${\cal N}$-extended supersymmetric Calogero models

We propose a new ${\cal N}$-extended supersymmetric $su(n)$ spin-Calogero model. Employing a generalized Hamiltonian reduction adopted to the supersymmetric case, we explicitly construct a novel rational $n$-particle Calogero model with an arbitrary even number of supersymmetries. It features ${\cal N}n^2$ rather than ${\cal N}n$ fermionic coordinates and increasingly high fermionic powers in the supercharges and the Hamiltonian.


Introduction
The original rational Calogero model of n interacting identical particles on a line [1], pertaining to the roots of A 1 ⊕ A n−1 and given by the classical Hamiltonian 1) has often been the subject of "supersymmetrization". In this endeavor, extended supersymmetry has turned out to be surprisingly rich. After the straightforward formulation of N = 2 supersymmetric Calogero models by Freedman and Mende [2], a barrier was encountered at N = 4 [3]. An important step forward then was the explicit construction of the supercharges and the Hamiltonian for the N = 4 supersymmetric three-particle Calogero model [4,5], which introduced a second prepotential F besides the familiar prepotential U . However, it was found that quantum corrections modify the potential in (1.1), and that F is subject to intricate nonlinear differential equations, the WDVV equations, beyond the three-particle case. These results were then confirmed and elucidated in a superspace description [6]. Finally, extending the system by a single harmonic degree of freedom (su(2) spin variables [7]) it was possible to write down a unique osp(4|2) symmetric four-particle Calogero model [8]. 1 A detailed discussion concerning the supersymmetrization of the Calogero models can be found in the review [9].
It seems that a guiding principle was missing for the construction of extended supersymmetric Calogero models. Indeed, while for n ≤ 3 translation and (super-)conformal symmetry almost completely defines the system, the n ≥ 4 cases admit a lot of freedom which cannot a priori be fixed. In the bosonic case, such a guiding principle exists [10]. The Calogero model as well as its different extensions (see, e.g. [11,12,13]) are closely related with matrix models and can be obtained from them by a reduction procedure (see [14] for first results and [15] for a review). If we want to employ this principle also for finding extended supersymmetric Calogero models, then the two main steps are • supersymmetrization of a matrix model • supersymmetrization of the reduction procedure or proper gauge fixing.
This idea is not new. It has successfully been employed in [16,17,18,19]. The resulting supersymmetric systems feature • a large number of fermions -far more than the 4n fermions expected in an N = 4 n-particle system within the standard (but unsuccessful!) approach • a rather complicated structure of the supercharges and the Hamiltonian, with fermionic polynomials of maximal degree • a variety of bosonic potentials, including su(2) spin-Calogero interactions but they do not contain a genuine N = 4 supersymmetric Calogero model, i.e. one with a mere pairwise inversesquare no-spin bosonic potential.
Here we use the same guiding principle and start with the bosonic su(n) spin-Calogero model in the Hamiltonian approach. We then provide an N -extended supersymmetrization of this system. It is important that we do not a priori fix a realization for the su(n) generators. Finally we generalize the reduction procedure to the N -extended system and find the first N -extended supersymmetric Calogero model, for any even number of supersymmetries.

Bosonic Calogero model from hermitian matrices
It is well known that the rational n-particle Calogero model [1] can be obtained by Hamiltonian reduction from the hermitian matrix model [10,14]. Adapted to our purposes, the procedure reads as follows. One starts from the su(n) spin generalization [12] of the standard Calogero model, as given by (2.1) The particles are described by their coordinates x i and momenta p i together with their internal degrees of freedom encoded in the angular momenta (ℓ ij ) † = ℓ ji with i ℓ ii = 0. The non-vanishing Poisson brackets are The Hamiltonian (2.1) follows directly from the free hermitian matrix model (for details see [15]).
To get the standard Calogero Hamiltonian (1.1) from (2.1) one has to reduce the angular sector of the latter, in two steps. Firstly, one (weakly) imposes the constraints They commute with the Hamiltonian (2.1) and with each other, hence are of first class. To resolve them one introduces auxiliary complex variables and realizes the su(n) generators ℓ ij asl Secondly, passing to polar variables r i and φ i defined as the constraints (2.3) are resolved by putting Plugging this solution into the Hamiltonian (2.1) one may additionally fix n−1 angles φ i , say At this stage the 2n variables {r i , φ i } are reduced to the two variables r n and φ n . However, the reduced Hamiltonian does not depend on φ n and has the form Therefore {H red , r n } ≈ 0 and r 2 n ≈ const =: g , (2.10) and the reduced Hamiltonian H red coincides with the standard n-particle rational Calogero Hamiltonian. We note that in the bosonic case most reduction steps are not needed, because the Hamiltonian (2.1) does not depend on the angles φ i at all. However, in the supersymmetric case all reduction steps will be important.
In what follows we will construct an N -extended supersymmetric generalization of the Hamiltonian (2.1) and perform the supersymmetric version of the reduction just discussed, finishing with an N -extended supersymmetric Calogero model, for N = 2M and M = 1, 2, 3, . . ..

