Quantum SU(2$|$1) supersymmetric Calogero-Moser spinning systems

${\rm SU}(2|1)$ supersymmetric multi-particle quantum mechanics with additional semi-dynamical spin degrees of freedom is considered. In particular, we provide an $\mathcal{N}{=}\,4$ supersymmetrization of the quantum ${\rm U}(2)$ spin Calogero-Moser model, with an intrinsic mass parameter coming from the centrally-extended superalgebra $\widehat{su}(2|1)$. The full system admits an ${\rm SU}(2|1)$ covariant separation into the center-of-mass sector and the quotient. We derive explicit expressions for the classical and quantum ${\rm SU}(2|1)$ generators in both sectors as well as for the total system, and we determine the relevant energy spectra, degeneracies, and the sets of physical states.


Introduction
The many-particle Calogero-Moser systems [1,2,3,4,5] and their generalizations occupy a distinguished place in the contemporary theoretical and mathematical physics. Apart from such notable mathematical properties, as the classical and quantum integrability, these systems possess a wide range of physical applications which are hard to enumerate. Among these applications, it is worth to mention, e.g., a close connection between the algebra of observables in the Calogero system and the higher-spin algebra, as was pointed out in [6,7]. In the same papers, there was revealed an important role of Calogero-like models for describing particles with fractional statistics. Another widely known applications of the Calogero-Moser systems concern the black hole physics. It was suggested [8] that the Calogero-Moser systems can provide a microscopic description of the extreme Reissner-Nordström black hole in the nearhorizon limit. It was argued that, from the M-theory perspective, an important role in this correspondence should be played by various N = 4 supersymmetric extensions of the Calogero-Moser models. Supersymmetric Calogero-Moser systems have also further applications in string theory (see, for example, [9]) and N = 4 super Yang-Mills theory [10,11].
Keeping in mind these physical and mathematical motivations, it seems of great interest to construct and study new versions of supersymmetric Calogero-type systems.
The matrix superfield X = X (ζ H ) is defined on the SU(2|1) harmonic superspace ζ H ≡ t A , θ ± ,θ ± , w ± i (i = 1, 2), while the analytic superfields Z + , Z + and V ++ on the analytic harmonic subspace ζ A = t A ,θ + , θ + , w ± i ⊂ ζ H . The relevant superfield action is written as where the invariant integration measures are written as 2) The mass-dimension parameter m is encoded in the centrally-extended superalgebra su(2|1) as the contraction parameter to the flat N = 4, d = 1 superalgebra. 2 It does not explicitly appear in (1.1) but comes out in the component action from the measure µ H and the θ-expansion of the superfields X as a result of solving the appropriate SU(2|1) covariant constraints (for more details, see [12,16]).The local U(n) transformations of the involved superfields are given by The superfield V 0 (ζ A ) is a prepotential for the singlet part Tr (X) of the matrix superfield X (see details in [12]). The constant c in (1.1) is a parameter of the model. After quantization, it specifies external SU(2) spins of the physical states, c → 2s + 1 ∈ Z >0 , which implies that the set of these states splits into irreducible SU(2) multiplets. The matrix d = 1 superfield X has the physical component fields X = (X a b ) = X † , Ψ k = (Ψ k a b ) and auxiliary bosonic component fields, the superfields Z + a , Z +a have the bosonic components Z ′k = (Z ′k a ),Z ′ k = (Z ′a k ) = (Z ′k ) † and auxiliary fermionic fields. Choosing the WZ gauge V ++ = 2i θ +θ+ A(t A ), eliminating auxiliary fields and redefining the spinor fields as Z ′i a → Z i a / (Tr(X)) 1/2 , we obtain from (1.1) the on-shell component action (1.6) Here, and (∇Z k Z k ) := ∇Z a k Z k a . The U(n) gauge-covariant derivatives in (1.5), (1.6) are defined by (1.8) The basic novel feature of the action (1.4) as compared to the more conventional actions of supersymmetric mechanics is the presence of the semi-dynamical spin variables Z k a [21], 3 which has a drastic impact on the structure of the relevant space of quantum states. These variables define extra SU(2) symmetries with the generators (1.7), with respect to which the physical states carry additional spin quantum numbers and so form the appropriate SU(2) multiplets. The diagonal su(2) algebra is an essential part of the "internal" algebra su(2) ⊂ su(2|1). Also, note the presence of the oscillator-type terms in (1.5) and (1.6), with the intrinsic parameter m as the relevant frequency. The simplest one-particle (n=1) case of the system (1.4) was quantized in a recent paper [24]. Here we consider the quantum version of the system (1.4) for an arbitrary n.
The quantization of the Calogero-type multi-particle systems can be accomplished by the two methods, basically leading to the same result. One method [31,6,7,28,27,30] is based on the construction of the Dunkl operators for a given system. Using such operators makes it possible to represent a multiparticle system as an oscillator-like system for which the Dunkl operators play the role of generalized momentum operators. Another way of quantizing multiparticle systems is based on considering matrix systems with additional gauge symmetries [34,27,28,35,36,29,30]. The elimination of some degrees of freedom in such matrix systems results in the standard multi-particle Calogero-type systems. Due to the oscillator nature of matrix operators, the quantization of matrix systems is simpler and the main task of this approach consists in finding solutions of the constraints generating gauge symmetries. In this paper, we will mainly stick to the second method. We will present the explicit expressions of the multi-particle operators of deformed N = 4 supersymmetry, in the matrix case and for the reduced system.
The plan of the paper is as follows. In Section 2 we construct the Hamiltonian formalism for the matrix system (1.4) and show that the model indeed describes SU(2|1) supersymmetrization of the U(2)-spin Calogero-Moser model [25,26,27,28,29,30]. In Section 3 we find, by Noether procedure, the supercharges of the underlying su(2|1) superalgebra, in matrix case and for a system with the reduced phase variables space. In the latter case su(2|1) is closed up to the constraints generating some residual gauge invariances. In Section 4 we construct a quantum realization of the deformed N = 4, d = 1 superalgebra su(2|1) for the multi-particle Calogero-Moser system. In the case of the reduced system with n bosonic position coordinates such a superalgebra is closed up to the generators of the [U(1)] n gauge symmetry, like in the classical case. This su(2|1) superalgebra is represented as a sum of two su(2|1) superalgebras. One su(2|1) acts in the center-of-mass sector, whereas the other operates only on the supervariables parametrizing the quotient over this sector. The spin operators are common for both these superalgebras. In Sections 5 -7 we analyze the energy spectrum in all cases: for the center-of-mass subsystem, for the system with relative supercoordinates and in the general case, when all position operators are included. The last Section 8 contains a Summary and outlook.

