Supersymmetric higher spin models in three dimensional spaces

We review the component Lagrangian construction of the supersymmetric higher spin models in three dimensional (3D) Minkowski and anti de Sitter (AdS) spaces. The approach is based on the frame-like gauge invariant formulation, where massive higher spin fields are realized through a system of massless ones. We develop a supersymmetric generalization of this formulation to the Lagrangian construction of the on-shell N=1, 3D higher spin supermultiplets. In 3D Minkowski space we show that the massive supermultiplets can be constructed from one extended massless supermultiplet by adding the mass terms to the Lagrangian and the corresponding corrections to the supertransformations of the fermionic fields. In 3D AdS space we construct massive supermultiplets using a formulation of the massive fields in terms of the set of gauge invariant objects (curvatures) in the process of their consistent supersymmetric deformation.


Introduction
As soon as the supersymmetry (see e.g. the books [1], [2], [3]) has been discovered it was aroused an immediate interest to finding the supersymmetric generalization of the known theories. In short time there appeared the supersymmetric extensions of such famous theories as the Standard Model, the Einstein's gravity, string theory. Construction of supersymmetric models and the study their properties on classical and quantum levels became one of the most attractive trends in the modern theoretical physics.
In the last decades there was an essential progress in the higher spin field theory (see e.g. the reviews [4][5][6]). Purpose of this review is to describe a recent development of Lagrangian construction for massless and massive supersymmetric higher spin models in three dimensional Minkowski and anti de Sitter spaces.
In the papers [7,8] the massless N = 1 supersymmetric higher spin field theory was formulated in four dimensional (4D) Minkowski space. The basic results of these papers were the global on-shell N = 1 supertransformations leaving invariant the pair of (Fang)-Fronsdal Lagrangians for free massless higher spin-(s, s + 1/2) fields [9,10]. The off-shell formulation of such system was given in [11,12] where the N = 1 superfield extension of (Fang)-Fronsdal Lagrangians in 4D Minkowski space was obtained. Later on this result was generalized for 4D AdS space [13]. In both cases the constructed superfield models, up to eliminating of auxiliary fields, reduce to the sum of spin-s and spin-(s + 1/2) (Fang)-Fronsdal Lagrangians thus describing N = 1, 4D massless higher spin supermultiplets. Later making use the same technique the off-shell formulation of 4D, N = 2 massless higher spin supermultiplets was found [14].
There exist much less results in the study of supersymmetric massive higher spin models. The reason is that shifting from massless component formulation to the massive one we have to introduce the very complicated higher derivative corrections to the supertransformations. Moreover the higher the spin of the fields entering supermultiplet the higher the number of derivatives one has to consider. The problem of supersymmetric description of 4D massive higher spin supermultiplet was resolved explicitly only in 2007 for the case N = 1 on-shell Poincare superalgebra [15]. The solution was based on the generalization of the gauge invariant formulation of the massive higher spin fields [19][20][21] to the case of massive supermultiplets. In such formulation the massive supermultiplets are described as a system of the appropriate massless ones coupled by local symmetries. On the other hand this system of massless supermultiplets should be invariant under the initial massless supertransformations corrected in a certain way. In [15] it was shown that to obtain the massive deformation it is enough to add the non-derivative corrections to the supertransformations for the fermions only. Complicated higher derivative corrections to the supertransformations reappeared if one tries to fix all local symmetries breaking gauge invariance 1 . Surprisingly but in 4D the above results are still the main results in massive supersymmetric higher spin theory till now.
Taking into account the difficulties in constructing the Lagrangian formulation for 4D massive higher spin supermultiplets it is natural to study the same problems in simpler case, for example to consider the massive higher spin supermultiplets in three dimensions. An indeed in the last years much attention in the supersymmetric higher spin theory has been focused on 3D spaces where higher spin theory is much more simple (see e.g. [4,22]). Here it is important to emphasize that in general a supersymmetry in different dimensions is realized quite differently. The matter is that the supersymmetry operates with spinor fields which are formulated separately for each space-time dimension. Therefore the 3D supersymmetry is independent type of symmetry and should be considered by itself.
It it known that in 3D the massless higher spin fields (k ≥ 3/2) do not propagate any physical degrees of freedom and one of the reasons to study such models can be a possibility to consider their deformation to a massive theory. In its turn the massive higher spin fields in 3D do propagate two physical degrees of freedom [23]. It is important to note that massive higher spin fields can be realized by different ways. One of the possibilities is to generate the mass for 3D massless gauge fields by adding Chern-Simons-like term [24] generalizing 3D topologically massive gravity [25]. Recently for these models the off-shell N = 1 and N = 2 superfield extensions have been constructed in 3D Minkowski space [26,27].
Another possibility to describe the 3D massive higher spin fields is to add the usual mass terms being quadratic in physical fields without derivatives. It mimics the corresponding construction for 4D massive higher spin fields. This possibility has been realized in the papers [28][29][30]. The goal of this review is to present the general methods of supersymmetric Lagrangian construction for massive higher spin fields in 3D Minkowski and AdS spaces. These methods are based on the gauge invariant description of the massive fields [29,30] and are realized in the component approach for the case of on-shell N = 1 supersymmetry. Main content of this review is based on the papers [31,32].
The review organized as follows. In the rest of the introduction we fix our notations and conventions on 3D field variables. In sections 2 and 3 we present the Lagrangian formulation of 3D free bosonic and fermionic higher spin fields respectively. In section 4 we construct the Lagrangian formulation for higher spin supermultiplets in 3D Minkowski space. Here we show that it is possible to construct one extended massless supermultiplet and then smoothly deform it into the massive one. Another approach is used in section 5 for the construction of the massive supermultiplets in 3D AdS. It is based on the Lagrangian formulations in terms of the explicitly gauge invariant objects and their consistent supersymmetric deformation. Such approach is more elegant but it requires the introduction of the so-called extra fields. Notations and conventions. In this review we use a language of differential form when all the objects are some p-forms Ω (p=0,1,2,3). It is defined as In particular derivative is defined as 1-form d = θ µ ∂ µ .
In 3D it is more convenient to use a frame-like multispinor formalism where all the objects have totally symmetric local spinor indices. To simplify the expressions we will use the condensed notations for the spinor indices such that e.g.
Also we always assume that spinor indices denoted by the same letters and placed on the same level are symmetrized, e.g.
In flat space usual derivative d commute d ∧ d = 0 while for AdS space we use following normalization of the covariant derivative: Basis elements of 1, 2, 3-form spaces are e α(2) , E α (2) , E respectively where the last two are defined as double and triple wedge product of e α(2) : Also we write some useful relations for these basis elements Further on the sign of wedge product ∧ will be omitted.

