From $\mathcal{N}{=}\,4$ Galilean superparticle to three-dimensional non-relativistic $\mathcal{N}{=}\,4$ superfields

We consider the general $\mathcal{N}{=}\,4,$ $d{=}\,3$ Galilean superalgebra with arbitrary central charges and study its dynamical realizations. Using the nonlinear realization techniques, we introduce a class of actions for $\mathcal{N}{=}\,4$ three-dimensional non-relativistic superparticle, such that they are linear in the central charge Maurer-Cartan one-forms. As a prerequisite to the quantization, we analyze the phase space constraints structure of our model for various choices of the central charges. The first class constraints generate gauge transformations, involving fermionic $\kappa$-gauge transformations. The quantization of the model gives rise to the collection of free $\mathcal{N}{=}\,4$, $d{=}\,3$ Galilean superfields, which can be further employed, e.g., for description of three-dimensional non-relativistic $\mathcal{N}{=}\,4$ supersymmetric theories.


Introduction
In recent years, one can observe the growth of interest in the non-relativistic (NR) fieldtheoretic models, in particular those describing NR gravity and NR supergravity, e.g., in the framework of the so-called Newton-Cartan geometry [1,2,3,4,5].Until present, the NR supersymmetric framework [3,4,5] has been basically developed for D = 2+1-dimensional case, 1 which corresponds to the exotic version of Galilean symmetry with two central charges [6,7,8,9,10].In this paper we address the next physically interesting case of N = 4, D = 3+1 supersymmetric extension of Galilean symmetries.Due to the distinguished role of N = 4 supersymmetric Yang-Mills theory (see e.g.[11]), this kind of extended supersymmetry merits as well an attention in the NR case.
Similarly as in the relativistic case, one can study the NR N = 4, D = 3+1 supersymmetric theories by following several paths: i) One can start with NR N = 4, d = 3 Galilean superalgebras (for arbitrary N see [12]) and then construct their superspace and superfield realizations.In this way we obtain the universal tool for constructing NR supersymmetric field theories.
ii) The NR field theories can be reproduced by performing the non-relativistic contraction c → ∞ (c is the speed of light) in the known relativistic non-supersymmetric, as well as supersymmetric field theory models (see, e.g., [4,13,14]).One of the advantages of such a method is the possibility to derive the proper NR contractions of relativistic action integrals.
iii) For a definite type of (super)symmetric framework one can consider the dynamics associated with particles, fields, string, p-branes, etc.An important role in such a list is played by the free classical and first-quantized (super)particle models, with the property that their first quantization leads to the classical (super)field realizations (see, e.g., [15,16]).
In our case, we will look for the free superparticle models invariant under N = 4, d = 3 Galilean supersymmetry.One can mention that in the relativistic case this way of deriving free superfields from the classical and first-quantized superparticles with extended N = 4, D = 4 Poincaré supersymmetry was already proposed in [17]. 2n this paper we will follow the path iii).We will consider the most general NR N = 4, d= 3 Galilean superalgebra, introduce the corresponding N = 4 Galilean supergroup and its cosets, construct the relevant nonlinear realizations and use the associated Maurer-Cartan (MC) oneforms to build NR N = 4 superparticle models.They will be subsequently quantized to obtain the NR superfields providing realizations of N = 4 Galilean supersymmetries.Note that in such a setting the original coset parameters are treated as the D=1 world-line fields.However, the whole formalism could be equally applied along the lines of path i), with the coset parameters treated as independent NR superspace coordinates.
As a prelude to our considerations, we will describe the Galilean symmetries and their supersymmetrizations in a short historical survey.
The Galilean theories describe the low energy, non-relativistic dynamical systems 3 , which can also be obtained as non-relativistic limit (c → ∞) of the corresponding relativistic theories (see, e.g., [26]- [30]).Such a contraction limit, applied to D = 4 Poincaré algebra (P µ , M µν ; P µ = (P 0 , P i ), M µν = (M i = 1  2 ε ijk M jk , N i = M i0 )), after shifting and rescaling where H stands for non-relativistic Hamiltonian and B i for the Galilean boosts, yields "quantum" d = 3 Galilean algebra [31] 4 [J i , J j ] = iε ijk J k , (1. 2) The central charge M = m 0 describes a non-relativistic mass which can be identified with the relativistic rest mass.
Because bosons and fermions occur in both relativistic and non-relativistic settings, one can consider the non-relativistic supersymmetry as well.The first proposal for supersymmetrization of Galilei algebra (1.2) was given in [32], where N = 1 and N = 2, d= 3 Galilean superalgebras were presented.The N = 1, d= 3 Galilean superalgebra is an extension of relations (1.2) by complex NR USp(2) ≃ SU(2) supercharges S α , Sα ∶= (S α ) † (A → A † denotes Hermitian conjugation) which satisfy the relations 5  {S α , Passing to the N = 2 d= 3 Galilean superalgebra [32] is accomplished by adding to the N = 1 Galilean superalgebra generators (J i , P i , B i , H; S α , Sα , M) the second pair of complex SU(2) supercharges Q α , Qα ∶= (Q α ) † , subject to the following relations: (1.4) In the relations (1.4), (1.3), besides the central charge M, there appears the new central charge Y.The N = 2, d= 3 Galilean superalgebra can be derived from N = 2, D= 4 Poincaré superalgebra (a = 1, 2) (plus the commutation relations with Poincaré and internal R-symmetry U(2) generators) by taking the c → ∞ contraction limit with M = m 0 .In general, the N = 2, D= 4 Poincaré superalgebra is endowed with one complex central charge Z or, equivalently, two real central charges, Z = X + iY . 6.Before taking the NR limit c → ∞, the Galilean supercharges Q α and S α (see (1.3), (1.4)) should be identified with the following linear combinations of two N = 2 Weyl supercharges in (1.5) where and Q± α = (Q ± α ) † .Also, we should postulate the following c-dependence of the central charges in (1.5) If X is finite in the contraction limit, it merely generates the shift H → H + X in the relations of the N = 2, d= 3 Galilean superalgebra (see the first relation in (1.4)).
In the present paper we consider N = 4, d= 3 Galilean superalgebra with all possible central charges.It will be obtained by the c → ∞ contraction procedure from the general N = 4, D= 4 relativistic Poincaré superalgebra [37] which involves 6 complex central charges Z AB = −Z BA (A, B = 1, 2, 3, 4).Correspondingly, the N = 4, d= 3 Galilean superalgebra obtained in the c → ∞ limit involves 12 real central charges 7 .If these central charges are numerical, then, using a suitable redefinition of supercharges by an unitary 4 × 4 matrix, one can cast the antisymmetric 4×4 complex matrix of six central charges Z AB = −Z BA (A, B = 1, 2, 3, 4) into a quasi-diagonal Jordanian form [38,39] 6 Since N = 2 D= 4 Poincaré superalgebra is covariant under the phase transformation of Weyl supercharges one could think that one real central charge is enough in N = 2 case.However, as was found by studying concrete dynamical models [33,34], it is the complex N = 2 central charge Z = X 1 + iX 2 what actually matters.It amounts to two physical real central charges: the topological magnetic charge X 1 and the non-topological electric charge X 2 .Only if these charges take constant eigenvalues, i.e. are numerical, they can be rotated to the single central charge by the phase transformations just mentioned. 7In fact, the NR N = 4 Galilean superalgebra involves 13 central charges if we take into account the Bargmann central charge M = m 0 obtained from the leading terms in the asymptotic expansion of P 0 and X in c (see (1.1) and (1.8)).