Hamiltonian analysis and gauge fixing
The action (1.4) yields the canonical Hamiltonian (2. 2) The Hamiltonian (2.1) involves the matrix momentum P a b ≡ (∇X) a b and another matrix quantity 3) The action (1.4) also produces the primary constraints The constraints (2.4), (2.5) are second class and so we introduce Dirac brackets for them. As the result, we eliminate the momenta P Z a k , P Ψk a b and their c.c. The residual variables obey the Dirac brackets Requiring the constraints (2.6) to be preserved by the Hamiltonian (2.1) generates secondary constraints Despite the presence of the constant c in (2.3) these constraints are first class: with respect to the Dirac brackets (2.7) they form u(n) algebra, and so produce the U(n) invariance of the action (1.4) where α a b (t) ∈ u(n) are d=1 gauge parameters. In the first-order formulation, the system (1.4) is represented by the action where H matrix was defined in (2.2).
Let us fix a partial gauge for the transformations (2.10). To this end, we introduce the following notation for the matrix entries of X and P : x a := X a a , p a := P a a (no summation over a) , x a a := 0 , p a a := 0 (no summation over a) , x a . Note that In the notation (2.13) the constraints (2.8) take the form for the diagonal elements of G, with Provided that the Calogero-like conditions x a = x b are fulfilled, we can impose the gauge for the constraints (2.14). Then we introduce Dirac brackets for the constraints (2.14), (2.17) and eliminate x a b by (2.17) and p a b by (2.14): Due to the resolved form of gauge-fixing conditions, new Dirac brackets for the remaining variables coincide with (2.7): In the gauge (2.17), the constraints (2.15) become where A a = A a a (no summation over a) and the generalized Calogero-Moser Hamiltonian is defined as The action ( With respect to the Dirac brackets (2.7) the objects S ak j for each index a form u(2) algebras The object S k j forms the "diagonal" u(2) algebra in the product of above ones The triplets of the quantities (2.25) (see also (1.7)) generate su(2) algebras (2.30) Below we will also use the brackets One more matrix present in the action (2.24) is T a b defined in (2.16). These quantities form u(n) algebra (2.9) with respect to the Dirac brackets: The odd matrix variables are transformed by adjoint u(n) representation: These u(n) transformations commute with u(2) transformations generated by S ak j : Let us consider the bosonic core of the system (2.24) and demonstrate that it corresponds just to the spin Calogero-Moser model. Omitting terms with fermionic variables, we find The Hamiltonian (2.38) contains a potential in the center-of-mass sector with the coordinate X 0 (the last term in (2.38)). Modulo this extra potential, the bosonic limit of the system constructed is none other than the U(2)-spin Calogero-Moser model which is a massive generalization of the U(2)-spin Calogero model [25,26,28,29,30]. Thus the system (2.24) with the Hamiltonian (2.23) describes SU(2|1) supersymmetric extension of the U(2)-spin Calogero-Moser model.