Free higher spin bosonic models
In this section we review the Lagrangian description of the arbitrary spin bosonic fields in three dimensional Minkowski and its cosmological extension AdS spaces [29]. Both for massless and massive fields we present the gauge invariant formulation using the framelike field variables. These fields generalize the tetrad and Lorentz connection in the frame formulation of gravity. Such approach allows us to construct gauge invariant objects (we will call them curvatures) similar to the gravitational curvature and torsion and use them to simplify many constructions.

Massless fields
It is well known that all massless fields with spin k ≥ 1 are gauge ones therefore the gauge invariant formulation for them is a natural form of description. As the gravity, which can be described in term of the metric g µν field or in terms of the frame field e µ a and the Lorentz connection ω µ a,b , the massless higher spins can be described in two ways: metric-like or frame-like. In 4D Minkowski or AdS spaces the metric-like approach leads to the Fronsdal formulation of massless integer spin-k in terms of totally symmetric tensors ϕ µ 1 µ 2 ...µ k subject to the double tracelessness condition ϕ σ σ ρ ρµ 5 ...µ k which becomes nontrivial for k ≥ 4. In the frame-like approach such field is described by generalized frame and Lorentz connection here µ is curved world index and a, b are flat tangent indices. World and flat indices are related by the background Minkowski or AdS frame e µ a . The flat indices of these generalized fields correspond to the irreducible so(3, 1) Lorentz tensors. The Fronsdal formulation is recovered by eliminating the auxiliary field Ω µ a 1 ...a k−1 ,b and considering symmetric combination In 3D it is convenient to use a dual higher spin connection where ε abc is a totally antisymmetric tensor. So in 3D the frame-like and Lorentz-like higher spin gauge fields have the same index structure and their local indices form irreducible so(2, 1) Lorentz tensors. Due to the isomorphism so(2, 1) ∼ sp (2) it is very convenient to use multispinor formalism in which our gauge fields take form where α = 1, 2 are spinor indices. Below we make use of language of differential forms, considering the above higher spin field variables as 1-forms (we omit index µ), and condensed notations for the spinor indices given in introduction.
In the frame-like formalism it is described by the physical 1-form f α(2k) and the auxiliary 1-form Ω α(2k) . Lagrangian in 3D AdS looks like One can construct a pair of the gauge invariant curvatures Using these curvatures the Lagrangian can be rewritten as follows In order to obtain formulation in 3D Minkowski space one should put λ → 0. Thus kinetic terms for higher spin fields is just the first line in (2.1) where D → d. Let us also present kinetic terms for the lower spin fields. Frame-like formulation for them is just the first-order formalism. Spin-1 is described by the physical 1-form A and the auxiliary 0-form B α (2) . The Lagrangian has form and it is invariant under the gauge transformations with 0-form parameter ξ δA = dξ Spin-0 is described by the physical 0-form ϕ and the auxiliary 0-form π α (2) . The expression for the Lagrangian looks like L = −Eπ αβ π αβ + π αβ E αβ dϕ As it will be seen below all these constructions play a role in the gauge invariant formulation of the massive bosonic fields.

