Superfield approach to interacting N=2 massive and massless supermultiplets in 3d flat space

Massive arbitrary spin supermultiplets and massless (scalar and spin one-half) supermultiplets of the N=2 Poincare superalgebra in three-dimensional flat space are considered. Both the integer spin and half-integer spin supermultiplets are studied. For such massive and massless supermultiplets, a formulation in terms of light-cone gauge unconstrained superfields defined in a momentum superspace is developed. For the supermultiplets under consideration a superspace first derivative representation for all cubic interaction vertices is obtained. A superspace representation for dynamical generators of the N=2 Poincare superalgebra is also found.


Introduction
In view of simplicity and aesthetic features of field theories in three dimensions these theories have attracted a considerable interest during long period of time. Lagrangian formulation of free arbitrary spin massive bosonic and fermionic irreducibe fields propagating in the flat R 2,1 space was obtained in Ref. [1]. 1 In Ref. [9], by using the light-cone gauge formalism, we begun a systematic study of interacting massive arbitrary spin and massless (scalar and spin one-half) fields in three dimensions. 2 We studied both the fermionic and bosonic fields. We introduced a classification for cubic interactions and, by using a first-derivative representation for cubic interactions, obtained the explicit expressions for all cubic interaction vertices. In this paper, we study interacting arbitrary spin massive supermultiplets and massless (scalar and spin one-half) supermultiplets of the N=2 Poincaré superalgebra in the R 2,1 . Both the integer spin and half-integer spin supermultiplets are studied. For such supermultiplets, our aim in this paper, is to provide a first-derivative superspace representation for all cubic interactions. To this end we use a light-cone momentum superspace and unconstrained light-cone gauge superfields defined in such superspace. We note that, it is the light-cone gauge unconstrained superfields we introduce in this paper that allow us to construct a simple superspace representation for all cubic interaction vertices of the supermultiplets under consideration. For the reader convenience, we note that, in the past, the light-cone momentum superspace has been used successfully in various interesting studies of supergravity and superstring theories. We mention the use of the momentum superspace in 10d extended supergravity in Ref. [10] and superstring field theories in Refs. [11]. Application of the light-cone momentum superspace for the study of light-cone gauge 11d supergravity may be found in Ref. [12].
In conclusion of the Introduction, let us briefly mention our main two long term motivations for our study of supersymmetric theories of massive and massless fields in the R 2,1 space. First, we believe that our results in this paper may be helpful in the search of yet unknown models of supersymmetric higher-spin massive fields in the R 2,1 space and their cousins in higher dimensions. Second, one expects that a spectrum of states of the superstring in the AdS 3 space is realized by arbitrary spin massive fields and low spin massless fields. We think then that our superspace representation for cubic vertices of light-cone gauge fields in the R 2,1 space might be a good starting point to understand cubic interactions of light-cone gauge fields in the AdS 3 space and hence might find applications in various studies of superstring in the AdS 3 space. 3 Further discussion of our long term motivations for our research in this paper can be found in Conclusions.
This paper is organized in the following way. In Sec.2, we start with the presentation of a light-cone frame of the N=2 Poincaré superalgebra. After that, by using light-cone gauge components fields, we review integer spin and half-integer spin massive and massless supermultiplets of the N=2 Poincaré superalgebra.
Sec.3 is devoted to a superfield description of the N=2 Poincaré superalgebra supermultiplets. We introduce a momentum superspace and define the corresponding light-cone gauge unconstrained superfields. In terms of our superfields, we describe then a realization of the N=2 Poincaré superalgebra on a space of the massive and massless supermultiplets under consideration. 1 Discussion of various interesting aspects of massless and massive higher-spin dynamics in three dimensions may be found in the (incomplete) list of Refs. [2]- [8]. 2 In 3d, all massless unitary irreps of the Poincaré algebra can be described by the spin-0 and spin-1 2 fields. For this reason, for the case of massless fields, we deal only with spin-0 and spin- 1 2 fields. 3 In the framework of light-cone gauge approach, superstring in the AdS 3 space was studied in Ref. [13]. See also Ref. [14] for studying 3-point correlator functions of AdS superstring. Interesting discussion of the N = 2 superstring model in the R 2,1 space may be found in Ref. [15].
In Sec.4 we discuss n-point interaction vertices. We present constraints on n-point interaction vertices which are obtained by using kinematical symmetries of the N=2 Poincaré superalgebra.
Sec.5 is devoted to cubic vertices. We consider constraints on cubic vertices which are obtained by using dynamical and kinematical symmetries of the N=2 Poincaré superalgebra. After that, we formulate a light-cone gauge dynamical principle and discuss the complete system of equations which are required to determine the cubic interaction vertices uniquely.
In Sec.6, we present a superspace form for all our cubic vertices which describe interactions of arbitrary spin massive and massless supermultiplets of the N=2 Poincaré superalgebra. We introduce a notion of critical and non-critical masses of fields entering the cubic vertices.
In Sec.7, we present solution for so called V -vertices entering superspace form of cubic vertices for the case of the non-critical masses, while, in Sec.8, we present solution for V -vertices entering superspace form of cubic vertices for the case of the critical masses.
Sec.9 is devoted to the study of a superspace representation of cubic vertices for the case when all three N=2 supermultiplets entering the cubic vertices are massless.
In Sec.10, we present our conclusions. Our notation and conventions are presented in Appendix A. Properties of our unconstrained superfields are discussed in Appendix B. In appendix C, we outline the derivation of the superspace representation of our cubic vertices.
2 Light-cone gauge formulation of free massive and massless N = 2 supermultiplets We use a method suggested in Ref. [19]. According to this method, the finding of a new light-cone gauge dynamical system amounts to the finding of solution for (anti)commutators of a symmetry algebra. For N=2 supersymmetric theories of massive and massless fields in the flat space R 2,1 , basic symmetries are associated with the N=2 Poincaré superalgebra. In this section, we start therefore with the presentation of (anti)commutators of the N=2 Poincaré superalgebra in lightcone frame. After that, we review N=2 supermultiplets in light-cone frame and realization of Poincaré algebra on space of massive and massless component fields. Realization of the N=2 Poincaré superalgebra on space of supermultiplets will be discussed in the next section by using unconstrained light-cone gauge superfields. Light-cone frame for N=2 Poincaré superalgebra in 3d. For the flat space R 2,1 , the N=2 Poincaré superalgebra consists of the three translation generators P µ , the three generators of the so(2, 1) Lorentz algebra J µν , the U(1) R-symmetry generator denoted as U, and four supercharges. For the Poincaré algebra generators, we assume the following commutators: where η µν stands for the mostly positive Minkowski metric. Explicit form of the remaining (anti)commutators of the N=2 Poincaré superalgebra will soon be given below.