where Such a structure of the internal sectors survives in the non-relativistic limit; one can therefore consider N = 4, d= 3 Galilean supersymmetric theories with the internal sectors USp(2)⊗USp(2) (four real Galilean central charges) or USp(4) (a pair of real Galilean central charges). 10n the most general N = 4 case, when we deal with six complex central charges, the central charge 4×4 matrix can be written as follows The central charges, besides bringing in the mass parameters, are also capable to simplify the formulation of N ⩾ 2 supersymmetric gauge theories.In particular, recall that N = 4, D= 4 Yang-Mills theory with one central charge and internal symmetry broken to O(5), contrary to N = 4, D= 4 supersymmetric Yang-Mills theory with SU(4) R-symmetry and without central charges, permits an off-shell superspace formulation which does not require harmonic variables [40,41].
The plan of the paper is as follows.In Sect.2, following [12], we derive the general N = 4, d= 3 Galilean superalgebra, which contains 12 independent real central charges and the additional thirteenth Bargmann central charge describing the rest mass.As in [26,27,28,29], in this derivation we employ the NR contraction c → ∞ of relativistic N = 4, D= 4 Poincaré superalgebra.In Sect. 3 we calculate the MC one-forms on the coset G H, where G = SG(3; 4 12) (see footnote 9) and stability subgroup H is given by SU(2) ≃ O(3) and USp(4) generators.In Sect. 4 we study the G-invariant actions linear in MC one-forms associated with central charges.For different choices of the central charges these actions describe various models of N = 4, d= 3 Galilean superparticles.We consider the phase superspace formulation of these superparticle models and present complete set of first and second class constraints.The first class fermionic constraints generate the non-relativistic N = 4 κ-gauge transformations which act in the non-physical part of the Grassmann coordinate sector.In Sect. 5 we quantize the model.Using super Schrödinger realization of quantum phase superspace algebra, we obtain as the quantum solutions of the model a set of free N = 4, d= 3 Galilean superfields.In Sect.6 we present an outlook, in particular, we describe briefly the alternative ways of constructing the N = 4 Galilean superparticle models.Concluding, we hope that our paper will contribute to the issue of superfield description of the interacting non-relativistic N = 4, d= 3 supersymmetric field theories. 11  2 General Galilean N = 4, d= 3 superalgebra with central charges The N = 4, D= 4 Poincaré superalgebra is spanned by the following generators iv) The set of 6 complex central charges Z AB = −Z BA , ZAB = (Z AB ) † , or equivalently the set of 12 real central charges X AB = −X BA , Y AB = −Y BA , where ) 11 For examples of supersymmetric extensions of QED and Yang-Mills Galilean theories see [42]- [44]. 12We define D = 4 sigma-matrices as follows: . Always in this paper we use weight coefficient in (anti)symmetrization: and the remaining non-zero commutation relations read Here α is some real parameter.If we choose α = 1 , it defines the chirality of supercharges (see (2.8)) and so identifies A as the generator of axial symmetry. 13n order to perform the non-relativistic contraction of N = 4, D= 4 Poincaré superalgebra to the limit describing N = 4, d= 3 Galilean superalgebra one should rewrite the superalgebra (2.3)-(2.8) in the new fermionic Weyl basis14 where the real 4×4 matrix In this paper we choose the following explicit form of Ω: where The relations (2.10) break manifest Lorentz symmetry O(3, 1) to O(3) (spinorial scalar product a α b α is U(2)-invariant) and the internal symmetry U(4) is broken to its subgroups which depend on the choice of central charges [39].
The supercharges (2.10) by definition satisfy the subsidiary symplectic-Majorana conditions [47] ( (2.12) The full set of supercharges Q ±a α , Q± αa ; Q ±ã α , Q± αã can be split into the holomorphic sector (Q ±a α , Q ±ã α ) and the antiholomorphic one Q± αa , Q± αã ; these both sectors are related by the subsidiary conditions (2.12), thus revealing the quaternionic structure of the pairs of complex supercharges related by Hermitian conjugation (see [48,45]).Due to the constraints (2.12) one can choose as unconstrained sets of linearly independent supercharges the generators from either holomorphic or antiholomorphic sectors.The N = 4 superalgebra spanned by the generators from holomorphic sector is however not self-conjugate.In order to define the complete selfconjugated Hermitian basis one should choose the full set of pairs of supercharges, which are related by Hermitian conjugation.For the choice (2.11) of the matrix Ω these self-conjugated pairs are In this paper we will use the supercharges belonging to the holomorphic sector, i.e.Q ±A α (A = (a, ã) = 1, ..., 4).They transform linearly under the USp(4) ≃ O(5) R-symmetry, which defines the compact R-symmetry sector of N = 4, d= 3 supersymmetry with one central charge Z corresponding to the following choice of 4×4 central charge matrix (1.10) In the holomorphic basis the non-vanishing relations (2.3), (2.4) can be represented as where The relations inverse to (2.17) are where It can be pointed out that X a b and Y a b are "real" with respect to the symplectic pseudoreality conditions similar to (2.12) and following from (2.18), The commutators (2.5) for the generators (2.10) can be rewritten as follows Further, let us decompose the generators of internal symmetry SU(4) as The projections (2.24) of T A B satisfy the relations The constraint (2.26) amounts to the conditions for the generators T ±A B , So, the set of generators T + contains 10 independent generators which are symmetric in their indices The set T − involves 5 independent operators forming a traceless antisymmetric matrix Using (2.1), we find that operators (2.28), (2.29) satisfy the following algebra Thus the original internal SU(4) symmetry generators T A B , decomposed according to the relations (2.24), do split into the ones generating USp(4) and the coset SU(4) USp(4): This decomposition of the su(4) algebra provides an example of symmetric Riemannian pair (h (3) , k (3) ): [h (3) , h (3) ] ⊂ h (3) , [h (3) , k (3) ] ⊂ k (3) , [k (3) , k (3) ] ⊂ h (3) . (2.33) The commutators (2.7) are rewritten in the new basis as where the 4×4 matrix defines the fundamental 4 × 4 representations of the USp(4) algebra given by the supercharges enlarge the matrices (U A B ) to the fundamental representations of SU(4) algebra which interchange the + and − projections.
Let us make a comment on the case of α ≠ 0 in (2.8), (2.9).Choosing α = 1 , one finds (2.37) Now we are prepared to define the N = 4, d= 3 Galilean superalgebra by making use of the NR contraction procedure.One rescales the relativistic supercharges as The physical rescaling of the bosonic generators of the algebra o(1, 3)⊕u(4), where (P µ , M µν ) ∈ o(1, 3) and (T ±A B , A) ∈ u(4), is performed as follows (2.39) where m 0 is the relativistic rest mass.The rescaling of the central charges is given by the formulas (see also (1.8)) where X AB , Y AB are defined in (2.17) and the operators X AB = −X BA , Y AB = −Y BA satisfy the symplectic pseudoreality conditions .We will firstly perform the c → ∞ contraction for a simple choice of the central charge matrix.