Massive fields
In the gauge invariant form the massive spin s field can be described as a system of the massless fields with spins s, (s − 1), ..., 0. In the frame-like approach the corresponding set of fields consists of The Lagrangian for the free fields with mass m in 3D AdS has the form Let us briefly discuss the structure of the Lagrangian (2.5). The first two lines are kinetic terms. It is just the sum of the massless Lagrangians for spins s, (s − 1), ..., 0 where ordinary derivatives are replaced by the AdS covariant ones. The third and the fourth lines contain cross-terms for neighboring spins. These cross-terms couple the individual massless fields into the whole system describing the massive spin-s field. Last line in (2.5) contains the mass terms. The coefficients in (2.6) are determined by the invariance of the Lagrangian under the following gauge transformations This gauge invariant formulation of the massive theory in 3D AdS space possesses some remarkable features. At first, we can consider a flat limit λ → 0 and immediately obtain the description of the massive fields in 3D Minkowski space. At second, there is a correct massless limit m → 0 without the gap in the number of physical degrees of freedom. In such limit our system decomposes into two systems describing the massless spin-s and the massive spin-(s − 1) fields. At last, in dS space when λ 2 < 0 one can consider the socalled partially massless limits a k → 0. In such limit the system decomposes into the two subsystems describing the partially massless spin-s field and the massive spin-k field. Now let us return to the general case of 3D AdS massive spin-s field. Having at our disposal the explicit expressions for the gauge transformations (2.7) we can construct the gauge invariant curvatures. After the change of the normalization the first part of curvatures looks like There is a peculiarity when we try to construct the curvatures for the B α(2) andπ α(2) fields. Namely, in order to achieve gauge invariance for them we should introduce the so-called extra fields B α(4) , π α(4) with the following gauge transformations: Then the corresponding gauge invariant curvatures look like In turn to construct gauge invariant curvatures for the B α(4) , π α(4) we should introduce the extra fields B α(6) , π α(6) and so on. The procedure ends when we construct curvatures for B α(2s−2) , π α(2s−2) . Thus the full set of extra fields is B α(2k) , π α(2k) , 2 ≤ k ≤ s − 1 with the following gauge transformations and the gauge invariant curvatures In three dimensions it is possible to rewrite the Lagrangian in terms of the curvatures only [33]. In the case of arbitrary integer spin field, the corresponding Lagrangian (2.5) can be rewritten in the following simple form Thus there are two approaches to the construction of the supersymmetric (or interacting in general case) higher spin models. According to one of them one can work with the explicit field variables and the Lagrangian in the form (2.5). It is straightforward but rather cumbersome way. We will use it in section 4 for the more simple case of 3D Minkowski space. According to the other way one can work in terms of the gauge invariant curvatures and Lagrangian in form (2.12). It is more elegant way and we use it in section 5 to study the supersymmetric higher spin models in 3D AdS.

Free higher spin fermionic models
In this section we review the Lagrangian description of arbitrary spin fermionic fields in three dimensional Minkowski and its cosmological extension AdS spaces [30]. As in the bosonic case we present frame-like gauge invariant formulation. It is natural form of description for massless fields and for massive fields gauge invariant formulation is realized as a system of massless fields coupled by Stueckelberg symmetries.