In place of the Lorentz frame coordinates x µ , µ = 0, 1, 2, we use the light-cone basis coordinates x ± , x 1 , where the coordinates x ± are defined as and the coordinate x + is taken to be an evolution parameter. In the light-cone frame, the so(2, 1) Lorentz algebra vector X µ is decomposed as X + , X − , X. A scalar product of the so(2, 1) Lorentz algebra vectors X µ and Y µ takes the form Relation (2.3) tells us that, in the light-cone frame, the metric η µν has the following non-vanishing elements: η +− = η −+ = 1, η 11 = 1. This implies the rules X + = X − , X − = X + , X 1 = X 1 .
In the light-cone frame, commutators of the Poincaré algebra can simply be obtained from (2.1) by using the flat metric η µν which has the following non-vanishing elements η +− = η −+ = 1, η 11 = 1. Using notation Q ±R , Q ±L for the four supercharges, we then note that, in the light-cone frame, the remaining (anti)commutators of the N=2 Poincaré superalgebra takes the form, 4 where, we recall, that the U appearing in (2.7) is a generator of U(1) symmetry (R-symmetry).
For the generators, we assume the following hermitian conjugation rules: Generators of the N=2 Poincaré superalgebra can be separated into the following two groups: In a field realization, with the exception of J +− , the kinematical generators (2.11) are quadratic in fields 5 , while, the dynamical generators (2.12) involve quadratic and higher order terms in fields.
To build a field theoretical realization of the N=2 Poincaré superalgebra generators on massive and massless fields, we use a light-cone gauge formulation of the fields. Let us first explain our 4 We use the N=2 Poincaré superalgebra without central charges. Dimensional reduction of the N=1 Poincaré superalgebra from 4d to 3d leads to the appearance of central charges. Therefore we do not expect that our vertices in this paper can be obtained via dimensional reduction from light-cone gauge vertices of N=1 massless supermultiplets obtained in Ref. [16]. Study of light-cone gauge vertices for N-extended massless supermultiplets in 4d may be found in Refs. [17,18]. For discussion of Lorentz covariant vertices for various massless supermultiplets in 4d, see, e.g., Refs. [37]- [38] (and references therein) 5 For interacting fields, the J +− takes the form J +− = G 0 + ix + P − , where a generator G 0 is quadratic in fields, while the dynamical generator P − (Hamiltonian) involves quadratic and higher order terms in fields. notation we use for component fields entering the N=2 supermultiplets. Component fields we use in this paper are denoted as φ m,λ,u (x) and ψ m,λ,u (x), where x stands for the space time-coordinates x + , x − , x 1 , while the labels m, λ, and u denote the respective mass parameter, spin, and U(1) charge of the fields. We recall that, for fields in three dimensions, the mass parameter m is allowed to be negative or positive. Sometimes, in the literature, fields with m > 0 and m < 0 are refereed to as the respective self-dual and anti-self-dual massive fields. Obviously, the self-dual and antiself-dual massive fields are associated with different irreps of the Poincaré algebra iso(2, 1). For bosonic fields, the spin parameter λ takes integer values, λ = s, s ∈ N 0 , while for fermionic fields, the λ takes half-integer values, λ = s + 1 2 , s ∈ N 0 . The U(1) charge takes values u ∈ Z. We now proceed with the description of massive and massless supermultiplets of the N=2 Poincaré superalgebra. N = 2 massive spin-s supermultiplet. N = 2 massive spin-s supermultiplet which we denote as {m, s, u} N =2 is described by the following set of massive fields: ,u are fermionic fields. 6 For the fields in (2.13), we assume the following hermitian conjugation rules (2.14) From (2.14), we see that, for u = 0, the fermionic fields are real-valued and the supermultiplet (2.13) is self-adjoint. For u = 0, all fields in (2.13) are complex-valued. N = 2 massive spin-(s + 1 2 ) supermultiplet. N = 2 massive spin-(s + 1 2 ) supermultiplet denoted as {m, s + 1 2 , u} N =2 is described by the following set of massive fields: 2 ,u are fermionic fields. For the fields in (2.17), we assume the following hermitian conjugation rules (2.18) 6 For s = 0, the field φ ⊖ m,s− 1 2 ,u in (2.13) should read as φ −m, 1 2 ,u .
Fields in (2.17) are decomposed into two invariant subspaces. These invariant spaces will be described in Sec.3. N = 2 massless spin-1 2 supermultiplet. Fields of N = 2 spin-1 2 massless supermultiplets are introduced by analogy with the ones for the massive supermultiplets in (2.15), where ψ ⊕ 0,0,u , ψ ⊖ 0,0,u are bosonic fields, while ψ ⊕ 0, 1 2 ,u−1 , ψ ⊖ 0, 1 2 ,u+1 are fermionic fields. For the fields in (2.19), we use the following hermitian conjugation rules Fields in (2.19) are decomposed into two invariant subspaces. These invariant spaces and some equivalencies between the massless spin-0 and the spin-1 2 supermultiplets are described in Sec. 3. In what follows, in place of the fields defined in x-space, we prefer to use fields defined in a momentum space. Namely, using the shortcut notation χ m,λ,u (x) for all fields above discussed, we introduce fields χ m,λ,u (x + , p, β) by using the Fourier transform with respect to the coordinates In what follows, in place of the field χ m,λ,u (x + , p, β), we use the shortcut χ m,λ,u (p). Note also that, throughout this paper, the field χ † m,λ,u (p) is defined as χ † m,λ,u (p) ≡ (χ m,λ,u (p)) † . Realization of the Poincaré algebra in terms of the fields. We now recall a field theoretical realization of the Poincaré algebra and U(1) symmetry on the space of the component fields above discussed. As is known, the realizations of the Poincaré algebra (2.1) and U(1) symmetry in terms of differential operators acting on the component fields χ m,λ,u (p) (2.22) is given as follows.