i) Jordanian quasi-diagonal form of central charge matrix
Let us consider the special case with central charge matrix in the reduced form (1.9) where the central charge matrix (2.14) is recovered at The rescaling (2.40) takes the more explicit form for this choice Substituting these expressions, as well as (2.38) and (2.39), into the superalgebra relations (2.20), (2.21) with Z a b = Z ãb = 0 , and making there the c → ∞ contraction, we obtain (2.46) , and the indices are chosen so that a = 1, 2 correspond to A = 1, 2 and ã = 1, 2 to A = 3, 4.

ii) General central charge matrix
In the general case with non-zero off-diagonal central charges X a b = −X ba and Y a b = −Y ba , the last lines in (2.44) and (2.46) are replaced, respectively, by the relations It is easy to check that the rescaling (2.38) preserves the symplectic-Majorana conditions (2.10) and in the limit c → ∞ one obtains the following Galilean form of N = 4 symplectic-Majorana conditions 15( Due to (2.18), off-diagonal central charges satisfy the following pseudo-reality conditions We point out that the constant m 0 can be considered as an additional thirteenth central charge, i.e. in fact the superalgebra (2.44)- (2.46)

iii) Internal symmetry sectors
After c → ∞ contraction (2.23), the covariance relations of the supercharges with respect to NR O(3) rotations J ij and Galilean boosts B i are written as Using substitutions (2.38), (2.39), the contraction of the relations (2.34) leads also to the covariance relations of supercharges with respect to the internal symmetry generators h For what follows, it will be useful to have the generators T +A B ∈ usp(4) in the splitting usp( 2)⊕usp( 2) basis.This notation corresponds to the following coset decomposition such that From (2.39) it follows that the coset generators k (3) are rescaled (h ) and in the limit c → ∞ we get the inhomogeneous extension of usp(4) ≅ o(5) internal algebra [h (3) , h (3) ] ⊂ h (3) , The five commuting generators T −A B of k (3) describe a kind of curved internal momenta.Thus in the contraction limit c → ∞ one gets the following N = 4 Galilean internal inhomogeneous symmetry algebra (2.59) We will denote the corresponding inhomogeneous group by IUSp(4) .The abelian generator A can be added to the ideal k (3) , so extending it to six-dimensional one. 16he action of the IUSp(4) generators in the USp(2) ⊗ USp(2) splitting basis, , as well as on the central charges, can be easily found from the relations (2.52).For instance, the commutation relations between T +a b and central charges are given by and by similar formulas for Y a b , Y 1 , Y 2 .It follows from these relations that the full set of central charges splits into two USp(4 The first relation in (2.53) amounts to the following set of relations in the splitting basis (2.61) The commutation relations between T −A B and the central charges or, in the splitting basis, The commutation relations between the U(1) axial generator A and the central charges have a similar structure.: Our last remark concerns the N = 4 Galilean algebra with the diagonal choice (1.9), (2.41) for the central charge matrix.Recalling (2.44) -(2.46), we observe that in this case N = 4 Galilean algebra (with suitable restriction of R-symmetry algebras taken into account) reduces to the sum of two N = 2 Galilean superalgebras spanned by the supercharge pairs (Q a α , S a α ), (Q ã α , S ã α ), with common generators H and P i .The only way to avoid such a splitting is to switch on the off-diagonal central charges as in (2.47).If we consider an extended N = 4 Galilean algebra, with the R-symmetry generators T +A B included, the N = 2 subsectors in (2.44) -(2.46) will be intertwined by the generators T +a b , e.g., In this case, the splitting into two N = 2 algebras arises only when we eliminate the generators T +a b from the R-symmetry algebra.
To avoid a possible confusion, note that the R-symmetry is described by the group of outer automorphisms of superalgebras and its generators do not appear in the r.h.s. of the (anti)commutators (distinctly from central charges).Therefore, when constructing the specific models, we can restrict the R-symmetry group to some of its subgroup.The maximal Rsymmetry group USp(4) ∼ O(5) can be ensured in two distinct cases: either for the choice (2.14) with Z being USp(4) ∼ O( 5) invariant (the same if Z is an operator or a number), or for the generic choice (1.10), with 5) vectors (see (2.60)), and with two O(5) singlets (X 1 + X 2 ), (Y 1 + Y 2 ) accommodating the remaining two central charges.
In the second case one has an additional freedom to eliminate, without breaking O(5) covariance, either all Y central charges or all X central charges, and further choose, e.g., X 2 + X 1 = 0 or Y 2 + Y 1 = 0 .As was already mentioned, with the general option (1.10) the choice of numerical central charges necessarily breaks O(5) R-symmetry down to O(3) .