Supercharges
In this Section we will find the classical expressions for the generators of the deformed N =4 supersymmetry (SU(2|1) supersymmetry) for the n-particle systems, both in the matrix formulation and in the case of the reduced system with n position coordinates.

Matrix system
The odd SU(2|1) transformations of the component matrix fields entering (1.4) are as follows 4 whereas the supertranslations of the spin fields are represented by the SU (2) rotations with the composite parameters Under the transformations (3.1), (3.2) and δA = 0 the action (1.4) transforms as Using (3.1), (3.2) and (3.3) we obtain the following expressions for Noether supercharges: where P = ∇X. The generators (3.4) constitute an su(2|1) superalgebra with respect to the Dirac brackets (2.7) Here, H = H matrix , where H matrix was defined in (2.2), and also the su(2) and u(1) generators are present: The Hamiltonian H commutes with all other generators and so can be identified with the central charge operator of su(2|1). The rest of Dirac brackets among the generators (2.2), (3.4), (3.6), (3.7) is given by the relations Note that the first-order action (2.11) is invariant, up to the surface term δS matrix = dtΛ 1 (with the substitution ∇X = P in Λ 1 ), under the transformations (3.1), (3.2), δA = 0 and It is worth pointing out that δH = 0 and δG a b = 0 under these transformations.

Reduced system in the standard Calogero-Moser representation
Let us compute the su(2|1) charges for the reduced system (2.24) which follows from the matrix formulation after imposing the gauge (2.17).
On the pattern of (2.13), we introduce the following notation for the entries of Ψ k andΨ k : In the gauge (2.17), supertranslations are a sum of the transformations (3.1), (3.2) and the additional compensating gauge transformations (2.10) with the composite parameters (3.14) These transformations preserve the conditions (2.17) and have the following explicit form An important property is that the constraints (2.20) are invariant with respect to these supersymmetry transformations, δT a = 0 . Also, δ a A a (T a − c) = 0 . The variation of the action (2.24) under the supersymmetry transformations (3.15) -(3.18) reads The corresponding Noether supercharges are found to be where p a =ẋ a . These expressions can be also obtained by inserting ( Here we used that the last relation in (2.19), being cast in the notation (2.13), (3.13), amounts to the relations ψ i a ,ψ b k (2.23), and the generators I i k , F were defined in (3.6), (3.7). The Hamiltonian (2.23) commutes with the supercharges (3.21) modulo the first-class constraints (2.20): The generators I i k , F satisfy the same Dirac brackets as in (3.8), (3.9), (3.10), and (3.11).

Quantum multi-particle su(2|1) superalgebra
Quantum su(2|1) superalgebra obtained by quantizing the Dirac brackets (3.5), (3.8), (3.9), (3.10), (3.11), is formed by the following non-vanishing (anti)commutators : (4.1) The second-and third-order Casimir operators of su(2|1) are defined by the expressions [18] In this Section we will present the explicit form of this deformed N =4 supersymmetry algebra for multiparticle system constructed in the previous Sections. We will do it for the matrix formulation of this system and for the reduced system with n position coordinates. In the matrix formulation, the n-particle system is described by quantum operators i which satisfy the quantum counterpart of the Dirac brackets algebra (2.7): The quantum supercharges are uniquely restored by the classical expressions (3.4): where the su(2) generators are These generators form the quantum algebra of the corresponding diagonal external algebra (2.30). The closure of the generators (4.5) is the full su(2|1) superalgebra (4.1) with the following even generators The set of physical states of the matrix system is singled out by the n 2 constraints which are quantum counterparts of the classical constraints (2.8) (the subscript "W" denotes Weyl-ordering) and should be imposed on the wave functions. The constant (2q + 1) present in (4.12) differs from the classical constant c due to ordering ambiguities. The operators (4.12) It is important that all constants appearing in the diagonal part of G a b , i.e. at a=b, are equal to (2q + 1). A corollary of (4.12) is that u(1) generator includes spin Z-operators only. As we will see below, the u(n) constraints (4.12) have a transparent meaning: The physical states are su(n) singlets. The constraint (4.14) fixes the homogeneity degree of the physical states with respect to spin variables, whence 2q ∈ Z >0 .