Realizations on space of fields χ m,λ,u (p): where partial derivatives ∂ β , ∂ p , and symbol e λ are defined as e λ = 0 for integer λ , e λ = 1 for half-integer λ . (2.28) Making use of the relations (2.23)-(2.28) we can build a field representation for the symmetry algebras. Namely, to quadratic order in the component fields, a field-theoretical realization for generators of the Poincaré algebra (2.1) and the u(1) algebra is given by where G diff are given in (2.23)-(2.28), while G [2] stands for the field-theoretical realization for generators of the Poincaré algebra and the u(1) algebra. To avoid unnecessary complicated expressions in terms of the component fields, the field-theoretical realization of the supercharges Q ±R,L will be given in Sec.3, by using superfield formulation of the N=2 supermultiplets. The fields χ m,λ,u satisfy the Poisson-Dirac equal-time (anti)commutation relations where the notation [a, b] ± is used for a graded commutator, Free light-cone gauge action for the component field χ m,λ,u (p) takes the well known form where ∂ − ≡ ∂/∂x + , while e λ is given in (2.28).
3 Superfield formulation of free massive and massless N = 2 supermultiplets To develop a light-cone gauge superfield formulation of the N=2 supermultiplets under consideration we introduce two Grassmann-odd momenta denoted by p θ and p η . Our momentum superspace is parametrized then by the light-cone time x + , the momenta p, β and the two Grassmann momenta p θ , p θ , x + , p , β , p θ , p η .
We now introduce various unconstrained superfields on the superspace (3.1) in turn. Superfield for N=2 massive spin-s supermultiplet. For N=2 massive spin-s supermultiplet (2.13) we introduce Grassmann even superfield Φ m,s,u defined as where component fields entering the Grassmann momenta expansion of the superfield in (3.2) are expressed in terms of the component fields of the supermultiplets (2.13) as 3) An argument p of superfields and fields stands for the momenta p and β entering the momentum superspace (3.1). In terms of the component fields in (3.1), the hermitian conjugation rules in (2.14) take the form We recall that, for all fields, we use the convention χ † (p) ≡ (χ(p)) † . Superfield for N=2 massive spin-(s + 1 2 ) supermultiplet. For N=2 massive spin-(s + 1 2 ) supermultiplet (2.15) we introduce Grassmann odd superfield Ψ m,s+ 1 2 ,u defined as Ψ m,s+ 1 2 ,u (p, p θ , p η ) = βψ(p) + βp θ ψ θ (p) + βp η ψ η (p) + p θ p η ψ θη (p) , (3.6) where component fields entering the Grassmann momenta expansion of the superfield in (3.6) are expressed in terms of the component fields of the supermultiplets defined in (2.15) as In terms of the component fields (3.6), the hermitian conjugation rules (2.16) take the form Superfield for N=2 massless spin-0 supermultiplet. For N=2 massless spin-0 supermultiplet (2.17), we introduce Grassmann even superfield denoted as Φ 0,0,u . This superfield takes the Grassmann momenta expansion identical to the one in (3.2), where fields entering the Grassmann momenta expansion of the superfield Φ 0,0,u are expressed in terms of component fields of the supermultiplets defined in (2.17) as In terms of the component fields entering the superfield Φ 0,0,u , hermitian conjugation rules in (2.18) take the same form as in (3.5). Note that the massless fields φ, φ θ and φ η , φ θη in (3.10), (3.11) constitute two invariant subspaces under transformations of the N = 2 Poincaré superalgebra. Superfield for N=2 massless spin-1 2 supermultiplet. For N=2 massless spin-1 2 supermultiplet (2.19), we introduce Grassmann odd superfield denoted as Ψ 0, 1 2 ,u . This superfield takes the Grassmann momenta expansion identical to the one in (3.6), where fields entering the Grassmann momenta expansion of the superfield Ψ 0, 1 2 ,u are expressed in terms of the component fields of the supermultiplets defined in (2.19) as In terms of the component fields entering the superfield Ψ 0, 1 2 ,u , hermitian conjugation rules in (2.20) take the same form as in (3.9). The massless fields ψ, ψ θ and ψ η , ψ θη in (3.12), (3.13) constitute two invariant subspaces under transformations of the N = 2 Poincaré superalgebra. We note also some equivalencies between the N=2 massless spin-0 and spin-1 2 supermultiplets. Namely, under action of the N=2 Poincaré superalgebra generators, the fields φ, iφ θ entering the superfield Φ 0,0,u transform in the same way as the fields iψ η , ψ θη entering the superfield Ψ 0, 1 2 ,u−1 , while, the fields iφ η , φ θη entering the superfield Φ 0,0,u transform in the same way as the fields ψ, iψ θ entering the superfield Ψ 0, 1 2 ,u+1 . This implies the following equivalence for N=2 massless supermultiplets. (3.14) In order to treat the above discussed superfields Φ and Ψ on an equal footing and to simplify our analysis we collect our superfields into new superfields denoted as Θ m,λ,u , where, depending on the values of λ and m, the new superfields Θ are identified with the Φ and Ψ in (3.2),(3.6) in the following way: We note that, for the integer λ, the new superfield Θ m,λ,u is Grassmann even, while, for the halfinteger λ, the new superfield Θ m,λ,u is Grassmann odd. Therefore, making use of the notation GP(Θ λ ) for the Grassmann parity of the new superfields Θ m,λ,u and recalling the definition of e λ (2.28), we note the obvious relation, N=2 Poincaré superalgebra realization on superfield Θ m,λ,u . We find the realization of the N=2 Poincaré superalgebra in terms of differential operators acting on the superfield Θ m,λ,u (p, p θ ): where operator M +− is defined as while the definition of ∂ β , ∂ p and e λ may be found in (2.27) and (2.28). Quantities ∂ p θ , ∂ pη stand for the left derivatives of the respective Grassmann momenta p θ and p η .
To discuss a light-cone gauge action, we find it convenient to use, in addition to the superfields Φ m,λ,u and Ψ m,λ,u , superfields which we denote as Φ * m,λ,u and Ψ * m,λ,u respectively. The superfields Φ * m,λ,u , Ψ * m,λ,u are built in terms of the hermitian conjugated component fields appearing in (3.2),(3.6). Namely, for m = 0, the Grassmann even superfield Φ * m,λ,u and the Grassmann odd superfield Ψ * m,λ,u are defined by the relations For the massless case, we introduce Φ * 0,0,u and Ψ * 0, 1 2 ,u which take the same Grassmann momenta expansion as in (3.28) and (3.29) respectively. As before, in order to treat the superfields Φ * , Ψ * on an equal footing, we introduce superfields Θ * m,λ,u , where, depending on the values of m and λ, the superfield Θ * is identified with Φ * , Ψ * in the following way Grassmann parity of the superfield Θ * m,λ,u is given by the relation which tells us, for the integer λ, the Θ * m,λ,u is Grassmann even, while, for the half-integer λ, the Θ * m,λ,u is Grassmann odd. Relations (3.19)-(3.26) allow us to build a superfield representation for the N=2 Poincaré superalgebra. Namely, to quadratic order in the superfields, a field-theoretical superspace representation for the generators of the N=2 Poincaré superalgebra is given by where G diff are given in (3.19)-(3.26). A realization of G diff on space of the superfield Θ * m,λ,u and some other properties of the superfield Θ * m,λ,u are discussed in Appendix B. The superfields Θ, Θ * satisfy the Poisson-Dirac equal-time commutation relations Making use of the relations (3.35), (3.36), it is easy to verify the standard equal-time (anti)commutator between the generators and the superfields where expressions for the G diff may be found in (3.19)-(3.26).
