Quantum SU$(2|1)$ supersymmetric $\mathbb{C}^N$ Smorodinsky--Winternitz system

We study quantum properties of SU$(2|1)$ supersymmetric (deformed ${\cal N}=4$, $d=1$ supersymmetric) extension of the superintegrable Smorodinsky--Winternitz system on a complex Euclidian space $\mathbb{C}^N$. The full set of wave functions is constructed and the energy spectrum is calculated. It is shown that SU$(2|1)$ supersymmetry implies the bosonic and fermionic states to belong to separate energy levels, thus exhibiting the"even-odd"splitting of the spectra. The superextended hidden symmetry operators are also defined and their action on SU$(2|1)$ multiplets of the wave functions is given. An equivalent description of the same system in terms of superconformal SU$(2|1,1)$ quantum mechanics is considered and a new representation of the hidden symmetry generators in terms of the SU$(2|1,1)$ ones is found.


Introduction
The models of supersymmetric mechanics were initially introduced as toy models for supersymmetric field theories [1]. Respectively, their study was mainly limited to the models exhibiting one-dimensional Poincaré supersymmetry defined by the relations where Q A are N real supercharges and H is the Hamiltonian. Some decade ago, there started a wide activity related to the study of field-theoretical models with the "rigid supersymmetry on curved superspaces" (see, e.g., [2]). These studies motivated two of us (E. I. and S. S.) to investigate the one-dimensional (i.e. mechanical) analogs of these theories [3], [4]. The main subjects of our interest were systems in which d = 1, N = 4 Poincaré supersymmetry is deformed by a mass-dimension parameter m into su(2|1) supersymmetry, with the following non-vanishing (anti)commutators 1 : Here, the generator H is the Hamiltonian (coinciding with the U(1) generator), while the remaining bosonic generators I i j (i = 1, 2) define SU(2) R-symmetry. As one of the results of this study, it was observed that the so-called "weak N = 4 supersymmetric Kähler oscillator" models (which are the particle models living on Kähler spaces and interacting with the specific potential field and a constant magnetic field) suggested earlier in [5,6] supply nice examples of SU(2|1) supersymmetric mechanics 2 , and they can be reproduced from the SU(2|1) superfield approach worked out in [4]. This observation further entailed the construction of a few novel SU(2|1) supersymmetric superintegrable oscillator-like models specified by the interaction with a constant magnetic field. They include superextensions of isotropic oscillators on C N and CP N [5], [8], as well as of C N Smorodinsky-Winternitz system in the presence of a constant magnetic field [9] and CP N Rosochatius system [10].
In a recent paper [11] we noticed that switching on an interaction with a constant magnetic field in the ordinary N = 4 supersymmetric mechanics on Kähler manifold breaks either N = 4 supersymmetry or isometries of the initial bosonic system, including their hidden symmetries (cf. [12]). We demonstrated that this drawback can be overcome by performing, instead of the d = 1 Poincaré supersymmetrization, its deformed variant, i.e. SU(2|1) supersymmetrization. Examining the SU(2|1) supersymmetric models listed above, we observed that the SU(2|1) supersymmetrization preserves all kinematical symmetries of the initial systems, and, in a number of cases, also the hidden ones (the explicit expressions for the "super-counterparts" of the hidden symmetry generators were constructed so far for C N oscillator and C N Smorodinsky-Winternitz system in the presence of a constant magnetic field). It should be pointed out that the hidden symmetries play an important role in the quantum domain: e.g., in the standard supersymmetric quantum mechanics they amount to an additional degeneracy of the (higher) energy levels as compared to N -Poincaré supersymmetry which enhances it by the factor 2 N /2 .
Keeping in mind these reasonings, it is desirable to better understand, on the concrete examples, how SU(2|1) supersymmetry affects the energy spectrum of the systems considered. The basic peculiar feature of the supersymmetric Kähler oscillator models is that their Hamiltonian is in fact identified with the U(1)-generator in (1.2). As was shown in [4], an extra U(1) R-charge ("fermionic number") can be introduced in addition to the Hamiltonian in this class of models only in the limit 1 These su(2|1) superalgebra relations differ from those in [11] by rescaling Q → √ 2 Q. We mostly follow the conventions of [4]. 2 Another version of this kind of supersymmetric mechanics was studied in [7], with SU(2|1) termed as "weak supersymmetry". m = 0. One can expect that the impossibility to separate the Hamiltonian from the fermionic number in such models at m = 0 could have an essential impact on the structure of the relevant energy spectra.