Separation of the center-of-mass sector
Let us split the matrix quantities as with being the center-of-mass operators and the traceless parts of matrix operators.
In terms of the variables (4.16), (4.17) the supercharges (4.5) are represented as where involve only the center-of-mass operators (4.16) and spin variables, whereaŝ depend on the traceless parts (4.17). The even operators (4.7), (4.8), (4.9), (4.10), (4.11) admit a similar splitting Here, The sets (Q k 0 ,Q 0k , H 0 , I 0 i k , F 0 ) and (Q k ,Q k ,Ĥ,Î i k ,F) form su(2|1) superalgebras (4.1) on their own, with the vanishing mutual (anti)commutators: Q i 0 ,Q k = Q i 0 ,Q k = 0, etc. Thus, we have singled out the center-of-mass sector from the total system. Note that the su(2|1) generatorsQ k ,Q k ,Ĥ,Î i k ,F have no action on the spin operators Z which in fact remain in the center-of-mass sector.
It is of importance that the constraints (4.12) involve in fact only the traceless parts (4.17) of the matrix operators (apart from the spin variable operators). Indeed, they can be rewritten in the form However, due to the presence of the same spin variables in the center-of-mass sector, these constraints are applicable also to the corresponding quantum states and so accomplish a link between the two sectors. The quantum counterpart of the multiparticle system from Sect. 3.2 is described by the quantum operators (4.29) Performing the Weyl-ordering in the quantum counterpart of (3.21), we obtain the quantum supercharges: where are quantum counterparts of (2.16) at a = b and Computing the anticommutators of the supercharges (4.30), we find the explicit form of the quantum even generators The commutators of the generator H with odd generators Q i ,Q i are a quantum generalization of (3.23). The remaining generators I i k , F obey the same commutation relations as in (4.1). From the (anti)commutators obtained we observe that the generators (4.30), (4.36), (4.37), (4.38) form the su(2|1) superalgebra (4.1) up to the differences (T a − T b ). However, recalling the constraints (2.20), this reduced system is specified also by the conditions which must be superimposed on the physical states. Therefore, the differences (T a − T b ) are vanishing on the physical states, and the physical sector of the relevant Hilbert space is closed under SU(2|1) symmetry. It is important that the quantum constraints (4.39) commute with the su(2|1) generators: In addition, the quantities (4.31) satisfy the algebra where T a a = T a at fixed a. It is instructive to be convinced that the numerator in the last term in (4.36) is indeed reduced to that for U(2) spin Calogero-Moser system [30], when applied to the bosonic wave functions Φ bos defined by the conditions It is easy to check that in this case Now, taking into account that in the numerator in (4.36) a = b, it is easy to check that

Division into subsystems
Using the simple identity which is valid for arbitrary n-vector operators K a , M a , a = 1, . . . , n, and introducing the center-of-mass quantities (4.32) and we can represent the charges (4.30) as the sums The first items Q k 0 ,Q 0k in these sums were defined in (4.19), and they involve only the centralof-mass supercoordinates, whereas the second items Q k ,Q k depend only on the differences of the supercoordinates: This implies that the bosonic generators (4.36), (4.37), (4.38) can also be represented as similar sums, (4.47) where H 0 , I 0 i k and F 0 are given by eqs. (4.22), (4.23), (4.24) and so involve only the center-ofmass coordinates, while the rest of operators is defined by the expressions The sets of the generators (Q k 0 ,Q 0k , H 0 , I 0 i k , F 0 ) and (Q k ,Q k , H, I i k , F) form two separate mutually (anti)commuting su(2|1) superalgebras. Note that second set generates an su(2|1) superalgebra up to the constraints, as in (4.33), (4.34), (4.35). Also, note that the "internal" SU(2) generators (4.49) (appearing in the anticommutator of supercharges) act on the indices i, j of the fermionic operators ψ i , while the indices i, j of the spin operators Z i a ,Z a i are subject to the action of the external SU(2) generators (4.6).