Finally we note that, in terms of the superfields above discussed, the light-cone gauge action takes the form where ∂ − ≡ ∂/∂x + and P − int stands for light-cone gauge Hamiltonian which describes interacting fields.
4 Kinematical symmetries of n-point dynamical generators of N =2 Poincaré superalgebra For interacting fields, the dynamical generators of the N=2 Poincaré superalgebra receive corrections having higher powers of the superfields. Namely, for the dynamical generators, one has the following expansion in the superfields is a functional having n powers of the superfields Θ * . We now describe restrictions imposed on the dynamical generators G dyn [n] , n ≥ 3, by the kinematical symmetries of the N=2 Poincaré superalgebra. We discuss the kinematical symmetry restrictions in turn. Kinematical P 1 , P + , Q +L symmetries. Making use of the (anti)commutators between the dynamical generators (2.12) and the kinematical generators P 1 , P + , Q +L , we find that the dynamical generators G dyn [n] with n ≥ 3 can be cast into the form: , , where we are using the following notation , ≡ dp η 1 dp θ 1 . . . dp ηn dp θn δ n a=1 p θa , (4.8) , and Θ [n] |j −1 [n] are shortcuts which are defined as where a label σ a is a shortcut for mass parameters, spins, and U(1) charges and indices a, b = 1, . . . , n are used to label superfields entering n-point vertices. Throughout this paper the p a as argument of delta functions (4.7) and superfields (4.13) stands for the momenta p a and β a . Sometimes, the quantities p − σ 1 ...σn , q −R,L σ 1 ...σn , and j −1 σ 1 ...σn (4.2)-(4.5), will shortly be denoted as g [n] , The quantities g [n] (4.15) will be referred to as n-point densities. Often the density p − [n] will be referred to as an n-point interaction vertex, while, for n = 3, the density p − [3] will be referred to as cubic interaction vertex.
In general, the n-point densities g [n] (4.15) depend on the momenta p a , β a , Grassmann momenta p θa p ηa , masses m a , spins λ a , and U(1) charges u a , a = 1, 2 . . . , n, (4. 16) Also note that, in (4.5), the differential operators X 1 [n] , X [n] θ act only on the arguments of the superfields. For example, the expression X 1 Θ * [n] |g [n] should read as Kinematical J +− -symmetry. Making use of the commutators between the dynamical generators P − , Q −R,L , J −1 and the kinematical generator J +− , we find the equations for the densities given by: , (4.19) Kinematical J +1 , Q +R -symmetries. Making use of the (anti)commutators between the dynamical generators P − , Q −R,L , J −1 and the kinematical generators J +1 , Q +R , we verify that the dependence of the densities g [n] (4.15) on the momenta p a and the Grassmann momenta p θa is realized through new momentum variables P ab and P θ ab defined by the relations Thus the densities g [n] (4.16) turn out to be functions of the new momenta P ab and P θ ab in place of the generic momenta p a , p θa , Kinematical U(1)-symmetry. Making use of the commutators between the dynamical generators P − , Q −R,L , J −1 and the kinematical generator U, we find the equations for the densities given by: where we use the notation given in (4.20).
Let us now summarize the kinematical symmetry restrictions on the n-point densities. i) The kinematical P 1 , P + , and Q +L symmetries lead to delta-functions in expressions for the dynamical generators P − , Q −R,L , J −1 (4.2)-(4.5). These symmetries imply the conservation laws for the momenta p a , β a and the Grassmann momenta p θa .
iii) The kinematical J +1 and Q +R symmetries imply that the n-point densities , j −1 [n] turn out to be functions that depend on the new momenta P ab and P θab (4.19) in place of the generic momenta p a and p θa respectively. Conservation laws for the momenta p a , β a , p θa imply that there are n − 2 independent momenta P ab and n − 2 independent Grassmann momenta P θab (4.19). For example, if n = 3, then there is one independent momentum P and one independent Grassmann momentum P θ (see below).

N=2 Poincaré superalgebra restrictions for cubic vertex and light-cone gauge dynamical principle
In this Section, firstly, we represent kinematical J +− symmetry equations (4.18), (4.19) in terms of the momenta P ab and P ab θ . Secondly, we find restrictions imposed by dynamical symmetries. Finally, we discuss light-cone gauge dynamical principle and write down the complete system equations which allows us to determine the cubic vertices unambiguously. Kinematical symmetries of the cubic densities. Making use of the momentum conservation laws it is easy to check that P 12 , P 23 , P 31 and Grassmann momenta P θ 12 , P θ 23 , P θ 31 can be expressed in terms of momenta P, P θ , where the momenta P and P θ are defined by the relations The use of the momenta (5.3) is convenient for us because these momenta turn out to manifestly invariant upon cyclic permutations of the external line indices 1, 2, 3. Making use of the simplified notation for the cubic densities, we note that the cubic densities p − [3] , q −R,L [3] , and j −1 [3] are functions of the momenta β a , P, the Grassmann momenta P θ , p ηa and σ a = m a , λ a , u a , a = 1, 2, 3, Thus we see that the dependence of the cubic densities on the momenta p a and p θa is realized through the respective momenta P and P θ . In view of this, the study of cubic densities is considerably simplified. Let us now represent equations given in (4.18), (4.19) and (4.23), (4.24) in terms of the cubic densities given in (5.5). J +− -symmetry equations: Making use of the representation for cubic densities in (5.5), we verify that, for n = 3, equations (4.18), (4.19) can be represented as where J +− is defined by the relations U(1)-symmetry equations: Making use of the representation for the cubic densities in (5.5), we verify that, for n = 3, equations (4.23),(4.24) can be represented as for g [3] = p − [3] , j −1 [3] , (5.9) where we use the notation as in (5.8).