In the present paper we construct SU(2|1) supersymmetric extension of the C N Smorodinsky-Winternitz (in what follows, S.-W.) quantum system in the presence of a constant magnetic field as an instructive and simple example of the general Kähler oscillator models. In particular, we focus on the interplay of SU(2|1) supersymmetry and hidden symmetry in forming the energy spectrum of this system. An analogous analysis of its purely bosonic sector was performed in [9]. An interesting unique feature of the SU(2|1) C N S.-W. model as compared to other models of SU(2|1) supersymmetric Kähler oscillators 3 is its implicit superconformal SU(2|1, 1) symmetry. At the classical level this follows from the results of ref. [13], while here we extend the correspondence with a complex SU(2|1, 1) superconformal mechanics to the quantum domain.
As a preamble, let us remind that the C N S.-W. system considered in [9] 4 amounts to a sum of N two-dimensional isotropic oscillators with a ring-shaped potential interacting with a constant magnetic field (orthogonal to the plane). It is defined by the Hamiltonian [9] where B is a constant strength of the magnetic field. Besides the standard Liouville integrals of motion, the model possesses the additional ones generated by the so-called Uhlenbeck tensor, and thus provides an example of superintegrable system. The C N S.-W. system interacting with a constant magnetic field belongs to the class of Kähler oscillators with the following Kähler potential [11] It admits SU(2|1) supersymmetric extension given by the superalgebra (1.2), with the deformation parameter m = 4ω 2 + B 2 .
It is convenient to summarize here at once the basic results of the present paper.
• The energy spectrum of the deformed SU(2|1) supersymmetric system constructed reveals an interesting feature: SU(2|1) supersymmetry gives rise to the separation of bosonic and fermionic states in the spectrum. Bosonic states are associated with the even levels, while all fermionic states with the odd ones. So the energy spectrum exhibits "even-odd" feature: the adjacent bosonic and fermionic states carry different energies shifted by half-integer numbers. The intrinsic reason for such a splitting is that the Hamiltonian in (1.2) hides in itself the fermionic number operator F and so does not commute with the supercharges. The ground state belongs to a non-singlet representation of SU(2|1), so the latter is spontaneously broken. These features are in contrast with the standard supersymmetric mechanics models where there is a degeneracy between the fermionic and bosonic wave functions.
• The system exhibits superconformal SU(2|1, 1) symmetry with the central charge given by the sum of the generators of kinematical U(1) symmetries. The SU(2|1) supersymmetric Hamiltonian H can be split into a sum of superconformal Hamiltonian and a central charge generator accounting for the constant external magnetic field. The whole set of wave functions is closed under the action of the superconformal algebra su(2|1, 1), which can be treated as a spectrum-generating algebra of the model considered.
• Due to an additional hidden symmetry generated by the proper superanalog of Uhlenbeck tensor commuting with the Hamiltonian, the spectrum of quantum SU(2|1) C N S.-W. system reveals an extra degeneracy. The superextended Uhlenbeck tensor is shown to admit a nice representation in terms of bilinears of the SU(2|1, 1) generators.
• Furthermore, a generalization of the super Uhlenbeck tensor was found, such that it commutes with the full set of the generators of the properly rotated SU(2|1) supergroup. It is also expressed through SU(2|1, 1) generators and gives rise to an additional degeneracy of the eigenvalues of the SU(2|1) Casimirs.
The paper is organized as follows. In Section 2 we formulate SU(2|1) supersymmetric extension of the superintegrable C 1 S.-W. quantum system and determine its energy spectrum and wave functions. We also analyze the space of quantum states of this supersymmetric system from the standpoint of the SU(2|1) representation theory. In Section 3 we show that the system considered admits an equivalent description in terms of some quantum superconformal SU(2|1, 1) mechanics. In Section 4 we define the SU(2|1) supersymmetric C N S.-W. quantum system as a sum of N copies of the C 1 -systems. We reveal its hidden symmetry given by the supersymmetric counterpart of the Uhlenbeck tensor and show that it is responsible for an additional degeneracy of the spectrum of the SU(2|1) Casimir operators. A new representation for the superextended Uhlenbeck tensor in terms of the superconformal SU(2|1, 1) generators is found. The summary and outlook are the contents of Section 5. In Appendices A and B we present some details related to the non-linear algebra of hidden symmetries. In Appendix C, it is briefly discussed how conformal SL(2,R) symmetry manifests itself in the spectrum of quantum bosonic C N S.-W. system.