Subsystems of N =4 supersymmetric Calogero-Moser model
We have found that the N =4 supersymmetric n-particle Calogero-Moser system is a direct sum of two subsystems with different realizations of the su(2|1) generators. The generators (Q k 0 ,Q 0k , H 0 , I 0 i k , F 0 ) act in the sector of the center-of-mass operators (X 0 , P 0 , Ψ i 0 ,Ψ 0 i ) and the spin operators (Z i a ,Z a i ). The second set of the su(2|1) generators (Q k , Q k ,Ĥ,Î i k ,F) act, in the matrix formulation, within the sector of the traceless operators (X, P,Ψ i ,Ψ i ). Physical states in this subsystem are specified also by the spin operators (Z i a ,Z a i ) which are present in the u(n) constraints (4.28). These constraints also specify physical states in the center-of-mass sector involving the same spin operators. In the reduced formulation, the generators (Q k ,Q k , H, I i k , F) are spanned by the set of operators ( It should be pointed out that the spin operators Z i a ,Z a i have a non-zero action on the physical states with q = 0 for all subsystems defined above and listed below. The just described structure of the considered system suggests that we can consider three subsystems:

I)
The center-of-mass sector spanned by the quantum operators (X 0 , P 0 , Ψ i 0 ,Ψ 0 i , Z i a ,Z a i ) and the symmetry operators (Q k 0 ,Q 0k , H 0 , I 0 i k , F 0 ); II) The pure Calogero-Moser multi-particle sector with the center-of-mass sector separated. It is spanned by the quantum operators (X,P,Ψ i ,Ψ i ) in the matrix formulation or by in the reduced formulation. In both formulations, this subsystem also involves the spin operators Z i a ,Z a i . The SU(2|1) symmetry generators are (Q k ,Q k ,Ĥ,Î i k ,F) or (Q k ,Q k , H, I i k , F); III) The full Calogero-Moser multi-particle system which contains the center-of-mass sector and so is spanned by the set of all quantum operators. The SU(2|1) symmetry generators are sums of the SU(2|1) generators acting in the two previously defined sectors.
Now we are prepared to determine the energy spectrum of all these systems.

Center-of-mass subsystem with n sets of spin variables
In this section we consider the subsystem I) which describes the center-of-mass sector with the Hamiltonian H 0 (4.22). The center-of-mass supercoordinates (4.32), (4.43) satisfy the following (anti)commutation relations while those for the spin variables read We will use the following realization of the operator relations (5.1), (5.2) where x 0 is a real commuting variable, z i a are complex commuting variables and ψ i 0 are complex Grassmann variables. In this realization the Hamiltonian (4.22) takes the form The number s will be associated with the diagonal SU(2) group generated by 8) and, in what follows, will be referred to as "SU(2) spin s". Since Φ (2q) is transformed in the direct product of n spin q SU(2) representations, the maximal external SU(2) spin is just s = nq. It will be convenient to expand Φ (2q) into irreducible multiplets of the diagonal SU(2), with spins running in the intervals 0, 1 . . . nq (for 2nq even) or 1/2, 3/2 . . . nq (for 2nq odd).
As an illustration, we dwell on two lower-n cases.
In this case the wave function Φ 0 (x 0 , z i 1 , z i 2 , ψ i 0 ) is subject to two constraints Their general solution is As a prerequisite, we adduce the following eigenvalue relations . . . z k 2s 2 A (k 1 ...k 2s ) , (5.14)   Thus, the energy spectrum reads It is worth pointing out that the solution for N = 4 supersymmetric harmonic oscillator (5.18) was originally given in [13]. The equations (5.19) for the fields A ±(i 1 ...i 2s ) , B (i 1 ...i 2s−1 ) and C (i 1 ...i 2s+1 ) have the generic form where γ is a constant. It is the well-known equation describing quantum states of non-relativistic particle moving in a sum of the one-dimensional oscillator and conformal inverse-square potentials, and it has the following general solution (see, e.g., [1,3,37]) is a generalized Laguerre polynomial. The corresponding energy levels are The general solution (5.25) was used in ref. [24] to reveal the energy spectrum of the oneparticle system with one set of the spin variables. Each equation in the set (5.19) has the form of (5.24), the parameter γ being s + 1, s + 2 and s, respectively. Thus, the energy of the states of spins s and s + 1/2 described by the wave functions A +(i 1 ...i 2s ) and C (i 1 ...i 2s+1 ) , is equal to The q = 1 case encompasses the same states as for q = 1/2 (s = 1) depicted in Fig. 1, but also additional states with higher spins and higher energies. The similar pictures persist at lager q.
Summarizing the above discussion, we observe the basic distinction between the one-particle system of ref. [24] and the center-of-mass sector of n-particle system considered here. In the former case, the energy spectrum arises as a solution of the eigenvalue problems of the type (5.24), with the Hamiltonians involving a sum of the oscillator and the inverse square potentials. In the latter case, the energy spectrum contains as well pure oscillator excitations due to the presence of the eigenvalue problems of the type (5.20).
In this case there are three constraints of the type (5.6) and for integer q they lead to the following dependence of the wave function on the spin variables Figure 1: The degeneracy of energy levels of H 0 for n = 2 and q = 1/2 . Circles and crosses represent bosonic and fermionic states, respectively. On the left from the dotted vertical line the degeneracy corresponding to harmonic oscillator [13,16] is shown. On the right side there is shown a sum of SU(2|1) representations specified by their spin values s and coinciding with those found in [24] for the relevant spin. For the considered simplest case of q = 1/2, spin s takes only one value, s = 1 .
The component wave functions in the expansion (5.28) are functions of x 0 and ψ i 0 , and they display the dependence on ψ i 0 similar to that in (5.12). In the three-spinor case, the relations analogous to (5.14), (5.15), (5.16), (5.17) are also valid, the difference is that now an additional spin variable z i 3 appears in the products. As a result, in the energy spectrum we find the same states as in the n = 2 case, though with a bigger multiplicity (due to extra indices ab in (5.28)), as well as the states of higher spins due to the presence of the additional spin variable z i 3 . In the case of half-integer q, the n = 3 wave function has an expansion in which the component wave functions carry odd numbers of spinor indices, as opposed to the expansion (5.28). For example, for q = 1/2 wave function is The superwave functions Φ a i (x 0 , ψ 0 ) display the energy spectrum of the one-particle system of ref. [24], this time with the three-fold degeneracy.
The pictures for higher n are similar to those for n = 2 and n = 3, such that the number of states and the values of admissible spins are increasing at increasing n. 6 Calogero-Moser system without center-of-mass sector As was mentioned in Introduction, there are two methods of finding the quantum energy spectrum of multiparticle Calogero-type systems: either by considering matrix models which produce physically equivalent Calogero-type systems after gauge-fixing and the corresponding reduction of phase space, or through introducing Dunkl operators and passing to a generalized oscillator system. In this section we apply the first method to quantize the N = 4 spin Calogero-Moser model under consideration in the matrix formulation, with the center-of-mass sector detached (Sect. 6.1). The case of the reduced-phase space is briefly addressed in Sect. 6.2.