We now proceed with studying the restrictions imposed by the dynamical symmetries.
Light-cone gauge dynamical principle. Equations in (5.6),(5.9) and (5.14)-(5.16) do not fix the cubic densities unambiguously. In order to fix the cubic densities unambiguously we should impose some additional restrictions on the cubic densities. Such additional restrictions will be referred to as light-cone gauge dynamical principle. We formulated the light-cone gauge dynamical principle as follows: i) All cubic densities p − [3] , q −R,L [3] , j −1 [3] should be polynomial in the momentum P and β-analytic; 7 ii) The cubic vertex p − [3] should satisfy the following restriction p − [3] = P − W , W is polynomial in P and β-analytic , where P − is defined in (5.17). We note that the restriction in (5.23) is motivated by the freedom of field redefinitions. Namely, upon field redefinitions, the cubic vertex p − [3] is changed by terms proportional to P − (see, e.g., Appendix B in Ref. [43]). Therefore ignoring restriction (5.23) leads to cubic vertex p − [3] which can be removed by field redefinitions. However we are interested in the cubic vertex p − [3] that cannot be removed by field redefinitions. For this reason we impose the restriction (5.23). For the reader convenience we note that the restriction in i) is the light-cone counterpart of locality condition used in Lorentz covariant and gauge invariant formulations. Complete system of equations for cubic vertex. To summarize the discussion in this section, we note that, for the cubic vertex given by Light-cone gauge dynamical principle: p − [3] , q −R,L [3] , j −1 [3] should be polynomial in P and β-analytic; (5.30) p − [3] = P − W, W should be polynomial in P and β-analytic.
It is the equations (5.26)-(5.31) that constitute our basic complete system of equations. If some vertex p − [3] fix satisfies our system of equations, then the vertex p − [3] obtained from p − [3] fix via the field redefinition p − [3] = p − [3] fix + P − f also satisfies our system of equations. This implies that in order to fix the vertex uniquely we should chose some representative of the vertex by using field redefinitions. Only after choosing a representative of the vertex our system of equations allows us to find all possible solutions for the cubic vertex p − [3] and the densities q −R,L [3] , j −1 [3] uniquely. Below, we use a representative of cubic vertex which we refer to as first-derivative representation of the vertex. 7 If function f = f (β 1 , β 2 , β 3 ) can be presented as f = g/h where g, h are polynomials in the momenta β 1 , β 2 , β 3 , then we say the function f is β-analytic. Otherwise, we say that the function f is β-nonanalytic. Let us consider Taylor series expansion for the vertex p − [3] = K n=0 f n P n . If all f n are β-analytic, then we say the vertex p − [3] as β-analytic. Otherwise, we say that the vertex p − [3] is β-nonanalytic.
6 Superspace first-derivative representation for cubic vertices Superspace first-derivative representation for cubic vertex. In general, the vertex p − [3] is a degree-K polynomial in the momentum P, where K < ∞. However, as noted in Ref. [9], in the framework of light-cone gauge approach in 3d, by making use of field redefinitions, any such vertex p − [3] can be cast into a degree-1 polynomial in the momentum P. Throughout this paper, the representation for the vertex p − [3] in terms of degree-1 polynomial in the momentum P is referred to as first-derivative representation of vertex. Thus our first-derivative vertices are polynomials of degree-1 in the momentum P. In this section, we present our superspace first-derivative representation for all cubic vertices p − [3] and the corresponding densities q −R,L [3] , j −1 [3] . In this section and Secs.7,8, we consider cubic vertices which involve at least one massive superfield. Cubic vertices for the case of three massless superfields are considered separately in Sec.9. Derivation of the results below presented may be found in Appendix C. We now present our results.
We find that equations (5.14)-(5.16) lead to three types of cubic vertices which we refer as type-k cubic vertices, where k = 1, 2, 3. We note that the type-k cubic vertex is realized as a homogeneous degree-k polynomial in the Grassmann momenta P θ and p η 1 , p η 2 , p η 3 , , for type-k vertex, (6.1) where we use the notation as in (5.8). 8 To clarify further our classification of vertices we use requirement that the Hamiltonian P − [3] (4.2) should be Grassmann even. To this end we note the following relations: The 1st relation in (6.2) tells us that the integration measure dΓ p θ pη [3] (4.6) is Grassmann odd, while the 2nd relation for the Grassmann parity of the product of three superfields Θ * [3] (4.13) is obtained by using (3.34). It is the 3rd relation in (6.2) that is obtained by requiring GP(P − [3] ) = 0. Note however that the Grassmann parity of the type-k cubic vertex is equal to GP(p − [3] ) = k. Equating this relation to the 3rd relation (6.2), we find the restriction Using (6.3), we can clarify further our classification of vertices. Namely, from (3.32)-(3.34), we note the relations GP(Φ * ) = 0, GP(Ψ * ) = 1. Using these Grassmann parities of the superfields Φ * , Ψ * , we conclude that, for k = 1, 3, relation (6.3) is satisfied by the product Φ * Φ * Φ * and Ψ * Ψ * Φ * while, for k = 2, relation (6.3) is satisfied by the product Φ * Φ * Ψ * and Ψ * Ψ * Ψ * . Omitting the asterisk of the superfields, we then conclude that we deal with the following vertices: As the type-1 and type-3 cubic vertices turn out to be related by hermitian conjugation, we first discuss these vertices and after that we discuss the type-2 cubic vertex.