iv) Hermitian basis
One can alternatively formulate NR N = 4, d = 3 superalgebra by using NR contraction of Hermitian pairs of supercharges which are self-conjugate with respect to Hermitian conjugation (see (2.13)).We define the set of unconstrained independent supercharges spanning N = 4 Galilean Hermitian superalgebra as follows (2.69) The Hermitian form of N = 4, d= 3 Galilean superalgebra (2.67)-(2.69)permits to obtain the generalized positivity conditions for the Hamiltonian H. From first two formulas in (2.67) one derives that for any normalized state Ψ⟩ belonging to the Hilbert space of physical states of the models the following conditions hold (2.70) In dynamical models (like those of Sect.4) the central charges X 1 , X 2 are represented on the normalized states Ψ⟩ by the mass-like parameters m 1 , m 2 , so from (2.70) one gets the lower bound on the energy values 3 Nonlinear realizations of N = 4, d= 3 Galilean supersymmetries In the nonlinear realization of N = 4, d= 3 Galilean supersymmetries we will assume that the linearization subgroup H involves the 3-dimensional space rotations generators J ik , the internal symmetry USp(4) generators T +A B and the abelian generator A 0 .All other generators are placed in the coset G = SG(3; 4 12) H.Some of the parameters belonging to G can be relocated into the linearization subgroup H just by nullifying the respective coset parameters.The coset element G can be written explicitly as where The factors The odd generators satisfy the symplectic-Majorana conditions (see also (2.12)) 3) The Grassmann coordinates dual to these odd generators satisfy similar pseudo-reality conditions Being dual to the relations (2.49), the reality conditions for the tensorial central charge coordinates read The full set of the left-covariant MC one-forms is given by where A straightforward calculation yields The remaining part of (3.7) is as follows We can write the formula (3.7) in the following way where T (K) stand for all coset G generators, and ω(K) denote the corresponding MC one-forms.We obtain where k 2 ∶= k i k i .The MC one-forms describing the whole coset G are defined as follows and can be calculated by the formulas (3.2) and (3.6).We observe that and, further, We see, in particular, that Let us find supersymmetry transformations of the coset coordinates.For this purpose we will use the well known formula iG −1 (ε ⋅ T )G = G −1 δG + δh, where T denotes the collection of coset generators and δh defines induced transformations of the stability subgroup h ind = 1 + δh (see, e.g., [49]).
Supersymmetry transformations generated by the left action of generators where ε α a , ε α ã are odd constant parameters, lead to the following transformations of the coset coordinates: The second half of the odd left shifts, those generated by S a α , S ã α , lead to the transformations where η α a , η α ã are the appropriate odd parameters.It follows from (3.18) and (3.20) that the three-vector k i and "harmonic variables" u are inert under all supersymmetry transformations.
From the form of the supersymmetry transformations (3.18) follows that the set of coordinates (t; ξ α a , ξ α ã ; h 1 , h 2 , h a b) is closed under the action of Q -supersymmetry, while this supersymmetry does not act on the remaining coordinates ( Alternatively, S -supersymmetry (3.20) leaves inert the subset (t; ξ α a , ξ α ã ; h 1 , h 2 , h a b) and transforms the remaining coordinates ( This split of the full set of coset parameters into two subsets, each closed under the action of one half of the supersymmetries and inert under another half, is due to the choice of coset parametrization (3.1), (3.2) with the particular order of the factors G 2 and G 3 , G = . . .G 2 G 3 . . . .In [36], there was used another parametrization, G = . . .G 3 G 2 . .., and the separation of Q -and S -transformations into two sectors could not be seen.
The closure of the transformations (3.18) and (3.20) generates all the bosonic transformations of G which do not belong to the stability subgroup H.The transformations of subgroup H are realized as some linear homogeneous maps of the coset fields and MC 1-forms.The abelian generators T −A B do not appear in the closure of fermionic generators, so the left shifts by these generators should be considered separately.The corresponding transformations of the coset parameters can be found explicitly, using the formulas (3.15).The coset parameters u A B are changed only by the pure shifts.Actually, in this paper we will not make use of these T −A B transformations.
We also observe that all MC forms ω(K) (see (3.12)) transform linearly under H transformations and are inert with respect to the odd transformations (3.18) and (3.20).4 The phase-space formulation of N = 4, d = 3 Galilean superparticle model and κ-gauge freedom Let us describe the mechanical system on the coset G with evolution parameter τ and with all parameters of G promoted to the d= 1 fields: t = t(τ ), We shall deal with the simplified situation, with all internal coordinates u A B being suppressed, which means that we transfer the generators T −A B into the stability subgroup and use the "truncated" MC one-forms ω(K) .We will not employ the strict invariance of the superparticle actions under these abelian outer automorphisms, as well under the full compact R-symmetry USp(4).Only the symmetries under some particular subgroups of the latter, as well as the O(3) space symmetry generated by J ij , will be respected.

Simplest bosonic case: Schrödinger NR particle
As the instructive step we consider the standard bosonic Schrödinger particle.We recall how to derive the action of non-relativistic massive particle, which, after quantization, leads to the non-relativistic Schrödinger equation.
Such an action is obtained from the MC one-form (4.5), which in the pure bosonic case is given by Selecting the rest mass as the normalization factor and omitting a total τ derivative, we obtain It leads to NR particle model studied in [26,36].
The action (4.7) provides the canonical momenta and the vanishing canonical Hamiltonian: The expressions (4.8) imply the first-class constraint defining free NR energy-momentum dispersion relation called free Schrödinger constraint which, after quantization in the Schrödinger realization, gives the non-relativistic Schrödinger equation for a free NR particle of mass m 0 .