Quantization in matrix formulation
We consider the quantization of the matrix subsystem in which su(2|1) superalgebra is formed by the generators (Q k ,Q k ,Ĥ,Î i k ,F) defined in (4.20), (4.25), (4.26) and (4.27). The basic operators of this system are spin operators Z i a ,Z a i and traceless matrix operatorsX a b ,P a b , Ψ a b ,Ψ a b subject to the constraints (4.28) (as usual, applied to the physical states). In this notation, the supercharges (4.20) are rewritten aŝ

Introducing creation and annihilation even operators
where quantum (anti)commutators of the involved operators are The constraints (4.28) take the form They involve the spin operators, with the non-vanishing commutator The operators Z i a , A + a b ,Ψ i a b form a full set of creation operators. Therefore, the general structure of the physical states is as follows In the holomorphic realization, we deal with the traceless objectsâ + andΨ i . Then the physical states (6.7) are rewritten as The constraint (6.5) indicates that all physical states are singlets of SU(n) (see [34,36,35,30]). This is also a direct consequence of vanishing of all Casimir operators on the states (6.7): On the other hand, the states (6.7) belong to irreducible representations of the group SU (2) with the generators S (ij) defined in (4.6) and the group SU (2) (6.11) Here we will basically limit our consideration to the pure bosonic case, without odd operatorŝ c . The set of fermionic states can be generated by action of the supercharges on the subset of bosonic states. Examples of fermionic states will be constructed below for few simple particular cases.
The trace part of the constraints (6.5) leads to homogeneity of the physical states of degree 2qn with respect to the spin operators Z . In addition, the property that physical states are the SU(n) singlets implies the following structure for them [34,36,35,30] where p 2 , p 3 . . . , p n are arbitrary integers and 0 ≤ l nr+1 ≤ l nr+2 . . . l nr+n < n. The wave function Φ (2q,s,ℓ) in (6.12) is given up to the coefficients C (i 1 i 2 ...i 2s ) , where the number s ≤ nq can be interpreted as SU(2) spin, with 2s being the number of symmetrized SU(2) indices of spin variables. It is worth pointing out that for l r < n/2 the coincident degrees, l r = l p , are permitted. Besides, such a degree cannot appear more than once in the products of monomials Degeneracy analysis of bosonic wave functions for the quantum spin Calogero model was considered in [33], where it was noticed that some of possible spin states may vanish. Here we consider the matrix construction for the system with the center-of-mass sector detached, 6 where some of these spin states may also vanish. Listing all admissible degree numbers l nr+1 , l nr+2 . . . , l nr+n is a rather complicated task.
On the states (6.12) the energy (6.11) take the values The energy is maximal for the choice (6.14): The minimal energy corresponds to the choice p 2 = p 3 . . . p n = 0 and l nr+1 = l nr+2 = 0, l nr+3 = l nr+4 = 1, l nr+5 = l nr+6 = 2, etc: , for even n , The fermionic states are constructed with the help of the operatorsΨ i b c , on the pattern of (6.12). Such physical states have additional contributions mN Ψ to the energy value (6.15). As was already mentioned, full wave functions can be generated from the bosonic states (6.7) by acting on them by the supercharges (6.3). Casimir operators (4.2), (4.2) take the following values on the states (6.7) and those produced from (6.7) by SU(2|1) supersymmetry transformation: Casimirs can take zero eigenvalues only for n = 2 at arbitrary q and for n = 3 at q = 1/2 (we consider q > 0 in this paper). The corresponding sets of the quantum states belong to atypical representations of SU(2|1). For illustration, we will consider here these two cases in some detail.