V P and q −R,L [3] for type-1 ΦΦΦ and ΨΨΦ vertices: (6.21) V P and q −R,L [3] for type-3 ΦΦΦ and ΨΨΦ vertices: (6.25) V P and q −R,L [3] for type type-2 ΦΦΨ and ΨΨΨ vertices: (6.29) For critical and non-critical masses, a structure of solutions of the equations for the V -vertices is different. Therefore, before discussing solutions of the equations for the V -vertices, we provide the definition of critical and non-critical masses we use in this paper. Critical and non-critical mass values. Consider a cubic vertex for three superfields which have masses m 1 , m 2 , m 3 . Let us introduce quantities D, P ǫm defined as ǫ a m a , (6.34) ǫ 2 1 = 1 , ǫ 2 2 = 1 , ǫ 2 3 = 1 , (6.35) where, relation (6.36) provides us an alternative representation for the D defined in (6.33) . If, for masses m 1 , m 2 , m 3 , we meet the inequality D = 0, then we refer to such masses as non-critical masses, while, if, for masses m 1 , m 2 , m 3 , we meet the equality D = 0, then such masses are referred to as critical masses. Note that the relation P ǫm = 0 implies the relation D = 0, while the relation D = 0 implies that there exist ǫ 1 , ǫ 2 , ǫ 3 (6.35) such that the relation P ǫm = 0 holds true. Thus we have the relations D = 0 , for non-critical masses; (6.37) D = 0 , P ǫm = 0 , for critical masses. (6.38) General structure of solutions to V -vertices. We find that, for the non-critical masses (6.37), general structure of β-analytical solutions to vertices V , V ′ , V a , and their bared cousins takes the form For non-critical masses: for type-1 ΦΦΦ and ΨΨΦ vertices; (6.39) for type-3 ΦΦΦ and ΨΨΦ vertices; (6.40) for type-2 ΦΦΨ and ΨΨΨ vertices; (6.41) where C, C ′ , C (1) , C (2) are coupling constants, while V κ , V (1), (2) a,κ are some fixed functions of β 1 , β 2 , β 3 . Explicit expressions for these functions are given in Secs.7.1-7.3. For the hermitian Hamiltonian P − [3] , the coupling constants entering vertices in (6.39)-(6.41) should satisfy the following rules: where I u stands for sign-inversion operator of the U(1) charges (see (A.1)), while the asterisk stands for the complex conjugation. 9 Let us make comments on the solution in (6.39)-(6.41). i) From (6.9), we see that the vertex V satisfies the two first-order differential equations with respect to β-momenta β 1 , β 2 , β 3 . Note however that, in view of the conservation law β 1 +β 2 +β 3 = 0, the vertex V depends only on two β-momenta. This implies that, up to one coupling constant C the vertex V is determined uniquely. The same holds true for the vertexV .
iii) From (6.16),(6.17), we see that the three vertices V a satisfy the six first-order differential equations with respect to β-momenta β 1 , β 2 , β 3 and one algebraic constraint. Note however that, in view of the conservation law β 1 + β 2 + β 3 = 0, the vertices V a depend only on two β-momenta. This implies that, the three vertices V a are determined uniquely up to two coupling constants denoted as C (1) and C (2) in (6.41) . The same holds true for the verticesV a . iv) Equations forV (6.10) are obtained from the ones for V (6.9) by the replacement κ → −κ. Therefore, if the solution for the vertex V is described by the function V κ , then the solution for the vertexV is given by the V −κ . The same holds true also for the vertices V ′ ,V ′ and V a ,V a . v) From (6.39), we see that the vertices V andV enter in the game with the one and the same coupling constant C. It is the requirement of the β analycity that leads to the one and same coupling constant for the vertices V andV . Namely, ignoring the β-analycity, we find that the general solution for the vertexV takes the formV =CV −κ . IfC = C, then the vertex p − [3] is βnon-analytic, while, ifC = C, then the vertex p − [3] becomes even function of κ, p − [3] (κ) = p − [3] (−κ). Explicit expressions for the V κ given in the next sections tell us that, in general, the V κ is a rational function of κ. This implies that, forC = C, the vertex p − [3] depends on κ 2 and hence is β-analytic. The same holds true also for the vertices V ′ ,V ′ and V a ,V a in (6.40), (6.41).
We now discuss vertices for the critical masses. We find that, for the critical masses (6.38), general structure of β-analytical solution to the vertices V , V ′ , V a , and their bared cousins takes the form For critical masses: for type-1 ΦΦΦ and ΨΨΦ vertices; (6.44) for type-3 ΦΦΦ and ΨΨΦ vertices; (6.45) for type-2 ΦΦΨ and ΨΨΨ vertices; (6.46) where quantities C, C ′ , C (1) , C (2) and their bared cousinsC,C ′ ,C (1) ,C (2) are coupling constants, while V ǫ , V (1), (2) a,ǫ are some fixed functions of β 1 , β 2 , β 3 . Explicit expressions for these functions are given in Secs.8.1 and 8.2. For the hermitian Hamiltonian P − [3] , the coupling constants entering vertices in (6.44)-(6.46), unless otherwise specified, should satisfy the following rules: Taking into account (6.49),(6.50), we see that equations forV (6.10) are obtained from the ones for V (6.9) by using the replacement ǫ 1 , ǫ 2 , ǫ 3 → −ǫ 1 , −ǫ 2 , −ǫ 3 . Therefore, if a solution for the vertex V is described by the function V ǫ , then a solution for the vertexV is given by the V −ǫ . The same holds true also for the vertices V ′ ,V ′ and V a ,V a .
ii) The κ given in (6.50) is β-analytic. Solutions for the V -vertices and their bared cousins turn also to be β-analytic. Therefore, in contrast to the relations in (6.39)-(6.41), there is no need to tune the coupling constants C, C ′ , C (1) , C (2) and their bared cousins in (6.44)-(6.46). As we said, for hermitian Hamiltonian P − [3] , the type-1 and type-3 vertices are related by hermitian conjugation rules. These rules are given explicitly by the relations (B.11),(B.12) in the Appendix B. For the type-2 vertices, hermicity of the Hamiltonian leads to the rules given in (B.13)-(B.15). Hermitian conjugation rules in (B.11)-(B.15) are given in terms of the vertices. In terms of the coupling constants these rules are given in (6.42),(6.43) and (6.47),(6.48). U(1) symmetry restrictions for type-k vertices. For the type-k vertices, by using the U(1) symmetry restriction (5.9), we can obtain constraints on the labels u 1 , u 2 , u 3 of the cubic vertex p − [3] = p − σ 1 σ 2 σ 3 , where σ a is defined in (4.14). Namely, using (5.9), we find that the u 1 , u 2 , u 3 of the cubic vertex p − σ 1 σ 2 σ 3 should satisfy the restrictions: U = 1 for type-1 vertices; U = −1 for type-3 vertices; (6.51) U = 0 for type-2 vertices; (6.52) In other words, the type-k cubic vertices are non-trivial only for the values of U given in (6.51),(6.52).

V κ -vertices for superfields with non-critical masses
In this Section,we deal with the non-critical masses defined by relations (6.37), where D is given in (6.33). Taking into account that some masses may be equal to zero, we will consider the following cases in turn: Note that, for m 3 = 0, the inequality D = 0 amounts to the inequality |m 1 | = |m 2 | in (7.2), while, for m 1 = 0, m 2 = 0, the inequality D = 0 amounts to the inequality m 3 = 0 in (7.3). General expressions for the vertices we deal in this section are given in (6.39)-(6.41). From these expressions, we see that all that is required is to provide explicit expressions for the functions V κ and V (1), (2) a,κ . Such expressions for the V κ and V (1), (2) a,κ are given below in Secs.7.1-7.3.