Massless fields
As in the integer spin case all massless fields with half-integer spin (k + 1/2) ≥ 3/2 are gauge ones and can be described according to the metric-like or the frame-like approaches. In 4D Minkowski or AdS spaces the metric-like approach leads to Fang-Fronsdal formulation of the massless half-integer spin-(k+1/2) fields in terms of totally symmetric spin-tensors ψ µ 1 µ 2 ...µ k ,α (here α, β = 1, 2 is spinor index) subject to the γ-tracelessness condition γ σ α β ψ ρ ρσ µ 3 ...µ k ,β which becomes nontrivial for k ≥ 3. In the frame-like approach such field is described by generalized frame-like field 3 Φ µ Again due to the isomorphism so(2, 1) ∼ sp (2) it is more convenient to use multispinor formalism in which our gauge field take the form Below we make use of the language of differential forms, considering the higher spin field variables as 1-forms, and using condensed notations for the spinor indices given in introduction. Spin k + 3/2 (k ≥ 0) In the frame-like formalism it is described by physical 1-form Φ α(2k−1) . The Lagrangian in 3D AdS has the following form It is invariant under the following gauge transformations The gauge invariant curvature looks like Using this curvature the Lagrangian can be rewritten as follows 3D Minkowski case corresponds to the flat limit λ → 0. Let us also write out kinetic term for the spin-1/2 field. Spin 1/2 is described by the physical 0-form φ α . It is not a gauge field and the Lagrangian looks like

Massive fields
To describe the massive spin-(s + 1/2) field in the gauge invariant form we have to consider a system of the massless fields with spins (s+1/2), (s−1/2), ..., 1/2. In the frame-like approach the corresponding set of fields consists of Lagrangian for the free field with mass m 1 in 3D AdS space looks like The structure of Lagrangian (3.5) is the same as in the bosonic case. The first line is kinetic terms, the second line is cross-terms and the third line is mass terms. The coefficients in (3.6) are determined by requirement of the invariance of the Lagrangian under the following gauge transformations In such formulation we can take correct massless limit m 1 → 0 in AdS (λ 2 > 0) and correct partially massless limits c k → 0 in dS (λ 2 < 0). Taking flat limit λ → 0 we obtain the description of the massive field in 3D Minkowski space. Note that the Lagrangian (3.5) describes massive Majorana left fermion carrying one physical degree of freedom. Let us return to the general massive fermion and reformulate the theory in terms of the gauge invariant curvatures. Having at our disposal the explicit expressions for the gauge transformations (3.7) we can construct the gauge invariant objects. After change of normalization they take the form As in the case of integer spins in order to achieve gauge invariance for C α we have introduced extra 0-form φ α(3) with the gauge transformations In turn to construct gauge invariant curvatures for φ α(3) field we should introduce extra 0form φ α(5) and so on. Iterations ends at the case of φ α(2s−1) so that the full set of extra fields we should introduce is φ α(2k+1) , 1 ≤ k ≤ (s − 1) with the following gauge transformations The gauge invariant curvatures for them have the form Finally the Lagrangian (3.5) can be rewritten in terms of these curvatures as follows In the next sections we study a supersymmetric higher spin models. In 3D Minkowski space we use the gauge invariant formulation in terms of the explicit fields and the Lagrangian in the form (3.5). In 3D AdS space we use the formulation in terms of the gauge invariant curvatures and the Lagrangian in the form (3.11).

Lagrangian construction of higher spin supermultiplets in 3D Minkowski space
In this section we show how to combine the bosonic and the fermionic higher spin fields into one supermultiplet in 3D Minkowski space and restrict ourselves with N = 1 on-shell supersymmetry. Our construction is based on the gauge invariant formulation given above in terms of the field variables. As we have shown in such formulation massive field in the massless limit decomposes into system of the massless fields. If we take such decomposition for each field in the massive supermultiplet we obtain in general its decomposition into supersymmetric system of the massless fields. So the main idea is to start with this supersymmetric system of the massless fields and construct smooth massive deformation. In other words we generalize the gauge invariant formulation to the supersymmetric case.