Supersymmetric C 1 Smorodinsky-Winternitz system
We proceed from the SU(2|1) supersymmetric S.-W. system on the complex Euclidian space C 1 . The quantum Hamiltonian constructed according to the generic prescription of [4,11] for the specific N = 1 Kähler potential (1.5), is defined by the expression, with the (anti)commutators 5 The operators of bosonic momenta are represented in the standard form while the fermionic operators ξ i ,ξ i can be represented as a 4 × 4 Euclidian gamma-matrices acting on four-component wave functions. However, we prefer a more formal, though equivalent representation, see, e.g., [15]. Namely, we consider one-component wave functions depending on z,z, ξ i (with ξ i , i = 1, 3 , being a doublet of Grassmann variables), and representξ j as a differential operator,ξ Respectively, the wave functions contain two bosonic components, ψ (z,z), ξ k ξ k ψ ′ (z,z), and two fermionic ones, Ψ i = ξ i ψ ′′ (z,z).
In what follows, it will be convenient to equivalently replace the parameters B and ω by λ and m, where m is a mass-dimension contraction parameter defined in (1.6), and λ is a dimensionless angle-type parameter defined by the relations (2.5) In the new notation, the Hamiltonian (2.1) is rewritten as where L is the angular momentum (or U(1)) operator The Hamiltonian (2.6) is manifestly invariant under U(1)-transformation z → e iκ z, ξ i → e iκ ξ i generated by this operator. The rest of SU(2|1) generators is expressed as It is straightforward to check that they satisfy the su(2|1) superalgebra relations (1.

Wave functions and spectrum
We define super wave functions as a ξ i -expansion of the wave functions depending on (z,z, ξ i ), with ξ j being an annihilation operator. This expansion amounts to the fermionic wave function Ψ i ∼ ξ i ψ ′′ (z,z) and two bosonic wave functions ψ (z,z), ξ k ξ k ψ ′ (z,z). As distinct from the fermionic function, the bosonic ones are not eigenstates of the Hamiltonian (2.6). The correct eigenstates are represented as their proper combinations: (see eq. (2.24) below). The simplest way to solve the eigenvalue problem is to firstly consider the action of (2.6) on the fermionic wave functions Ψ i ∼ ξ i ψ ′′ (z,z). On the bosonic factor ψ ′′ (z,z) the Hamiltonian (2.6) acts exactly as the bosonic one [cf. (1.3)], and this action reads (2.10) After solving the eigenvalue problem as in the pure bosonic case [9], the fermionic states are written as where L (l ) n−1 are generalized Laguerre polynomials, i.e. n = 1, 2, 3 . . . is a positive integer 6 . The integer number l is an eigenvalue of the angular-momentum operator L, The energy spectrum of the fermionic states is directly calculated to be The double degeneracy of the fermionic levels is due to the unbroken su(2) ⊂ su(2|1) with respect to which the wave function (2.11) transforms as a doublet. The bosonic wave functions can be now obtained by action of the supercharges on the fermionic wave function Ψ i : 2m Ω + (n,l) . (2.14) They are explicitly expressed as , L Ω ± (n,l) = l Ω ± (n,l) .

(2.15)
On the other hand, the action of supercharges on the bosonic wave functions yields the original fermionic wave functions Ψ i : Using these expressions, we derive the energy spectrum for the bosonic states Ω ± (n,l) and find that it differs from that for the fermionic states, Thus SU(2|1) supersymmetry creates additional energy levels, with the bosonic and fermionic states being separated. Note that the discrete energy spectrum bounded from below in the model under consideration is ensured by the oscillator term ∼ m 2 in (2.10). So the limit m = 0 cannot be taken in the quantum case (see also Section 3). 6 One could work with the Laguerre polynomials L (l ) n ′ (mzz) where n ′ = 0, 1, 2 . . . , but for further convenience we deal with n = n ′ + 1, such that n = 1, 2, 3 . . . (see the next Subsection).