The superparticle model with vanishing off-diagonal central charges
As the next step, we consider the action with the Lagrangian density taken as a linear combination of the MC one-forms associated with central charges described by Jordanian quasidiagonal form of the central charge matrix where a, m 1 , m 2 , µ 1 , µ 2 are real constants.The choice of these parameters specifies the explicit form of odd constraints, including the first class ones generating local κ-symmetries.
Using the expressions of the MC forms (3.12), (4.5) and omitting total derivative terms, we get from (4.12) the Lagrangian where α were defined in (4.3).Without the loss of generality, the terms proportional to a in (4.13) can be omitted because they can be re-absorbed by the redefinition of m 1 and m 2 .Therefore we will put a = 0 (see the same condition in [36], assumed, however, for another reason).Then the Lagrangian (4.13) produces the following bosonic momenta and the fermionic ones where p x αβ ∶= p xi (σ i ) αβ .Using the canonical Poisson brackets, {t, we find the non-vanishing Poisson brackets of the classical NR supersymmetry generators (4.17):The definitions (4.15) of fermionic momenta lead to the constraints: Using (4.18), we obtain the non-vanishing Poisson brackets for the system of constraints (4.21) and (4.22) We will be interested in the NR superparticle models possessing local fermionic κ symmetry [51,52], after imposing suitable relations between the parameters of the model (see, e.g., (4.26), (4.27) below).In the N = 2 , d = 2 case this kind of NR superparticles was considered in [36].We recall that in the phase space formulation, κ symmetry is generated by the first class odd constraints.
Let us determine the values of central charges in the model which imply the first class odd constraints.For that purpose we should calculate the determinant of the 16-dimensional matrix of the Poisson brackets of fermionic constraints (4.23), in the presence of the bosonic constraint (4.21), and assume that this determinant becomes zero. Defining we find that the determinant of the matrix P of the fermionic constraints (4.23) is given, modulo a multiplicative constant, by the expression Thus, first class odd constraints are present at least under one of the following two conditions If the condition (4.26) is valid, half of the odd constraints linear in D ξ a α , D θ a α are first class.Explicitly, these constraints are The Poisson brackets involving the constraints (4.28) form the following set We see that the full set of the original constraints (D ξ a α , D θ a α ) is equivalent to the set (F ξ a α , D θ a α ), where F ξ a α are first class, and D θ a α are second class.The analysis of the second half of the odd constraints (with tilded indices) is performed quite analogously.If the condition (4.27) holds, the constraints are first class, with the following Poisson brackets with all odd constraints The set of constraints (D ξ ã α , D θ ã α ) is therefore equivalent to the set (F ξ ã α , D θ ã α ), with F ξ ã α being first class and D θ ã α second class.In Section 5 we will study the quantization of the superparticle model defined by the action (4.12), (4.13) possessing κ-symmetries due to the presence of conditions (4.26) and (4.27).Using (4.22), we obtain the following explicit form of the first class constraints (4.28) and (4.30) generating κ-symmetries We notice that, if we specialize our discussion of constraints to one sector only, with either index a or index ã, we recover the model with smaller N = 2 Galilean supersymmetry.Such models in d = 2 case have been studied in [36] for the case of N = 2, d = 2 Galilean supersymmetry with only one central charge.The d = 2 models of [36] can be obtained as a special case of our model with only one of the N = 2 sectors retained.
The constraints (4.32) generate local κ-transformations of an arbitrary phase space function X by the following Poisson bracket where κ α a (τ ) and κ α ã (τ ) are local Grassmann parameters.The κ-transformations (4.33) of the variables in the Lagrangian (4.13) with a = 0 are as follows Under these transformations, the variation of Lagrangian (4.13) (with a = 0) is Using local transformations (4.34) one can choose the gauge ξ a α = 0, ξ ã α = 0 .In such a gauge, the rigid Q -transformations (3.18) should be accompanied by the appropriate compensating gauge transformations (see, e.g., [36]).In this case, as well as in other cases considered below, we will not impose such gauges, reserving it for more complicated N = 4 NR superparticle models still to be constructed, e.g., those formulated on external electromagnetic background.

The superparticle model with all central charges incorporated
We can add to the action (4.12) at a = 0 the additional terms associated with off-diagonal central charges where n a b and ν a b are constants with the reality conditions as for USp(2) ⊗ USp(2) ≃ O( 4) bispinors (see (2.49), (3.5)) These bi-spinorial constants can be represented as internal four-vectors (isovectors) where  4) invariance in the sector described by off-diagonal central charges, will be discussed in Sect.6.
The Lagrangian in (4.39) is given by the following expression In comparison with (4.13), the Lagrangian (4.40) contains two additional terms, which involve only derivatives of ξ's.Therefore, the momenta p t , p xi , p θ a α , p θ ã α are the same as in (4.14), (4.15), whereas the momenta p ξ a α , p ξ ã α acquire additional terms as compared with (4.15).Then it follows that the bosonic constraint T ≈ 0 (see (4.21)) and the fermionic constraints D θ a α ≈ 0, D θ ã α ≈ 0, defined by (4.22), remain the same, whereas fermionic constraints D ξ a α ≈ 0, D ξ ã α ≈ 0 will acquire additional terms in comparison with (4.22): For the model with Lagrangian (4.40) the Noether charges generating the NR supersymmetry transformations (3.18) and (3.20) contain, in comparison with the expressions (4.17), some additional terms and take the form

Analysis of the constrains
The determinant of the Poisson brackets matrix P of the fermionic constraints (4.41), (4.22) defined in (4.24) in the case under consideration looks more complicated than (4.25).It is equal, up to a multiplicative constant, to the following expression: where ν ∶= T is the first class constraint (4.21), and the 4-vector w a b = i(σ M ) a bw M is defined by It should be added that n = 1 2 n a bν a b = n M n M and ν, ŵ are the length squares of O(4) internal symmetry vectors, i.e. one gets that n ≥ 0, ν ≥ 0 and ŵ ≥ 0.Moreover, n = 0 leads to n a b = 0 as well as ν = 0 leads to ν a b = 0; similarly, ŵ = 0 implies w a b = 0 .
The odd first class constraints generating κ-symmetry, are present provided that The constants n a b, ν a b enter the expression (4.45) and the equation (4.48)only through two quantities, ν and ŵ defined by (4.46).Since (4.45) is not factorized, in contrast to (4.25), resolving eq.(4.48) is a more complicated task.
The condition (4.48) is necessary for (any number of) odd first class constraint.The full number of such constraints is found by solving the characteristic equation which determines the eigenvalues λ of the matrix P (see analogous consideration, e.g., in [53,54]).In (4.49), λ describes the spectral parameter and I is the unit matrix.The number of first class constraints is equal to the number of solutions λ = 0 of the characteristic equation (4.49).
In the presence of k odd first class constraints among sixteen constraints D A , the equation (4.49) has the form In the model considered here the characteristic equation (4.49) coincides with the equation (4.48) in which the substitutions p t → (p t − λ) and m 0 → (m 0 − λ) are performed.Using the expression (4.45)18 the characteristic equation (4.49) can be written in the following form where As we see from (4.51), the condition (4.48) implies the presence of at least four odd first class constraints in the total set of sixteen fermionic constraints (4.41), (4.22).
The condition A = 0, together with (4.48), lead to the presence of eight odd first class constraints.The condition A = 0 requires vanishing those terms in (4.52) which are proportional to p t : An additional condition which stems from A = 0 is the vanishing of the remaining constant term: For m 1 = m 2 it leads to the condition n a bw a b = 2n M ν M = 0. Further, one can show that B ≠ 0 and C ≠ 0 in (4.51) due to non-vanishing constant coefficients in (4.53) and (4.54) in front of (p t ) 2 and p t .
Thus, by definite choices of central charges, we can recover the cases, when the number of odd first class constraints is quarter or half the total number of odd constraints.Note that, up to a sign, the algebra of the fermionic constraints (4.23), (4.42) coincides with the NR superalgebra (4.19), (4.44) and the number of the first class constraints equals the number of preserved supersymmetries in BPS configurations.Therefore, respective models describe BPS configurations preserving 1 4 or 1 2 of NR supersymmetry.
In the last part of this Section, we will consider in detail two special cases.