3D vs 4D supermultiplets
For the first time the idea to construct the massive higher spin supermultiplets from the supersymmetric system of massless fields was realized for 4D Minkowski space [15]. Moreover it was shown that this supersymmetric system of the massless fields is perfectly combined into the system of the massless supermultiplets. Therefore it is useful to consider how the familiar massive 4D supermultiplets decomposes into the massless ones and then compare it with decomposition of the more specific massive 3D supermultiplets. In both cases it is used that in the gauge invariant formulation massive bosonic and fermionic fields decompose into a set of the massless ones Massive 4D, N = 1 supermultiplets with the half-integer spin-(s+1/2) as the highest one contains four massive fields s+1/2, s, s ′ and s−1/2. Recall that in 4D massive bosonic spin-s has 2s + 1 d.o.f. and massive fermionic spin-(s + 1/2) has 2s + 2 d.o.f. All 4D massless fields have 2 d.o.f. except for the spin 0 which has 1. In the massless limit massive supermultiplet decomposes into massless ones in the same way as (4.1). Simple counting d.o.f. gives So to construct massive supermultiplets we have to start with 2s + 1 massless ones and find a massive deformation. However in [15] it was shown that the crucial point in the whole construction is the possibility to make a dual mixing of the massless supermultiplets by rotating fields with spin k and spin k ′ In such mixing the massless bosonic fields with spin k and k ′ must have equal spins but opposite parities. Thus the structure of the decomposition of the 4D massive supermultiplets with higher half-integer spin into the massless ones looks like Analogously we obtain that the 4D massive supermultiplets with higher integer spin in the massless limit has decomposition Now let us consider 3D massive supermultiplets decomposition in the massless limit and compare it with 4D case. First of all recall that all 3D massless bosonic and fermionic higher spin fields do not propagate any degrees of freedom. Only massless fields with spin 1, 1/2 and 0 propagate one physical degree of freedom. In turn 3D massive higher spin fields do propagate physical degrees of freedom, two and one d.o.f. for bosons and fermions respectively. In gauge the invariant formulation this is clearly seen from (4.1).
As in 4D the minimal massless supermultiplets in 3D contain one boson and one fermion. However unlike 4D massless higher spin fields in 3D do not have physical degrees of freedom that is why one can extend massless supermultiplet adding such fields. In some sense it plays analogous role as the mixing between two massless supermultiplets in 4D case (4.3). As we will see in 3D one can construct extended massless supermultiplet which will correspond to the massless decomposition of the massive supermultiplet.
Further in this section we first of all construct minimal massless higher spin supermultiplets. They will play the role of initial blocks for the construction of the extended massless supermultiplet. Then we fined a gauge invariant massive deformation for them so that the resulting system describe massive supermultiplets.

Massless higher spin supermultiplets
Supermultiplet (k + 3 2 , k + 1), k ≥ 1. It contains one fermionic field with spin (k + 3 2 ) and one bosonic field with spin (k + 1). In the frame-like formulation corresponding field variables are the one 1-form Φ α(2k+1) for the fermion and two 1-forms Ω α(2k) , f α(2k) for boson. The Lagrangian describing this supermultiplet is just the sum of their kinetic terms (see subsections 2.1 and 3.1 for details): It is not hard to show that the Lagrangian is invariant under the following global supertransformations: Let us calculate commutator of the supertransformation on the bosonic field In the frame-like formulation the right side of the commutator corresponds to the translations and it means that In what follows we will not fix the normalization of supertransformations. Supermultiplet (k + 1, k + 1 2 ), k ≥ 1. It contains one fermionic field with spin (k + 1 2 ) and one bosonic field with spin (k + 1). In the frame-like formulation the corresponding field variables are one 1-form Φ α(2k−1) for fermion and two 1-forms Ω α(2k) , f α(2k) for boson. The Lagrangian describing this supermultiplets has the form: It is invariant under the following supertransformations So we have described the full set of the massless higher spin supermultiplets but in the massive case we will also need the massless lower spin supermultiplets (1, 1 2 ) and ( 1 2 , 0). Supermultiplet (1, 1 2 ) contains the fermionic zero-form ψ α as well as the bosonic zero-form B αβ and the one-form A. The sum of their kinetic terms One can show that on the auxiliary field B αβ equations we have E α β dB βγ = E γ β dB βα (4.10) Lagrangian (4.9) is invariant under the following supertransformations 4 : Supermultiplet ( 1 2 , 0) contains the fermionic zero-form φ α and two bosonic zero-forms π αβ and ϕ. The sum of their kinetic terms looks like: Using the fact that on the auxiliary field π αβ equations we have we can show that the Lagrangian is invariant under the following supertransformations: (4.14)