V κ -vertices for three massive superfields
ΦΦΦ vertex. Three arbitrary spin massive superfields Φ. We consider a cubic vertex for the following three massive superfields: Solution for the V κ entering the vertices V and V ′ (6.39),(6.40) takes the form We note that the helpful relations for L κ,a (7.6) may be found in Appendix A in Ref. [9]. ΨΨΦ vertex. Two arbitrary spin massive superfields Ψ and one arbitrary spin massive superfield Φ. We consider a cubic vertex for the following three massive superfields: Solution for the V κ entering the vertices V and V ′ (6.39),(6.40) takes the form where L κ,a are given in (7.6). ΦΦΨ vertex. Two arbitrary spin massive superfields Φ and one arbitrary spin massive superfield Ψ. We consider a cubic vertex for the following three massive superfields: Solution for the functions V (1), (2) κ,a entering the vertices V a (6.41) takes the form V (1), (2) a,κ = V (1), (2) a,κ L s 1 κ,1 L s 2 κ,2 L s 3 κ,3 , a = 1, 2, 3 , (7.10) where L κ,a are given in (7.6) and we use the notation ΨΨΨ vertex. Three arbitrary spin massive superfields Ψ. We consider a cubic vertex for the following three massive superfields: Solution for the V (1), (2) κ,a entering the vertices V a (6.41) takes the form where L κ,a are given in (7.6), while V (1), (2) a,κ take the same form as in (7.11)-(7. 16) 7.2 V κ -vertices for two massive superfields and one massless superfield ΦΦΦ vertex. Two arbitrary spin massive superfields Φ with masses |m 1 | = |m 2 | and one massless superfield Φ. We consider a cubic vertex for the following three superfields: Solution for the V κ entering the vertices V and V ′ (6.39),(6.40) takes the form ΨΨΦ vertex. Two arbitrary spin massive superfields Ψ with masses |m 1 | = |m 2 | and one massless superfield Φ. We consider a cubic vertex for the following three superfields: Solution for the V κ entering the vertices V and V ′ (6.39),(6.40) takes the form where L κ,1 , L κ,2 , and κ take the same form as in (7.20), (7.21). ΨΨΦ-vertex. One massless superfield Ψ, one arbitrary spin massive superfield Ψ and one arbitrary spin massive superfield Φ with masses |m 2 | = |m 3 |. We consider a cubic vertex for the following three superfields: Solution for the V κ entering the vertices V and V ′ (6.39),(6.40) takes the form ΦΦΨ vertex. Two arbitrary spin massive superfields Φ with masses |m 1 | = |m 2 | and one massless superfield Ψ . We consider a cubic vertex for the following three superfields: Solution for the V (1), (2) a,κ entering the vertices V a in (6.41) takes the form where L κ,1 , L κ,2 , and κ take the same form as in (7.20), (7.21). ΦΦΨ vertex. One massless superfield Φ, one arbitrary spin massive superfield Φ and one arbitrary spin massive superfield Ψ with masses |m 2 | = |m 3 |. We consider a cubic vertex for the following three superfields: Solution for the V (1), (2) a,κ entering the vertices V a in (6.41) takes the form where L κ,2 , L κ,3 , and κ take the same form as in (7.26), (7.27). ΨΨΨ vertex. Two arbitrary spin massive superfields Ψ with masses |m 1 | = |m 2 | and one massless superfield Ψ. We consider a cubic vertex for the following three superfields: Solution for the V (1), (2) a,κ entering vertices V a in (6.41) takes the form where L κ,1 , L κ,2 , and κ take the same form as in (7.20), (7.21).

V κ -vertices for two massless superfields and one massive superfield
ΦΦΦ vertex. Two massless superfields Φ and one arbitrary spin massive superfield Φ. We consider a cubic vertex for the following three superfields: Solution for the V κ entering the vertices V and V ′ (6.39),(6.40) takes the form ΨΨΦ vertex. Two massless superfields Ψ and one arbitrary spin massive superfield Φ. We consider a cubic vertex for the following three superfields: Solution for the V κ entering the vertices V and V ′ (6.39),(6.40) takes the form ΨΨΦ vertex. One arbitrary spin massive superfield Ψ, one massless superfield Ψ and one massless superfield Φ. We consider a cubic vertex for the following three superfields: Solution for the V κ entering the vertices V and V ′ (6.39),(6.40) takes the form ΦΦΨ vertex. Two massless superfields Φ and one arbitrary spin massive superfield Ψ. We consider a cubic vertex for the following three superfields: Solution for the V (1), (2) a,κ entering the vertices V a (6.41) takes the form where L κ,3 and κ take the same form as in (7.35). ΦΦΨ vertex. One massless superfield Φ, one arbitrary spin massive superfield Φ and one massless superfield Ψ. We consider a cubic vertex for the following three superfields: Solution for the V (1), (2) a,κ entering the vertices V a (6.41) takes the form ΨΨΨ vertex. Two massless superfields Ψ and one arbitrary spin massive superfield Ψ. We consider a cubic vertex for the following three superfields: Solution for the V (1), (2) a,κ entering the vertices V a (6.41) takes the form where L κ,3 and κ take the same form as in (7.35).

V κ -vertices for superfields with critical masses
In this Section, we deal with the critical masses defined by relations (6.38). Taking into account that some masses may be equal to zero, we will consider the following cases in turn: where P ǫm is defined in (6.34), (6.35). Note that, for m 3 = 0, the relation P ǫm = 0 amounts to the relation |m 1 | = |m 2 | in (8.2). General expressions for the vertices we deal in this section are given in (6.44)-(6.46). From these expressions, we see that all that is required is to provide explicit expressions for the functions V ǫ and V (1), (2) a,ǫ . Such expressions for the V ǫ and V (1), (2) a,ǫ are given below in Secs. 8.1,8.2. Throughout this section we use P ǫm and ǫ 1 , ǫ 2 , ǫ 3 defined in (6.34),(6.35).