The case when half of odd constraints is first class
This particular example is specified by the following condition on isotensorial central charges: or, equivalently, In this case the vanishing of the quantity (4.45) (with T ≈ 0) requires that The relation (4.59) is obeyed provided at least one of two conditions or both of them are fulfilled.The conditions (4.60), (4.61) are the obvious generalizations of (4.26) and (4.27).Now we present the full set of the constraints which occur when the conditions (4.60), (4.61) and (4.58) are valid.
The first class bosonic constraint (4.21) represents the Schrödinger equation as in the previous cases.
Fermionic constraints , where (F ξ a α , F ξ ã α ) are defined by the following expressions The complete set of non-vanishing Poisson brackets for the constraints (4.62), (4.63) reads We see that if the conditions (4.60), (4.60) and (4.58) are valid, the constraints (D θ a α , D θ ã α ) are second class, while the constraints (F ξ a α , F ξ ã α ) are first class and generate κ-symmetries.Substituting the expressions (4.41) into the κ-symmetry generators (4.62), (4.63), we obtain them in the following explicit form As opposed to the constraints (4.32) in Sect.4.2, in the considered case the constraints (4.65) mix two USp(2) sectors characterized by untilded and tilded USp(2)-indices.It turns out, however, that this model for m 1 ≠ m 2 and µ 1 ≠ µ 2 is just the model of Sect.4.2 in disguise.To show this, recall that the full Lagrangian (4.40) is formally invariant under the simultaneous O(5) rotation of the d = 1 fields and the set of coupling constants m 1 , m 2 , µ 1 , µ 2 , n a b, ν a b, which gives an opportunity to pass to O(5) frame where these constants are reduced to some minimal set. 19It is important that the coupling constants are divided into the O(5) singlets m 1 + m 2 , µ 1 + µ 2 and O(5) vectors (n a b, m 1 − m 2 ) and (ν a b, µ 1 − µ 2 ); then the condition (4.58) means the vanishing of some particular linear combination of the O(4) vector components of these two O(5) vectors.The fifth component of the O(5) vector containing the O(4) vector (4.58), is This quantity is zero just as a difference of the conditions (4.60), and(4.61)!The rest of these conditions, their sum, expresses the particular O(5) invariant (m 1 + m 2 ) through other one, To construct a system with eight first-class constraints, which would be non-equivalent to the system of Sect.4.2 and involve the constants n a b, ν a b which cannot be removed, one needs to break explicitly the USp(4) ≃ O(5) covariance in the space of coupling constants.The simplest option is to assume The USp(4) ∼ O(5) covariance is also explicitly broken in a system with four first-class constraints corresponding to 1 4 BPS states.We will consider it as the second example.

The case when quarter of odd constraints is first class
Our second example is characterized by non-vanishing off-diagonal central charges, with all quasi-diagonal ones vanishing:   5 Quantization of the model and N = 4, d = 3 Galilean superfields In this section we present the canonical operator quantization of our model.We will introduce the (super)Schrödinger realization of quantum phase coordinates and obtain the superfield description of N = 4, d = 3 Galilean states.For this purpose we will quantize the second class constraints by the Gupta-Bleuler (GB) procedure [55,56], without introducing for them the Dirac brackets.
We will consider two versions of our model: the first one with eight first class constraints (introducing 1  2 BPS states or fraction 1 2 of unbroken supersymmetry) and the second one with four first class constraints (introducing 1  4 BPS states or fraction 1  4 of unbroken supersymmetry).We will use, instead of the symplectic-Majorana real quantities, the complex Hermitian conjugate Grassmann coordinates, which are defined by (3.4) in the following way The corresponding complex momenta, which are explicit solutions of the conditions (4.16), are Poisson brackets of these phase superspace variables are given by {ξ α , p In the quantization procedure, we will use the graded coordinate representation with the following super-Schrödinger realization for the momenta: (5.4)
Even first class constraint (5.9) yields the Schrödinger equations for all component fields.Odd first class constraint (5.10) provide the following four superwave equations for the superfield (5.13) Here we have introduced the operators ∆ θα , ∆θ α , ∆θα , ∆θ α which do not depend on ξ-variables, and the covariant derivatives for N = 8 extended one-dimensional supersymmetry which form two mutually anticommuting with constants m 1 , m 2 playing the role of central charges.It is straightforward to check that the integrability condition for the equations (5.14) is just the Schrödinger equations (5.9) (we mention that the conditions (4.26), (4.27) are valid).
To summarize, physical states of the considered model are described by the two-chiral superfield Φ 0 (t, x i , θ α , θα ) with the following component expansion In (5.18) all component fields are complex functions of t and x i which satisfy the Schrödinger equation (5.9).Thus the presented model results in five complex scalar fields A(t, x i ), B(t, x i ), B(t, x i ), B [αβ] (t, x i ), C(t, x i ) which describe spin 0 states; one complex vectorial field B (αβ) (t, x i ), which accommodates spin 1 states; and four spinorial fields Ω α (t, x i ), Ωα (t, x i ), Λ α (t, x i ), Λα (t, x i ) corresponding to spin 1 2. It is easy to see that we obtain in such a way equal number of 8 bosonic and 8 fermionic component fields.
It is worth to emphasize that the description of physical states by the two-chiral superfield (5.18) is consistent with the possibility of imposing the gauges ξ a α = 0, ξ ã α = 0 on the local transformations (4.34), as was mentioned in Sect.4.2.