Massive higher spin supermultiplets
Supermultiplet (s + 1 2 , s) This supermultiplet contains spin-s + 1 2 fermion with mass m 1 and spin-s boson with mass m. In the massless limit it decomposes into the system of the massless fields with spins Correspondingly we begin with the appropriate sum of the kinetic terms for all fields This Lagrangian possesses the following supersymmetry: Strictly speaking this Lagrangian is invariant up to the terms proportional to the auxiliary field B αβ equation only (4.10). Thus there are two possible approaches here. From one hand one can introduce non-trivial corrections to the supertransformations for this auxiliary field. Another possibility, that we will systematically follow here and further on, is to use equations for the auxiliary fields in calculating all variations.
These supertransformations are determined by the structure of the supertransformations for the massless supermultiplets given above. So in some sense the Lagrangian (4.15) describes "large" massless supermultiplet which contain the full set of the massless fields.
To construct a massive deformation we have to add the lower derivative terms. For the bosonic terms we take the ones corresponding to the gauge invariant description of massive spin-s boson with mass m (see subsection 2.2) Note that after such deformation equations (4.10), (4.13) are also deformed. Now they have the form For the fermionic terms we take the ones corresponding to the gauge invariant description of the massive spin-s + 1/2 fermion with mass m 1 (see subsection 3.2) Calculating the variations we use auxiliary field equations: In this case to find a massive deformation for the supertransformation we have to add corrections for the fermionic fields only. From the variations with one derivative we found that we must introduce the following set of the corrections to the supertransformations: Taking into account that all the coefficients in the Lagrangian are fixed, it is straightforward to find solution for the parameters of the supertransformations which appears to be valid when m 1 = m only Supermultiplet (s, s − 1 2 ) We need the same set of fields as in the previous case except the field Φ α(2s−1) . So we take the same massless Lagrangian (4.15) with this field omitted and the same set of initial supertransformations (4.16) where now α s−1 = 0. As far as the low derivative terms, the bosonic terms again have the same form (4.17), while the fermionic terms correspond to the gauge invariant description of the massive spin-s −1/2 fermion (it coincide with (4.18) where Φ α(2s−1) is omitted). Since the structure of the bosonic and the fermionic terms is the same as in previous case the correction to the supertransformations also look the same that is δΦ α(2k+1) = γ k f α(2k) ζ α + δ k f α(2k+1)β ζ β δΦ α = δ 0 f αβ ζ β + γ 0 Aζ α +γ 0 ϕe α β ζ β (4.21) δφ α = ρ 0 ϕζ α Calculating all variations we obtain relation between masses m = m 1 and the following expressions for the parameters determining supertransformations: Thus we explicitly construct the Lagrangian description for the two massive supermultiplets (s+1/2, s) and (s, s−1/2). Both of these massive supermultiplets have two bosonic and one fermionic (left) degrees of freedom i.e. possess (1,0) supersymmetry. One can construct massive supermultiplets possessing (1,1) supersymmetry with equal number of bosonic and fermionic degrees of freedom if one introduces one more massive fermion (right). In our previous work [31] we have constructed the particular cases of such massive supermultiplets when supertransformations take the form of diagonal superposition of two (1,1) supertransformations.

Higher spin supermultiplets in 3D AdS space
In this section we study N = 1 supersymmetric construction for the higher spin models in 3D AdS space. As it have been shown by [34] N -extended supersymmetry in 3D AdS space exists in several incarnations. They are so-called (p, q) supersymmetries where p, q are integers and N = p + q. The simplest (1, 0), which we restrict ourselves with, is naturally associated with 3D AdS supergroup (2) so that we have supersymmetry in the left sector only. In practice this means that the massive higher spin supermultiplets as well as the massless ones contain only one bosonic and one fermionic degrees of freedom.
As in the previous section our construction is based on the gauge invariant description of the higher spin fields but in contrast to Minkowski space we work in terms of the gauge invariant curvatures and the Lagrangians in the form (2.12) and (3.11). Below we present a general scheme of our higher spin supermultiplet construction and apply it to the massless and massive cases. At the end we demonstrate how 3D AdS superalgebra is realized in our construction. In particular we show that the higher spin supertransformations satisfy (1,0) superalgebra for which the commutation relation of supercharges has the form where P α (2) and M α(2) are the generators of 3D AdS algebra.

Procedure of curvature deformation
Here we present the general scheme of our higher spin supermultiplet constructions in 3D based on the procedure of curvature deformation. As it has been shown in section 2 and 3 the Lagrangian description of the bosonic and fermionic higher spin fields can be reformulated in terms of the gauge invariant curvatures. To supersymmetrize the system of the bosonic and fermionic fields we covariantly deform the corresponding curvatures by a background non-dynamical gravitino 1-form Ψ α with global transformation in 3D AdS being where ζ α is a parameter of the supersymmetry. So such construction can be interpreted as a supersymmetric theory in terms of the background fields of the supergravity. Due to some difference in the structure of the gauge invariant curvatures for the massless and massive case we separately present the procedure of their deformation.