SU(2|1) representations
The SU(2|1) representations are specified by the eigenvalues of Casimir operators [16] On the quantum states Ψ j (n,l) , Ω ± (n,l) these eigenvalues are: The quantum number n uniquely defines SU(2|1) representation for a fixed l.
Therefore, the ground states for each l belongs to a non-trivial four-fold SU(2|1) multiplet. This means that SU(2|1) supersymmetry is spontaneously broken at any l.
The quantum states within the given SU(2|1) multiplet occupy different energy levels according to (2.13) and (2.17). Since the supercharges do not commute with the Hamiltonian (2.6), they are capable to decrease and increase its eigenvalues. One can designate levels occupied by bosonic states as even levels (including the zeroth level for the ground state), then the fermionic states occupy odd levels. As distinct from the standard d = 1 Poincaré supersymmetry, there is no any degeneracy between these two types of the levels. The relevant picture is drawn on Figure 1.
One can see that the bosonic states Ω + (n,l) and Ω − (n+1,l) have the same energy though they belong to different SU(2|1) representations, For example, the energy level E (1,l) + m/2 on Figure 1 can be alternatively denoted as E (2,l) − m/2. The hidden symmetry responsible for this degeneracy will be presented in the next section. As was already mentioned, the two-fold degeneracy of fermionic states is due to the SU(2) generators (2.8) acting on the fermionic variables ξ i . The wave functions Ω + (n,l) and Ω − (n+1,l) for n = 1, 2, 3 . . . can be represented as , where , These functions have just the structure (2.9) and are eigenfunctions of the Hamiltonian (2.6) with the eigenvalues (2.22), but they are not eigenfunctions of Casimirs (2.18) and (2.19). The only exception is the wave function Ω 1 (n,l) which can be naturally continued to n = 0 as It is directly related to the lowest bosonic state Ω − (1,l) 8 : In the next section we will show that the state Ω 1 (0,l) can be interpreted as a singlet ground state with respect to some new SU(2|1) superchargesQ,Q .
Let us summarize the basic peculiarities of the energy spectrum.

Superconformal symmetry
In this section, following [13], we relate the generic supersymmetric C 1 S.-W. system to the superconformal mechanics with the Hamiltonian We will show that the Hamiltonians of these two systems differ by a central charge generator, and, as a consequence, the Hamiltonian of conformal model inherits all the symmetries of the original Hamiltonian (and vice-versa).
As was proved in [13], the superconformal Hamiltonian of deformed supersymmetric mechanics should be an even function of the deformation mass parameter m : m → − m, H conf → H conf . In accord with this proposition we define the superconformal Hamiltonian (3.1) as Such a change of the Hamiltonian amounts to the effective elimination of the magnetic field 9 . One can equally choose the basis in which the conformal Hamiltonian is not deformed by the oscillator term, Then we introduce the dilatation generator D and the generator of conformal boosts K: These generators, together with the Hamiltonian (3.3), close on the conformal algebra so(2, 1) ∼ sl(2, R) [17]: The trigonometric type of (super)conformal mechanics involving the parameter m is defined by the following linear combinations [13]: The algebra (3.5) is then rewritten as With B = 0, i.e. λ = π/4, the deformation parameter coincides with the frequency, and the central charge Z 1 vanishes: Therefore, at this special choice of parameters the C 1 S.-W. Hamiltonian (2.10) just coincides with the superconformal Hamiltonian (3.1), H = H conf .
One could come back to the original Hamiltonian H according to (3.2), but in this case the conformal algebra will be deformed by a central charge. So, irrespective of whether we deal with superconformal symmetry or its bosonic limit, it is appropriate to use the Hamiltonian H conf containing no magnetic field. Note that the first relation in (3.6) implies that just H conf with m 2 = 0, as opposed to H conf , is the correct quantum Hamiltonian with the spectrum bounded from below, in accordance with the assertion in the pioneering paper [17]. The conformal algebra (3.7) can be extended to the superconformal algebra in the following way. Applying the discrete transformation m → − m to the supercharges Q i defined by (2.8) we obtain the new fermionic generators which can be identified with the conformal supercharges, (3.8) These new generators extend the superalgebra su(2|1) to the centrally extended superalgebra su(2|1, 1) : The superconformal algebra contains three central charges: Note that the quadratic operator constructed out of the central charges is reduced on the states to the square of the quantum numberl = l 2 + g 2 defined in (2.11), When g = 0 , this operator coincides with L 2 . Below we will show that the set of three central charges in (3.9) can be reduced to a single central chargeZ 1 =L . This agrees with the fact that the superalgebra su(2|1, 1) contains 15 generators: eight supercharges, three su(2) generators, three generators of so(2, 1) and one central charge, in accord with the decomposition su(2|1, 1) = psu(2|1, 1) ⊕Z 1 , where psu(2|1, 1) is a centerless superalgebra. Before proceeding further, we point out that the SU(2|1, 1) trigonometric superconformal model of the multiplet (2, 4, 2) with a superpotential term resulting in the Hamiltonian (3.1) was constructed within a manifestly SU(2|1) covariant (N = 4 deformed) superfield approach in [13]. In the N = 2, d = 1 superfield formalism, the model amounts to a system of the coupled (2, 2, 0) and (0, 2, 2) multiplets, with only N = 2 superconformal symmetry SU(1|1, 1) ⊂ SU(2|1, 1) being manifest. In such a formulation, the inverse-square terms with g = 0 in (3.1), (2.1) come out solely from the coupling of these two multiplets and disappear after decoupling of the fermionic multiplet (0, 2, 2). So in this limit (still respecting SU(1|1, 1) invariance) our model is reduced to the two-dimensional N = 2 superconformal oscillator model based on the chiral multiplet (2, 2, 0). In ref. [18] (see also [19]) there was considered a different SU(1|1, 1) superconformal model of the multiplet (2, 2, 0), with the Hamiltonian involving an inverse-square potential induced by some spin coupling. It cannot be obtained as any truncation of our model.