The models with non-vanishing off-diagonal central charges
Like in the previous subsection, we will use the complex variables (5.1), (5.2).

The models with eight odd first class constraints
We consider firstly the case with condition (4.58).The wave function has the form (5.8) and the wave equations are derived from even first class constraint (5.9) as well as odd first and second class constraints, the last ones quantized by the Gupta-Bleuler quantization method.
Second class constraints have the same form as in (5.7), while the solution of the constraints (5.11) leads to the reduced superwave function (5.12), (5.13).
For simplicity we will restrict our study to the case of parallel four-vectors n where (see (4.58)) Other choices of the constants n a b and ν a b satisfying (4.58) lead to the same set of physical states.
Using complex variables (5.1), (5.2), the odd first class constraints (4.65) can be brought to the form These expressions generalize the ones given in (5.6).Imposing the constraints (5.10) on the superwave function (5.12), we obtain that the superfield (5.13) satisfies the following generalization of four superwave equations (5.14) where and D ξ α , Dξ α , D ξ α , Dξ α are defined in (5.15).If the conditions (4.60), (4.61) and (4.58) are valid once again, the integrability condition for the system of equations (5.23), (5.24) is the free Schrödinger equations (5.9).The unconstrained superfield can be found as in the previous subsection: we apply the expansion (5.17) and again find that all superfield components in this expansion are expressed through the derivatives of single two-chiral superfield Φ 0 (t, x i , θ α , θα ) (see (5.18)).Therefore, the considered model with off-diagonal central charges has the same physical fields content as the previously studied model with the diagonal central charges only.
Finally, we would like to notice that both types of chiral superfields describing superwave functions in this Section are essentially on-shell since they require Schrödinger equation as the integrability condition of the relevant odd first class constraints.

Conclusions
In this paper we considered the N = 4, d= 3 NR superparticle models with twelve constant central charges transforming in certain representations of USp(4) ∼ O(5) and USp( 2 It should be added that rest mass m 0 , describing the Bargmann central charge in Galilean sector, can be treated as the thirteenth central charge, which does not break any R-symmetry. The superparticle action, constructed in Sect. 3 and Sect. 4 of the present paper, is linear in the MC one-forms associated with central charges.The numerical coefficients in front of the central charge MC one-forms provide the numerical values of central charges.In our further work we plan to consider also alternative action densities as nonlinear functions of MC forms, which would permit, e.g., a generalization of (4.36) to the action manifestly invariant under the O(4) internal symmetries.In particular, following the construction of the model for free relativistic massive particle 20 , one can replace the action (4.36) by the USp(2)⊗USp(2)-invariant action depending on all eight off-diagonal central charges where k 1 , k 2 , k 3 are constant and ω a b (X) , ω a b (Y ) are defined in (3.12).The system described by the action S 1 + S ′ 2 produces the same fermionic constraints (4.41) where, however, n a b and ν a b are not constant anymore: they become the canonical momenta for the tensorial central charge coordinates h a b and f a b (see Sect. 3, eqs.(3.2) and (3.12)).So, although the fermionic constraints have basically the same form in both models (see (6.1) and (4.36)), in the case of the action (6.1) the group parameters h a b and f a b are introduced as the dynamical tensorial central charges coordinates.In such a way we deal with an extension of the bosonic target space sector (t, x i ) describing NR space-time to an extended target space with auxiliary central charge coordinates (t, x i ; h a b, f a b).Additional coordinates h a b, f a b enter into new three bosonic constraints which fix n = n M n M , ν = ν M ν M , n M ν M (see (4.46)) by (k 1 ) 2 , (k 2 ) 2 , (k 3 ) 2 .In such a way we obtain a sort of Kaluza-Klein (KK) extension of the superparticle model, with auxiliary KK bosonic dimensions represented by central charge coordinates.Analysis of this modified N = 4 NR superparticle model will be given elsewhere (for an early attempt in this direction see [57]).
In the future we plan also to examine another way of preserving internal symmetry by using the harmonic type variables u b a and u b ã which occur in the coset space parametrization (3.2), as well as the "genuine" harmonic variables defined for the R-symmetry group USp(4). 21urther direction for the future study is to couple the NR superparticle presented here to electromagnetic, YM and supergravity backgrounds.It can be important for the following reason.The energy momentum dispersion relations in our model for arbitrary spin states are described by the free Schrödinger equation depending on the same mass parameter m 0 .One can argue that, after switching on the background fields, other central charges will also become dynamically active and will contribute to the modification of Schrödinger equation.
) where a, b = 1, 2 (ã, b = 1, 2) are the left (right) USp(2) ≃ SU(2) spinor indices.The four complex central charges Z a b constitute complex O(4, C) isovector Z M = 1 2i (σ M ) ãb Z bã , where (σ M ) ãb are D = 4 Euclidean Pauli matrices σ M = (σ i , i1 2 ).If Z M = 0 (i.e., the central charge matrix is reduced to (1.10)) we deal with the decomposition of N = 4 Galilean superalgebra into the direct sum of two N = 2 Galilean superalgebras, each possessing USp(2) automorphism; if Z M ≠ 0 the decomposition of N = 4 Galilean supersymmetry into such a sum of two N = 2 superalgebras is not possible.As we will see, in the absence of central charges the full compact internal R-symmetry in the NR case is U(1)⊗USp(4) as opposed to U(4) of the relativistic N = 4, D = 4 superalgebra.If the central charges take numerical values, the presence of off-diagonal supercharges (1.10) provides the breaking of USp(2)⊗USp(2) ≃ O(4) ⊂ USp(4) internal symmetry (still preserved by the diagonal central charges) down to the exact O(3) or O(2) internal symmetries which form diagonal subgroups in the product O(3)⊗O(3) = O(4).
generators and 10 symmetric ones G A (s) B = 1 2 (G A B + G B A ) describe the coset U(4) O(4).The axial U(1) generator A = G A (s) A can be separated out, i.e.U(4) = SU(4) ⊗ U(1), where SU(4) generators T A B = G A B − 1 4 δ A B A are traceless, T A A = 0, and satisfy the relation