Massless fields
Massless higher spin bosonic and fermionic fields are described by 1-forms (see subsections 2.1 and 3.1) which we denote as Ω and Φ respectively. Let us also denote corresponding gauge invariant curvatures as R and F which are 2-forms. They have the form (schematically)

Massive fields
In the gauge invariant description of the massive bosonic and fermionic higher spin fields the set of field variables contain 1-forms as well as 0-forms (see subsections 2.2 and 3.2). We denote them respectively as Ω A , B A for bosons and Φ A , φ A for fermions. Let us also denote the corresponding gauge invariant curvatures as R A , B A and F A , C A (R A and F A are 2-forms, B A and C A are 1-forms). Schematically they have the following structure Note that in 3D terms (eeB) A and (eeφ) A in the expressions for the 2-form curvatures disappear for higher spin components. It is typical for 3D higher spin models and is related to the fact that the massless higher spin fields do not propagate any degrees of freedom. Unlike the massless case the Lagrangian for the massive higher spins can be presented as the sum of the quadratic expressions in 2-form and 1-form curvatures (2.12), (3.11) Let us turn to a realization of the supersymmetric construction. Deformation of the curvatures by the terms containing the background gravitino 1-form Ψ α schematically can be written as Note here that the presence of (eΨφ) A and (eΨB) A terms in 2-form curvatures are relates with the presence of (eeB) A and (eeφ) A ones. Hence such terms in the deformations appear in the 2-form curvatures for the low spin components only. The requirement of covariant deformation defines uniquely supertransformations Finally the supersymmetric Lagrangian for the given supermultiplet is the sum of the free Lagrangians where initial curvatures are replaced by the deformed oneŝ Possible arbitrariness is fixed by the condition that the Lagrangian must be invariant under the supertransformations. Below we demonstrate how such general scheme can actually be realized for the massless and massive supermultiplets in 3D AdS.

Massless supermultiplets
Supermultiplet (k + 1, k + 3/2) Integer spin-(k + 1) field is described by two 1-forms Ω α(2k) , f α(2k) and half-integer spin-(k + 3/2) is described by one 1-forms Φ α(2k+1) . The initial curvatures for these system are defined by (). Let us begin with the deformation of the curvatures for the integer spin. There is a unique possibility where we have two arbitrary parameters σ k and α k . To construct covariant deformation we write out the corresponding supertransformations Explicit calculations of the covariant transformations for the deformed curvatures give us, on one hand and on the other hand Comparing these expressions we obtain the relation on the parameters Now let us turn to the curvature deformation for the half-integer spin fields. A unique possibility is where β k and γ k are another pair of arbitrary parameters. The structure of the deformed curvatures defines the form of the supertransformations Explicit verification of the covariant transformation for the deformed curvatures gives on one hand and on the other hand Comparing these expressions we conclude that So we constructed the covariant supersymmetric deformation for the curvatures but we still have two free parameters α k and β k . To relate them we construct the supersymmetric Lagrangian. We choose the following form for it It is just sum of the free Lagrangians for spin-(k + 1) and spin-(k + 3/2) fields where we have replaced curvatures. Using supertransformations for fields and curvatures we obtain for the Lagrangian variations In order to achieve invariance we must put Thus we showed in the explicit form how using curvature deformation procedure to construct supersymmetric Lagrangian formulation of the multiplet (k + 1, k + 3/2). Constructed model contain one free parameter β k which is related with the normalization of the superalgebra. Supermultiplet (k + 1, k + 1/2) In this supermultiplet the integer spin is the same as in the previous case and described by pair of 1-forms Ω α(2k) , f α(2k) and half-integer spin-(k + 1/2) is described by one 1-forms Φ α(2k−1) . Supersymmetric construction for the given supermultiplet is analogous and so let us present the final results only. The supersymmetric curvatures have the form They are covariant under the following supertransformations The supersymmetric Lagrangian looks as follows where