In this case the Hamiltonian is written aŝ Bosonic wave functions, from which the full set of the wave functions can be produced by the supercharges (6.4), are given by where C (i 1 i 2 ...i 2s ) are coefficients with 2s symmetric indices. The wave functions Φ (2q,s,ℓ) are eigenfunctions of the Hamiltonian (6.19), with the energy eigenvalues The wave functions on which Casimirs take zero values, i.e., those belonging to atypical representations of SU(2|1), correspond to the choice s = 0 , ℓ = 0: This ground state wave function is SU(2|1) singlet, since it is annihilated by both supercharges. There is still another atypical non-singlet bosonic state corresponding to s = 1 , ℓ = 0: which gives rise to the fundamental SU(2|1) representation. The other two components of this representation are generated from (6.24) by SU(2|1) supercharges: The n = 3 Hamiltonian readŝ For q = 1/2, the bosonic wave functions Φ (2q,s,ℓ) as eigenfunctions of this Hamiltonian are constructed as with the coefficients C k , C ′ k , C (ijk) . They have the following energy values where ℓ = 2p 2 + 3p 3 . The minimal energy is achieved on the state and it is equal to Casimir operators take zero values on this state. The action of the SU(2|1) supercharge, produces an additional fermionic state which, together with (6.29) and one more bosonic state generated by further action of supercharges on (6.31), constitute an atypical fundamental SU(2|1) supermultiplet.
In the simpler case g ab = ϑ (ϑ ∓ 1) , quantization was given in [31,6,7] via Dunkl operators defined as where K ab is a permutation operator, K ab x b = x a K ab . Below we consider the simplest case n = 2 , where g 12 = s (s + 1) and the operator K 12 becomes Klein-type operator acting on the relative coordinate x 1 − x 2 as Let us consider in details the two-particle system (n = 2). It is described by the algebra of quantum operators Below we use the following realization for them Here x, z i a and ψ i , ψ i we define the creation and annihilation operators a ± through the Dunkl operator D: Then eq. (6.46) is rewritten as where One can also choose an alternative ansatz The construction of wave functions via the creation and annihilation operators will give the same solution for the energy spectrum as (6.54). We skip details of this construction which is similar to the previous one.
7 Quantization of the full system (center-of-mass plus relative-coordinate sectors) The energy spectrum of the unified system, which is a sum of the center-of-mass sector of Sect. 5 and the relative coordinate system of Sect. 6, can be found as a tensorial product of the spectra of these two subsystems.
In the ungauged matrix formulation, the bosonic wave functions are a generalization of (6.12) in the holomorphic realization (6. Their general structure is quite specified by the three requirements: • The wave functions should be U(n) invariant as a consequence of the constraint (4.12) (or its equivalent form (4.28)). This means that all U(n) indices a should be contracted with the appropriate invariant tensors; • They should be of degree 2qn with respect to the whole set of spin variables in virtue of the constraint (4.14); • All free SU (2) indices of the spin variables should be symmetrized and contracted with the indices of f (i 1 ...i 2s ) (x 0 ). The energy spectrum of admissible spins of these functions extends from s = 0 to nq (for 2nq even) and from s = 1/2 to nq (for 2nq odd).
All the fermionic wave functions can be obtained by action of the total supercharges on (7.1). The basic distinctions of the total system from the multi-particle system of Sect. 6 concern the realizations of the SU(2) symmetry appearing in the anti-commutators of the supercharges as an internal subgroup of SU(2|1). In the system with the center-of-mass sector detached considered in Sect. 6, this SU(2) symmetry is given by (4.49), acts only on the fermionic operators and gives rise just to degeneracy of the energy spectrum. In the total system, the internal SU(2) symmetry acts on the indices i, j, . . . of all components of the wave functions (7.1) and their fermionic completion.
Taking into account the analysis of the previous section, we see that the problem of description of all states in the unified case (the option III) in Sect. 4.3) for an arbitrary n is rather complicated. At the same time, we can directly determine, for all possible cases, the full energy spectrum simply by applying the methods of the previous sections. Let us briefly describe the energy spectrum for the choice of n = 2.
In this simplest case the matrix system is described by the Hamiltonian The traceless part of the general constraints (6.5), requires wave functions to be SU(n) scalars, while its trace part fixes the degree of homogeneity with respect to spin variables: The complete set of the quantum states is recovered through the action of SU(2|1) supercharges on the complete set of these bosonic wave functions. The generic case in the reduced phase space formulation will be considered elsewhere.