Casimir operators
Let us consider the quadratic Casimir operator of D(2, 1; α) [20]: The definition of the superalgebra D(2, 1; α) in terms of these generators was given in [13], with m = −αµ. The limit α = −1 gives rise to the superalgebra D(2, 1; α = −1) → psu(2|1, 1) ⊕ su (2) , where psu(2|1, 1) is a centerless superalgebra and su (2) is an external automorphism generated by the generators F , C andC. Hence, the Casimir operator of psu(2|1, 1) reads The automorphism su (2) generators F , C andC commute with this operator. In our case we deal with the centrally extended superalgebra (3.9), so we are led to perform an alternative way of reaching the limit α = −1. It implies the following preliminary redefinition of the extra su(2) generators: Then, multiplying (3.12) by the factor ∼ (α + 1) and taking the limit α = −1 afterwards, we observe that only the last piece of (3.12) survives this limit: The generators (3.14) commute among themselves and with all other generators in the limit considered, so they form the triplet of central charges. Thus we are left with the superalgebra (3.9) for which the invariant operator (3.11) is a proper limit of the quadratic Casimir operator (3.12) and it is the genuine Casimir for the centrally extended su(2|1, 1) superalgebra. Below we will show that SU(2|1, 1) is a spectrum-generating supersymmetry acting on infinitedimensional irreducible SU(2|1, 1) multiplets labeled by the eigenvaluesl 2 of (3.11). However, we must take into account that the angular momentum operator L can take two eigenvalues l = ±|l| leading to the same (l) 2 . So, the operator L commuting with all su(2|1, 1) generators can be treated as the second Casimir operator of su(2|1, 1) and the full space of quantum states of the model is the collection of two copies of SU(2|1, 1) multiplets, with the same value ofl and two opposite-sign values of the quantum number l.

Passing to the new basis in SU(2|1, 1)
It is useful to bring the superconformal algebra (3.9) to the form containing only one central charge. To eliminate two of the original central charges, we perform the rotatioñ where 10 cos 2ϕ = g The only central chargeZ 1 we are left with reads and it takes the valuel = l 2 + g 2 on the quantum states (see (3.11)). In the new basis, the superalgebra is rewritten as form a different su(2|1) subalgebra, with the same (anti)commutators as in (1.2). So one can construct the new space of quantum SU(2|1) states just with respect to this transformed su(2|1) superalgebra. It should be pointed out that the su(2|1, 1) generators in the original and new bases, taking into account their explicit form, can be realized on the full set of the quantum states defined in Section 2, so that this set is closed under the action of these generators. Thereby, the construction using the transformed su(2|1) superalgebra will give rise to the same total set of quantum states, although with the energy spectrum calculated with respect to the HamiltonianH defined in (3.20).
The new SU(2|1) supersymmetry generated by the superchargesQ i andQ j is not spontaneously broken and the corresponding ground state is given by the SU(2|1) singletΩ + (0,l) . To prove this, let us consider the relevant Casimir operators On the statesΩ ± (n,l) and Ψ i (n,l) , they take the eigenvalues The states with n = 1 correspond to an atypical representation, since Casimir operators are also zero on these states. This representation is spanned by the three states The states Ψ i (1,l) ,Ω + (1,l) form the fundamental SU(2|1) representation. All excited states with n > 1 correspond to the simplest four-fold typical SU(2|1) representations.
Whereas the supersymmetry associated withQ i andQ j is not broken, the second pair of SU(2|1) superchargesS i andS j corresponds to the spontaneously broken supersymmetry (see Figure 2), with the minimal energyl m as the lowest eigenvalue of the relevant shifted Hamiltonian H conf + mZ 1 /2.
On Figure 2, we demonstrate how an action of SU(2|1, 1) supercharges mixes all bosonic and fermionic states with a fixed number of the angular momentum l. Thus, all these states at fixed l belong to a single infinite-dimensional SU(2|1, 1) representation labeled byl 2 = l 2 + g 2 as the square of the central charge. The states with − l belong to the other SU(2|1, 1) representation labeled by the samel 2 = l 2 + g 2 . These two representations are distinguished only by the eigenvalue of the operator L (see discussion in Subsection 3.1) and they exhaust the whole space of the quantum states of the model. Thus the conformal supergroup su(2|1, 1) acts as the spectrum-generating algebra on this space of quantum states 11 .
P P P P P P ✐ P q P P P P P P q P ✐ P P P P P P ✐ P q P ✐ P P P P P P q P ✐ P P P P P q These generators act on the bosonic wave functions only and form an exotic SU(2) symmetry [3] belonging to the universal enveloping of su(2|1, 1): The ground stateΩ + (0,l) is annihilated by these SU (2) generators. This SU(2) symmetry is responsible as well for the double-fold degeneracy of the initial wave functions Ω ± (n,l) , since the latter are related toΩ ± (n,l) via (2.23) and (3.25).

