for the coefficients a k , b k , c k and d k . Using these identities one can express the parameters α and γ in terms of β: Also we obtain recurrent relations on the parameters β k and β ′ k : Last but not least, we obtain four independent equations which relate β and β ′ as well as the bosonic and fermionic depth parameters: In the next two subsections we find the solutions of these equations for the two possible types of the partially massless superblocks.

Solution for the superblock (s + 1/2, s)
Let us consider a superblock containing a partially massless boson with spin s and depth t = (s − l − 1) and a partially massless fermion with spin s + 1/2 and deptht = (s −l − 1.
The explicit expressions for the bosonic coefficients have the form: while the fermionic ones look like: Recall that the parameters α k and γ k are determined in terms of β by (3.2)  Remind that a real (imaginary) values of β s−1 corresponds to a parity-even (parity-odd) bosonic field. Now we proceed with the solution of all remaining equations and obtain, for l = l and forl = l − 1 where β = ρ (β = iρ) takes pure real (imaginary) value. Schematically these four solutions can be presented as where ± signs for the bosons correspond to their parity, while for the fermions to the sign of d k .
3.3 Solution for the superblock (s, s − 1/2) Now let us turn to the second superblock which contain a partially massless boson with spin s and depth t = (s − l − 1) and a partially massless fermion with spin s − 1/2 and deptht = (s −l − 2). Thus for the bosonic field we still have the same expressions for the coefficients a k and b k as in the previous subsection, while for the fermion we obtain First of all, let us consider equation ( and forl = l − 1 where again β = ρ (β = iρ) takes pure real (imaginary) value. Schematically these four solutions can be presented as

Partially massless supermultiplets
As we have already mentioned, each partially massless supermultiplets contains two bosonic and two fermionic fields. As in the massive case, the two bosons must have opposite parities, and it turns out to be important that the two fermions have opposite signs of their mass terms. Moreover, the depths of the partial masslessness for each field must be properly adjusted. Schematically, the two possible types of such supermultiplets look like: s s s s s s s s s s s s s s s s s s s Let us use the notations (f + , Ω + ) and (f − , Ω − ) for the parity-even/parity-odd bosons and Φ + , Φ − for fermions according to their sign of d k . In these notations the supertransformations for the whole supermultiplets are the combination of four separate superblocks corresponding to the parameters ρ 1,2,3,4 shown above. Namely, we take for the bosons: and similarly for δf with the replacement ρ 1 → ρ 3 and ρ 2 → ρ 4 , while for the fermions we use and similarly for δΦ with the replacement ρ 1 → ρ 2 and ρ 3 → ρ 4 . So to construct a complete partially massless supermultiplet we have to adjust these four parameters ρ 1,2,3,4 so that the algebra of supertransformations be closed. It means that the commutator of the two supertransformations must produce a combination of translations and Lorentz transformations: The structure of the mass-shell condition (2.6) shows that, for example, the commutator on the bosonic field f + α(k−1)α(k−1) must only contain Ω + α(k)α(k−2) , Ω + α(k−2)α(k) , f α(k)α(k) + , f + α(k−1)α(k−1) and f + α(k−2)α(k−2) . This requirement leads to the number of relations on the parameters: If these relations are fulfilled the general form of the commutator looks like: For the bosonic field f − α(k−1)α(k−1) the commutator has the same form with the replacements ρ 1 → ρ 3 and ρ 2 → ρ 4 . Let us stress that the coefficients in this expression must be kindependent. This gives additional restrictions on the parameters and also serves as quite a non-trivial test for our calculations.

Supermultiplets with half-integer superspin
The partially massless supermultiplet with the half-integer superspin Y = (s − 1/2) has the following structure: s s s s s s s s s s We have only a handful of free parameters in our disposal and a large number of equations to fulfill, however, the closure of the superalgebra is achieved at: σ 2 = σ 1 , τ 2 = τ 1 + 1, ρ 1 2 = ρ 2 2 = ρ 3 2 = ρ 4 2 , ρ 1 ρ 3 = ρ 2 ρ 4 (4.2) Note that such relations between σ, τ parameters mean that two bosons and two fermions must enter with opposite norms of kinetic terms. This is in agreement with [1] where the same result was obtained for the case of the supermultiplet with partially massless spin-2 field.
The final form for the commutators of the supertransformations on parity-even spin-s f + and parity-odd spin-(s − 1) f − fields appears to be the same:

Summary
In this paper we have presented the component Lagrangian description of partially massless higher spin on-shell arbitrary N = 1 supermultiplets in AdS 4 space corresponding the classification given in [1]. The constructed supermultiplets are non-unitary and contain partially massless fields with appropriately chosen spins and depths. As in a massive case [9] we show that N = 1 partially massless supermultiplets can be constructed as a combination of four partially massless superblocks containing one partially massless boson and one partially massless fermion. As a result we have derived both the supertransformations for the components of the supermultiplet and the corresponding invariant Lagrangian. Also we show that a closure of superalgebra requires that two bosons and two fermions must enter supermultiplets with opposite norm of kinetic terms. All our results are in agreement with the results of [1] and extend them. The constructed Lagrangian formulation describes a dynamics of arbitrary superspin partially massless supermultiplets in AdS 4 space.