8.1 V κ -vertices for three massive superfields ΦΦΦ vertex. Three arbitrary spin massive superfields Φ. We consider a cubic vertex for the following three massive superfields: The V ǫ entering the vertices V and V ′ (6.44),(6.45) is given by whereβ a is defined in (5.3). We recall that the V −ǫ entering the verticesV ,V ′ (6.44),(6.45) is obtained from the V ǫ (8.4) by using the replacement ǫ 1 ,ǫ 2 ,ǫ 3 → −ǫ 1 ,−ǫ 2 ,−ǫ 3 . ΨΨΦ vertex. Two arbitrary spin massive superfields Ψ and one arbitrary spin massive superfield Φ. We consider a cubic vertex for the following three massive superfields: Solution for the V ǫ entering the vertices V and V ′ (6.44),(6.45) takes the form where we use the notation as in (8.5). ΦΦΨ vertex. Two arbitrary spin massive superfields Φ and one arbitrary spin massive superfield Ψ. We consider a cubic vertex for the following three massive superfields: where we use the notation as in (8.5), while V (1), (2) a,ǫ are defined as 14) Requiring hermicity of the Hamiltonian, we get the restrictions on the coupling constants, ΨΨΨ vertex. Three arbitrary spin massive superfields Ψ. We consider a cubic vertex for the following three massive superfields: Solution for the V (1), (2) a,ǫ entering the vertices V a (6.46) takes the form (8.19) where P ǫm , S ǫ are defined as in (8.5), while V (1), (2) a,ǫ are defined as in (8.10)- (8.16). To ensure hermicity of the Hamiltonian the coupling constants should satisfy the relations (8.17).
8.2 V κ -vertices for two massive superfields and one massless superfield ΦΦΦ vertex. Two arbitrary spin massive superfields Φ and one massless superfield Φ. We consider a cubic vertex for the following three superfields: ΨΨΦ vertex. Two arbitrary spin massive superfields Ψ and one massless superfield Φ. We consider a cubic vertex for the following three superfields: Solution for the V ǫ entering the vertices V and V ′ (6.44),(6.45) takes the form ΨΨΦ vertex. One massless superfield Ψ, one arbitrary spin massive superfield Ψ and one arbitrary spin massive superfield Φ. We consider a cubic vertex for the following superfields: Solution for the V ǫ entering the vertices V and V ′ (6.44),(6.45) takes the form ΦΦΨ vertex. Two arbitrary spin massive superfields Φ and one massless superfield Ψ. We consider a cubic vertex for the following three superfields: Solution for the V (1), (2) a,ǫ entering the vertices V a (6.46) takes the form ΦΦΨ vertex. One massless superfield Φ, one arbitrary spin massive superfield Φ and one arbitrary spin massive superfield Ψ. We consider a cubic vertex for the following three superfields: Solution for the V (1), (2) a,ǫ entering the vertices V a (6.46) takes the form ΨΨΨ vertex. Two arbitrary spin massive superfields Ψ and one massless superfield Ψ. We consider a cubic vertex for the following three superfields: Solution for the V (1), (2) a,ǫ entering the vertices V a (6.46) takes the form

Superspace first-derivative cubic vertices for massless fields
We now present cubic vertices of three massless superfields, m 1 = 0, m 2 = 0, m 3 = 0. We find that, for the three massless superfields, equations (5.14)-(5.16) lead to four types of cubic vertices which we refer as type-k cubic vertices, where k = 1, 2, 3, 4. By definition, the type-k cubic vertex is realized as homogeneous degree-k polynomial in the Grassmann momenta P θ and p ηa . Let us first present relations for the p − [3] and j −1 [3] which are valid for all types of cubic vertices. Namely, all cubic vertices p − [3] and the corresponding densities j −1 [3] can be presented as where V P is a vertex that depends only on the momenta β 1 , β 2 , β 3 and the Grassmann momenta P θ , p η 1 , p η 2 , p η 3 , while the operator is given in N ηE β is defined in (5.19). From (9.1),(9.2), we see that in order to get the cubic densities we should fix expressions for the V P and the supercharge densities q −R,L [3] . The expressions for V P and q −R,L [3] we find are presented below in (9.3)-(9.11). 10 Outline of the derivation of relations in (9.3)-(9.11) may be found at the end of Appendix C.
In conclusion of this section, we recall that, for the massive and massless fields having κ ≡ / 0, we found a finite number of the cubic vertices in Secs. 6,7,8. In contrast to this, for the three massless fields, we obtained infinite number of light-cone gauge cubic vertices (9.3)-(9.11) which are parametrized by the vandv-functions. To our knowledge, an infinite number of Lorentz covariant vertices associated with our infinite number light-cone gauge vertices are not available in the literature at the present time. This is to say that we encounter a mismatch between classifications of Lorentz covariant and light-cone gauge cubic vertices. This mismatch for the massless fields in 3d was noticed in Ref. [9]. We recall that, for the case of massless fields in 4d, the mismatch between classifications of Lorentz covariant cubic vertices and light-cone gauge cubic vertices is well known (see, e.g., Ref. [20]). 11

Conclusions
In this paper, we applied a superfield light-cone gauge approach for the study of the interacting N = 2 massive arbitrary (integer and half-integer) spin supermultiplets and massless (spin-0 and 11 Recent study of new Lorentz covariant vertices in the framework of 4d chiral higher-spin gravity may be found in Ref. [21]- [23]. We expect that adaptation of the twistor methods in these references to 3d may be helpful for the study of yet unknown new Lorentz covariant vertices in 3d. Recent discussion of twistors in 3d may be found in Refs. [24]. For interesting application of twistors to the study of a superparticle in AdS 5 × S 5 , see Refs. [25] . spin-1 2 ) supermultiplets in 3d flat space. Using the light-cone momentum superspace, we developed the light-cone gauge formulation in terms of the unconstrained superfields. Using our superfield formulation, we built all cubic vertices that describe interactions of the massive and massless supermultiplets under consideration. We believe that development made in this paper might be helpful for the following applications and generalizations. i) In this paper, we studied cubic vertices for the N=2 massive and massless supermultiplets in the flat 3d space. Light-cone gauge action for arbitrary spin free massive fields in AdS 3 space was obtained in Ref. [26], while the method for studying cubic vertices of light-cone gauge higher-spin massless AdS fields was developed in Ref. [27]. 12 Generalization of our studies in this paper and in Refs. [26,27] to the case of light-cone gauge interacting N=2 massive and massless supermultiplets in AdS 3 space could be of great interest. For the reader convenience, we note the recent Ref. [29] devoted to studying cubic vertices for fields in AdS 3 space by using covariant frame-like formulation. For discussion of methods for the investigation of supersymmetric theories in 3d, see Refs.