Invariant operators
The Casimir operator (3.13) allows us to guess the form of some new invariant operatorsĨ,M of the su(2|1) superalgebra. They commute withQ i ,Q j , I i j ,Z 1 , H conf and read 12 On the SU(2|1) representations generated byQ i andQ j they take the values The quadratic SU(2|1) Casimir can be written in terms of these operators and the central charge: These expressions will help us to construct generalizations of Uhlenbeck tensor in the next Section. Note that such invariants can be constructed only for su(2|1) generated by the transformed superchargesQ i andQ j . No their analogs commuting with Q i andQ j exist.

Summary of Section 3
The superconformal Hamiltonian H conf , eq. To avoid a possible confusion, we point out that the complete quantum consideration of the SU(2|1) supersymmetric C N S.-W. system, including energy spectrum, the structure of the Hilbert space of wave functions and their SU(2|1) representation contents, has been already given in Section 2. The basic aim of Section 3 was to demonstrate that the same results can be restored, starting from an equivalent description of this model in terms of complex SU(2|1, 1) superconformal mechanics associated with the supermultiplet (2,4,2). Many peculiar features of the original formulation become simpler in the superconformal formulation, including, e.g., simplifying the formula for the energy spectrum. The phenomenon of disappearance of the dependence on the parameter λ in the second formulation also deserves an attention.

Supersymmetric C N Smorodinsky-Winternitz system
We define the quantum SU(2|1) supersymmetric C N S.-W. system as a sum of N copies of the C 1 system with the Hamiltonians (2.1) involving the same parameters B, ω (equivalently, m, λ), The supercharges and R-charges which form, together with the Hamiltonian H, su(2|1) superalgebra, are also defined as sums of the relevant quantities of each particular C 1 -system. Clearly, the generators H a commute with each other, and thus define the constants of motion of the supersymmetric C N S.-W. system. In addition to N commuting integrals H a , this system possesses N manifest U(1) symmetries z a → e iκ z a , ξ a i → e iκ ξ a i , with the generators Hence, these generators provide the system to be integrable. The wave functions of this system are obviously given by the products of those of N onedimensional copies, and the energy spectrum -by the sum of energies (2.27), One observes the same distinction between the spectra of bosonic and fermionic wave functions as in the C 1 case. The SU(2|1) supersymmetric C N S.-W. system has an additional degeneracy of the spectrum. It is due to the existence of the additional constants of motion given by the components of the supersymmetric extension of the Uhlenbeck tensor generating a hidden symmetry in the bosonic case. The classical version of this supersymmetric Uhlenbeck tensor was constructed in [11], while its quantum counterpart can be written in the form 5) where no sum over a and b is assumed. These constants of motion, together with (4.2) and H a , provide the system with the superintegrability property. It turns out that this tensor admits a convenient representation in terms of the generators of the associated superconformal algebra su(2|1, 1).