Massive supermultiplets
Supermultiplet (s, s + 1 2 ) For the realization of the given massive supermultiplets let us first consider their structure at the massless flat limit m, m 1 , λ → 0. In this case the Lagrangian is described by the system of massless fields with spins (s + 1 2 ), s, ..., 1 2 , 0 in three dimensional flat space It is the same extended massless supermultiplet which we start the construction of the massive supermultiplets in Minkowski space with. We have shown that this Lagrangian is supersymmetric. If the equations of motion (4.10), (4.13) are fulfilled, the Lagrangian is invariant under the supertransformations (4.16) Here we take into account the normalization (2.8), (3.8). Thus, requiring that massive theory has a correct massless flat limit we partially fix an arbitrariness in the choice of the supertransformations. Parameters α k , β k , β 0 , α 0 ,δ 0 at this step are still arbitrary. As in the massless case we will realize supersymmetry deforming the curvatures by the background gravitino field Ψ α . We start with the construction of the deformations for the bosonic fields The corresponding ansatz for the supertransformations has the form It leads to relation M 1 = M + λ 2 between mass parameters M 1 and M and defines the parameters From the requirement that we obtainρ k = ρ k ,σ k = σ k ,β k = β k ,α k = α k The requirement of covariant transformations for the remaining curvatures gives the solutionρ Now let us consider the deformation of the curvatures for the fermions. We choose an ansatz in the form and the ansatz for the supertransformations in the form From the requirement that we have the same relation between masses M 1 = M + λ 2 . Besides, it leads to In turn, the requirement that gives usγ At last, the requirement for the other curvatures yields solution Now, all the arbitrary parameters are fixed. The supersymmetric Lagrangian is the sum of the free Lagrangians where the initial curvatures are replacement by the deformed oneŝ The Lagrangian is invariant under the supertransformations for curvatures up to equations of motion for the fields B α(2) , π α(2) Φ = 0, A = 0 ⇒ e γ(2) Π γ(2) = DΦ − 2sMA = 0 (5.26) The Lagrangian (5.25) is a final solution for the massive supermultiplet (s, s + 1 2 ). Supermultiplet (s, s − 1 2 ) In this section we consider another massive higher spin supermultiplet when the highest spin is boson. The massive spin-s field was described in section 3.1 in terms of massless fields. The massive spin-(s−1/2) field can be obtained for the results in section 3.2 if one makes the replacement s → (s − 1). So the set of massless fields for the massive field with spin s − 1/2 is Φ α(2k+1) , 0 ≤ k ≤ s − 2 and φ α . The gauge invariant curvatures and the Lagrangian have the forms (3.7) and (3.11) where the parameters are Following our procedure we should construct the supersymmetric deformations for the curvatures. Actually the structure of the deformed curvatures and supertransformations have the same form as in previous subsection for the supermultiplets (s, s + 1/2). There is a difference in parameters (5.27) only. Therefore we present here only the supertransformations for the curvatures. The requirement of covariant curvature transformations for the bosonic fields gives us the relation M 1 = M − λ 2 between masses M 1 and M. Besides, it leads to From the requirement of covariant supertransformations for the fermionic curvatures Supersymmetric Lagrangian have the form (5.25) and it is invariant under the supertransformations up to equations of motion for the auxiliary fields B α(2) , π α(2) (5.26).

Realization of (super)algebra
In this section we analyze the commutators of the (super)transformations and show how the (super)algebra is realized in our construction. All the considerations are valid both for the (s, s + 1/2) supermultiplets and the for (s, s − 1/2) one.

Summary
In this review we have presented and discussed the component supersymmetric formulations of the higher spin fields in three dimensions. We applied these formulations for the Lagrangian construction of the on-shell N = 1 massless and massive supermultiplets in 3D Minkowski and AdS spaces. Although the off-shell formulation of 3D massless higher spin supermultiplets is known long enough we discussed the on-shell massless formulation as well since it is interesting by itself. Besides such a formulation is a base for the deformation to the massive higher spin supermultiplets. To generate the massive terms in the Lagrangians we have used the approach based on the gauge invariant formulation of the massive higher spin fields where the massive fields are described as the system of the massless ones coupled to each other in a special way.
In 3D Minkowski space we generalized the gauge invariant formulation of massive higher spin fields to the case of the massive supermultiplets. In particular we showed that the massive supermultiplets can be constructed from the extended massless one. This extended massless supermultiplet is defined and its smooth mass deformation is explicitly constructed for the two case of the massive supermultiplets (s, s + 1/2) and the (s, s − 1/2). Both of these massive supermultiplets have two bosonic and one fermionic (left) degrees of freedom, i.e. possess (1,0) supersymmetry.
In 3D AdS space the construction of the N = 1 massive supermultiplets with (1,0) supersymmetry is realized by another more elegant way. We have used a technique of the gauge invariant curvatures. Namely, we found their supersymmetric deformation by the background gravitino field. It means that resulting theory describes the massive supermultiplets on the background of AdS supergravity. Constructed higher spin supermultiplets have a correct flat limit.