Concluding remarks and outlook
In this paper, we presented the full quantum description of the SU(2|1) supersymmetric multiparticle Calogero-Moser system with spin variables. It was constructed by making use of the matrix formulation of this system. Due to the presence of spin variables, the system under consideration involves internal spin degrees of freedom and so provides N =4 supersymmetrization of U(2) spin Calogero-Moser system, as opposed to the systems considered in refs. [38,39,40]. We obtained the explicit expressions for the classical and quantum charges of the massdeformed N =4 supersymmetry inherent to the multiparticle system considered. The crucial role in quantization of this system is played by the property that it became possible to single out the center-of-mass subsector in the full system. This allowed us to separately explore the case of the center of mass and the case without the center-of-mass variables. Knowing the energy spectrum in these two cases immediately allows one to derive the energy spectrum of the total system.
We computed the energy spectrum, exploiting the matrix formulation of the N =4 supersymmetric U(2) spin Calogero-Moser system. An alternative way of quantizing such systems is to deal with the reduced system, involving the dynamical position coordinates only. Such a method [31,6,7,28,27,30] (the "operator method" in the terminology by A. Polychronakos) widely uses the Dunkl operators for building the oscillator-like phase space of the multi-particle Calogero-type systems. Some simple examples of applying this equivalent method within the model considered here were already discussed in Sect. 6.2. In the next publication we are planning to develop, in full generality, the applications of the operator method to the systems with spin variables. On this way we expect, in particular, to find out some new generalizations of the Dunkl operators and obtain complete set of independent conserved quantities (integrals) for a rigorous proof of integrability. One more direction for the future study is to construct and quantize multi-particle Calogero-type models with higher-rank deformed supersymmetries of the kind SU(m|n) and to reveal their relationships with the integrable structures in N = 4 super Yang-Mills theory, e.g., along the lines of ref. [10,11].
One more interesting problem is to elucidate a possible hidden superconformal symmetry of the multi-particle system considered. In the one-particle case, the corresponding quantummechanical (massive) system [12] was found to possess such a hidden N =4 superconformal symmetry associated with the supergroup OSp(4|2) [24]. In the quantum domain, the corresponding superalgebra osp(4|2) acts as a spectrum-generating algebra. The existence of an analogous extension of SU(2|1) symmetry in the multi-particle case is an open question. In general, one could expect as well a hidden D(2, 1; α) supersymmetry for which OSp(4|2) is a particular case corresponding to the choice α = −1/2. However, this possibility would require, from the very beginning, some nonlinear sigma model action for the superfields X a b in (1.1) and, respectively, for the bosonic fields X a b in (1.5). The choice of OSp(4|2) is the unique one consistent with free kinetic terms for the bosonic fields, as long as one insists on the supercharges (4.30) being linear in fermionic variables [41]. Allowing for supercharge terms cubic in the fermionic operators will constrain their coefficient functions by the so-called WDVV equations [38,39,40,42,43,41]. It will be interesting to develop a superspace variant of this more general situation.