Superconformal view
Let us define the superconformal Hamiltonian on C N as a sum of N copies of superconformal Hamiltonians on C 1 , where H (conf) a is given by (3.1), with different parameters g a for each a (a = 1 . . . N ), but with the common parameters λ and m . So we deal with a direct sum of su(2|1, 1) algebras labeled by the index a. Then, we take sums of all these generators and obtain, once again, the conformal superalgebra (3.19) with the superconformal Hamiltonian (4.6). Here,Z 1a are defined asZ 1a = (L a ) 2 + g 2 a . (4.8) The wave eigenfunctions of (4.6) are obviously the products of N wave functions corresponding to a = 1 . . . N . Taking into account (4.4), the energy spectrum of the SU(2|1) HamiltonianH = H conf − mZ 1 /2 is given by the obvious generalization of the formula (3.27) where σ is a sum of σ a = 0, 1/2, 1 and it takes integer and half integer values ranged from 0 to N . Bosonic and fermionic states still occupy separate levels with integer and half-integer values of the energy (modulo the overall parameter m), respectively. The Uhlenbeck tensor (4.5) commutes with the superconformal Hamiltonian (4.6). In terms of the generators of the conformal algebra so(2, 1) it can be represented in the very simple form 10) or, in the basis (3.6), where M ab := 1 2m 2 T aTb . (4.12) The second form of I ab makes obvious its commutativity with the superconformal Hamiltonian (4.6), as well as with the Hamiltonian of C N S.-W. system (4.1). The non-linear algebra generated by I ab reads (no summation over repeated indices), where the function T cbd has a simple representation through the generators of conformal algebra: (4.14) Notice that for calculating the commutation relations (4.13) we do not need the explicit expressions for I ab in terms of the variables (z a ,z a , ξ ai ,ξ a i ) as in (4.5), now it suffices to make use of the standard commutation relations (3.5) or (3.7) of the conformal algebra so (2,1) .
Looking at the expressions (4.11) and (4.14), we observe that they involve, apart from N Hamiltonians H (conf)a , also N 2 bilinear generators M ab that commute with (4.6) and (4.1). Thus, what actually matters is the nonlinear closed algebra generated by M ab and H (conf) b : (4.16) (no summation over indices). One can add to this set the U(1) generators L a , which commute with everything. Note that the symmetric combination M ab + M bā can be directly expressed through H (conf) b and I ab from (4.11), but it is not true for the antisymmetric one M ab − M bā entering T abc . However, it is possible to express (M ab − M bā ) 2 through the rest of constants of motion: Thus the quantity T cbd defined in (4.14) is a function of the original hidden symmetry generators H (conf) a , I cd and so the relations (4.13), (4.14) constitute a closed non-linear algebra which is equivalent to the algebra (4.15), (4.16).
Like in the bosonic C 1 model (see (C.10)), the diagonal integrals I aa are yet expressed through other integrals: 20) The operators I 2 a are Casimirs for N copies of SU(2) symmetries acting only on fermionic variables. The additional new integrals of motion A a can be written in terms of the superconformal SU(2|1, 1) generators as In Appendix A we adduce some further details on the structure of these extra constants of motion. Degeneracy of the energy spectrum (4.9) can be attributed to any operator commuting with the Hamiltonian. One can construct many examples of such operators like (4.11), (4.12) or (3.33). Let us illustrate, on the simplest N = 2 example, how an action of the operators (4.12) creates (n + 1)-fold degeneracy of the bosonic wave functionsΩ + (n1,l1) ⊗Ω + (n2,l2) , with n = n 1 + n 2 . The action of M 12 on them is simple: It just increases the number n 1 as n 1 → n 1 + 1 and decreases n 2 as n 2 → n 2 − 1, so that the total number n = n 1 + n 2 is not altered. The action of M 21 is opposite: n 1 → n 1 − 1, n 2 → n 2 + 1.
A slight modification of the Uhlenbeck tensor (4.11) by other superconformal SU(2|1, 1) generators yields a generalization of the operator (3.36), such that it commutes also with the SU(2|1) generatorsQ i ,Q j , I i j andZ 1 defined by (4.7): In a similar way, the bilinear operator (4.12) is modified as Once again, these invariants can be constructed only for new superchargesQ i andQ j . No their analogs can be defined for su(2|1) generated by Q i andQ j . The algebra of the operators (4.25) and (4.26) is nonlinear and its closure lies in the universal enveloping of the superconformal algebra su(2|1, 1) (3.19). The non-zero commutators of the generators (4.25) and (4.26) are presented in the Appendix B. It is worth to point out that the crucial property for revealing various degeneracies of the su(2|1) multiplets of the wave functions is the commutativity ofM ab andĨ ab with the SU(2|1) generatorsQ i ,Q j , I i j andZ 1 and, hence, with the relevant Casimir operators. The precise structure of the closure of the hidden symmetry generators is not too important from this point of view.

Products of SU(2|1) representations
One can consider the degeneracy of eigenvalues of the Casimir operators (3.29) of the N dimensional system, though these eigenvalues cannot be presented by a generic formula and so each particular N ≥ 2 model requires a separate analysis. An additional degeneracy, besides the degeneracy with respect to SU(2|1) generators, comes out with respect to the hidden symmetry operators (4.25) and (4.26). Below we present their action as the hidden symmetry operators on the SU(2|1) multiplets of wave functions. We will always deal with the "superconformal" SU(2|1) generated by the generators Q i ,Q k and the relevant SU(2|1) multiplets.
The product of N one dimensional SU(2|1) representations can be decomposed as a non-trivial sum of irreducible SU(2|1) representations [22]. For simplicity we consider here only N = 2 case and present the decomposition for the levels n = 0, 1, 2.
The level n = 0. The lowest state with n = 0 is a product of the single states with n 1 = 0 and n 2 = 0: |0 ≡ |0, 0, 0, 0 = Ω + (0,l1) ⊗Ω + (0,l2) .  The product of the fundamental representations with n 1 = 1, n 2 = 1 has the total dimension 3 × 3 = 9: With respect to the generators (4.7) it splits into a sum of 4-dimensional typical representation and 5-dimensional atypical representation. The atypical representation is spanned by the triplet of bosonic states and the doublet of fermionic states: The operatorĨ 12 annihilates this state. The typical representation encompasses the states l2) .