. 43 )
The set of non-vanishing Poisson brackets between the classical supersymmetry generators (4.43) involves the relations (4.19) and, in addition, the following Poisson brackets{Q a α , Q b β } = 2i ǫ αβ n a b , {Q a α , S b β } = {Q b β , S a α } = −2 ǫ αβ ν a b .(4.44)The Poisson brackets (4.44) are classical counterparts of the anticommutators (2.47).We see that the constants n a b and ν a b of the general model (4.40) reappear at the level of Poisson brackets in place of the central charges X a b and Y a b .
.69) In this case it follows from (4.47) that w a b = n a b.Further, vanishing of the quantity (4.45) required for the presence of odd first class constraints leads to the condition (ν) 2 = 4(m 0 ) 2 n .(4.70)If we wish to have eight odd first class constraints with the conditions (4.69), the relations (4.55) and (4.70) (as consequences of the condition A = 0) are valid, and they imply that n = 0, ν = 0 , whence ŵ = 0 and, further, n a b = ν a b = 0 .Thus, if non-vanishing central charges n a b, ν a b are present and m 0 ≠ 0, we can only obtain four odd first class constraints.Let us separate now odd first and second class constraints.The initial fermionic constraints (D ξ a α , D ξ ã α ; D θ a α , D θ ã α ) are equivalent to the set (G a α , F ã α ; D θ a α , D θ ã α ), where (G a α , F ã α ) are defined by the following expressions

. 7 )
Second class constraints (5.7) form the Hermitian conjugate pairs D θα , Dθ α and D θα , Dθ α , what permits us to apply Gupta-Bleuler quantization.In accord with this quantization technique we impose on wave function half of second class constraints, i.e., Dθ α and Dθ α .

F
α Φ = Fξ α − 2m 0 n ν Fξ α Φ = Fξ α Φ = 0 .(5.31)Using (5.23),(5.24),we obtain that the equations (5.30), (5.31) amount to the following pair of equations for the superfield (5.29) Ω = 0 , (5.32)Dξ α − nξ α − 2 θβ p x β α − 2i νθ α Ω = 0 .(5.33)One can consider its general expansion with respect to Grassmann coordinates ξα , ξα Ω = Ω 0 + ξα Ω α + ξα Ωα + ξα ξα Ω 1 + ξα ξα Ω1 + ξα ξβ Ω αβ + ⋯ , (5.34) )⊗USp(2) ≃ O(4) ⊂ O(5) internal R-symmetry groups.The maximal U(4) R-symmetry group of relativistic N = 4 , D = 4 superalgebra in the NR contraction limit is reduced to a semi-direct product of the compact R-symmetry group USp(4) ≃ O(5) and some abelian six-dimensional commutative ideal.In the dynamical framework of our superparticle model, after quantization the central charges are identified with constant parameters of the underlying world-line Lagrangian.Depending on the specific non-vanishing values of these central charges, we are left, before any Hamiltonian analysis, with different fractions of unbroken internal symmetry, G int ⊂ USp(4) ≃ O(5), namely a) If only one central charge Z (see (2.14)) is present, the maximal R-symmetry O(5) of NR N = 4 superalgebra remains in the model; the central charge is O(5) singlet; b) If we have two quasi-diagonal central charges (see (1.9)) the internal symmetry is broken to G int = USp(2)⊗USp(2) ≃ SU(2)⊗SU(2) ≃ O(4).The central charges are presented by four USp(2) singlets m 1 , m 2 , µ 1 and µ 2 ; c) Adding the off-diagonal central charges described by two arbitrary constant O(4) isovectors n a b and ν a b (see (4.38)) radically changes the situation.At m 1 ≠ m 2 and µ 1 ≠ µ 2 the set of twelve constant central charges determines two O(5) vectors (n a b , m 1 − m 2 ) and (ν a b , µ 1 − µ 2 ) and two O(5) singlets (m 1 + m 2 ) and (µ 1 + µ 2 ).The O(5) frame can be fixed so that one of these O(5) vectors carries only one non-zero component, say m 1 − m 2 .There still remains O(4) covariance which can be further restricted in such a way that another O(4) vector ν a b (see (4.38)) will have only one non-zero component, ν a b = ǫ a bν .Thus in such R-symmetry frame we end up with five independent constant central charges and O(3) as the residual R-symmetry group; d) If m 1 = m 2 and/or µ 1 = µ 2 , the O(5) covariance is reduced to O(4).Hence we can choose the frame where O(4) isovector n a b contains only one non-zero component, n a b = ǫ a bn , and the residual R-symmetry group is O(3).The second non-parallel O(4) vector ν a b can be split into O(3) singlet and vector parts, with only one non-zero vector component, ν a b → (ǫ a b ν, δ a bν 2 ), that reduces R-symmetry O(3) to the minimally possible one, given by O(2).Thus in this particular frame we end up with six (or less) independent constant central charges and O(2) as the minimal exact internal symmetry in our model.
.19) The Poisson brackets (4.19) are the classical counterparts of the anticommutators (2.44)-(2.46).We see that in the model (4.13) the parameters m 1 , m 2 and µ 1 , µ 2 generate the constant central charges X 1 , X 2 and Y 1 , Y 2 . 17Canonical Hamiltonian of the model (4.13) is vanishing as in the bosonic case H 1 = p t ṫ + p xi ẋi + p ξ .67) Thus it follows that the conditions (4.58) and (4.60), (4.61) lead to the vanishing of one out of two independent O(5) vectors in the space of coupling constants and as well relate with each other some O(5) invariant combinations of these constants.Thus these conditions preserve O(5) covariance, and one can still use the O(5) rotations in order to choose the frame where n a b (or ν a b) are zero.In such a frame, due to (4.58), both O(4) vectors are zero and we are left with the constraints (4.26), (4.27) as the only remaining ones.So for the off-diagonal central charges satisfying the conditions (4.58) our model becomes identical to the one considered in Sect.4.2.
.73) Thus, if the condition (4.70) is valid, four constraints F ã α , defined in (4.72) are first class.The constraints D θ a α , D θ ã α and G a α , defined in (4.71) are second class.
as in the previous cases.The remaining odd constraints are second class constraints G a α , defined in (4.71), and first class constraints F ã α , defined in (4.72).The second class constraints (4.71) coincide with the constraints (5.21), (5.23) taken at m 1