Summary and outlook
In this paper we studied the quantum mechanics of SU(2|1) supersymmetric extension of the S.-W. system on the complex Euclidian space C N interacting with an external constant magnetic field [9]. This supersymmetric system can be considered as a unification of N non-interacting C 1 S.-W. systems. Accordingly, we first quantized the model on C 1 and then generalized the consideration to the case of C N . We constructed the complete space of the wave functions and found the relevant energy spectrum. We studied how all bosonic and fermionic states are distributed over the irreducible representations of the supergroup SU(2|1). We also showed that the bosonic S.-W. model possesses conformal symmetry SO(2, 1) (see Appendix C). In the supersymmetric case we redefined the Hamiltonian as (3.2) and showed that it exhibits SU(2|1, 1) superconformal symmetry which serves as the spectrum-generating symmetry on the full set of the quantum states.
The wave functions of supersymmetric quantum S.-W. system on C N were constructed as products of N wave functions of the C 1 models. Correspondingly, on these products irreducible SU(2|1) representations are realized. For simplicity we considered the case of N = 2 that already amounts to a non-trivial sum of irreducible SU(2|1) representations. Also, the generalization to C N reveals hidden symmetry generators (4.25) which correspond to a supersymmetrization of Uhlenbeck tensor [9,11]. It is responsible for the degeneracy of the wave functions belonging to irreducible SU(2|1) representations. The general expressions for the hidden symmetry generators in terms of products of the generators of the superconformal algebra su(2|1, 1) were found.
It would be interesting to consider, along the same lines, quantum deformed SU(2|1) extensions of other Kähler oscillator models, e.g., of the CP N one. These models are not superconformal, so their quantum analysis should be similar to what has been performed in Section 2. On the other hand, a non-trivial multi-particle extension of the SU(2|1) C 1 S.-W. model could be a complex N -particle interacting system of the Calogero-Moser type, hopefully preserving the superconformal invariance of the one-particle C 1 model. Then the whole consideration of Sections 3 and 4 based on the superconformal group SU(2|1, 1) could be applicable.
Finally, let us notice that the quantum C 1 S.-W. system without magnetic field (also known as a "circular oscillator with ring-shaped potential") was used in a more phenomenological setting for the study of the particle behavior in the two-dimensional quantum ring [23]. Respectively, the C N S.-W. system with coincident parameters g a can be interpreted as an ensemble of N free particles in a single quantum ring interacting with a constant magnetic field orthogonal to the plane. It would be interesting to reveal possible physical implications of the quantum SU(2|1) supersymmetric version of this system within such an interpretation.
where the totally antisymmetric tensorT abc is defined as with L playing the role of a central charge. The conformal algebra has the standard form (3.7) just in the basis with the conformal Hamiltonian, with L becoming an external generator commuting with all conformal generators. The generators T ,T and H conf act on the wave functions Φ (n,l) , as T Φ (n,l) = m Φ (n+1,l) ,T Φ (n,l) = n n +l m Φ (n−1,l) , H conf Φ (n,l) = m n +l 2 + 1 2 Φ (n,l) . (C. 8) We observe that the explicit dependence on the external magnetic field disappears in the spectrum of the conformal Hamiltonian H conf . The relevant Casimir operator of so(2, 1) defines an irreducible representation of the tower of states Φ (n,l) for n = 0, 1, 2, . . . and a fixed l: TT +T T Φ (n,l) = l2 − 1 4 Φ (n,l) . (C.9) So, at each l the quantum states constitute an infinite-dimensional irreducible representation of so(2, 1) ∼ sl(2, R). Hence the conformal algebra serves as the spectrum-generating algebra of the quantum C 1 S.-W. model. Note that the relation (C.9) immediately follows from the operator identity which holds upon substitution of the explicit expressions for the involved generators.
In the general case of C N S.-W. quantum system, the degeneracy with respect to (4.12) is given by the binomial coefficient C n n+N = (n+N )! n!N ! .