N -extended supersymmetric su(n) spin-Calogero model
On the outset we have to clarify what is the minimal number of fermionic variables necessary to realize an N = 2M supersymmetric extension of the su(n) spin-Calogero model (2.1). Clearly, as partners to the bosonic coordinates x i one needs N n fermions ψ a i andψ i a with a = 1, 2, . . . M . However, this is not enough to construct N supercharges Q a and Q b which must generate the N = 2M superalgebra The reason is simple: to generate the potential term in the Hamiltonian, the supercharges Q a and Q b must contain the terms respectively, where ρ a ij andρ ij a are some additional fermionic variables. These fermions cannot be constructed from ψ a i orψ i a . Hence, we are forced to introduce N n(n−1) further independent fermions ρ a ij andρ ij a subject to ρ a ii =ρ ii a = 0 for each value of the index i. In total, we thus utilize N n 2 fermions of type ψ or ρ, which we demand to obey the following Poisson brackets, ij † =ρ ji a and ρ a ii =ρ ii a = 0 . (2.13) The next important ingredient of our construction is the composite object (2.14) One may check that, with respect to the brackets (2.13), the Π ij form an su(n) algebra just like the ℓ ij , and they commute with the our fermions as follows, It is a matter of straightforward calculation to check that the supercharges obey the N = 2M superalgebra (2.11) with the Hamiltonian modulo the first-class constraints The supercharges Q a and Q b in (2.17) and the Hamiltonian H in (2.18) describe the N = 2M supersymmetric su(n) spin-Calogero model.
For N = 4 it essentially coincides with the osp(4|2) supersymmetric mechanics constructed in [16,17]. However, there are a few differences: • The Hamiltonian (2.18) has no interaction for the center-of-mass coordinate X = i x i . Correspondingly, the supercharges (2.17) do not include certain terms which appeared in [16,17]. • Working at the Hamiltonian level, we may keep the su(n) generators ℓ ij unspecified. Precisely this enables the minimal realization (2.5) with a minimal number of auxiliary variables v i ,v i . At the Lagrangian level this corresponds to using (2,4,2) supermultiplets for the auxiliary bosonic superfields instead of (4, 4, 0) superfields as in [16,17]. Now we are ready to reduce our N = 2M su(n) spin-Calogero model to a genuine N = 2M Calogero model.

N -extended supersymmetric (no-spin) Calogero models
As we can see from the previous subsection, the supersymmetric analogs (2.19) of the purely bosonic constraints (2.3) appear automatically. These constraints generate n−1 local U(1) transformations 2 of the variables {v i ,v i , ρ a ij ,ρ ij a }. In terms of the 2n polar variables r i and φ i defined in (2.6), the constraints (2.19) can be easily resolved as r 2 k ≈ r 2 n + Π kk − Π nn for k = 1, . . . , n−1 .

(2.21)
After fixing the residual gauge freedom as we obtain the supercharges and Hamiltonian which still obey the N = 2M superalgebra (2.11) and contain only the surviving pair (r n , φ n ) of the originally 2n "angular" variables. One may check that the supercharges Q a and Q b and the Hamiltonian H, with the generators ℓ ij replaced byl ij and with the constraints (2.21) and (2.22) taken into account, perfectly commute with r 2 n − Π nn . Thus, the final step of the reduction is to impose the constraint r 2 n − Π nn ≈ const =: g (2.23) and to fix the remaining U(1) gauge symmetry via The previous two relations are the supersymmetric analogs of (2.10). We conclude that the full set of the reduction constraints reads r 2 i ≈ g + Π ii and φ i ≈ 0 for i = 1, . . . , n . (2.25) With these constraints taken into account, our supercharges Q a and Q b and the Hamiltonian H acquire the form (2.26) It is matter of quite lengthy and tedious calculations to check that these supercharges and Hamiltonian form an N = 2M superalgebra (2.11). The main complication arises from the expressions √ g + Π ii present in the supercharges and the Hamiltonian. Due to the nilpotent nature of Π ij , the series expansion eventually terminates, but even in the two-particle case with N = 4 supersymmetry we encounter a lengthy expression, For n particles the series will end with a term proportional to (Π ii ) N (n−1) . Clearly, these terms will generate higher-degree monomials in the fermions, both for the supercharges and for the Hamiltonian. We can only speculate that the dread of such complexities impeded an earlier discovery of genuine N = 4 Calogero models.
2 Due to the relation n i χ i = 0 we have only n−1 independent constraints.

2.4
Simplest example: N = 2 supersymmetric two-particle Calogero model For N = 2 supersymmetry one has to put M = 1 in the expressions (2.26) for the supercharges and Hamiltonian. This somewhat reduces their complexity compared to the N = 4 case, but the real simplification occurs for two particles. Indeed, for n=2 we get for g = 0 .

Conclusion
We propose a novel N -extended supersymmetric su(n) spin-Calogero model as a direct supersymmetrization of the bosonic su(n) model [12]. In the case of N = 4 supersymmetry, our model resembles the one constructed in [16,17]. However, there are two main differences: • the center of mass is free • the su(n) generators are not specified in a particular realization.
Thanks to these features, we were able to generalize the reduction procedure to the no-spin Calogero model from N = 4 supersymmetry to any number N = 2M of supersymmetries. This lead to the discovery of a genuine N = 2M supersymmetric rational Calogero model for any number of particles.
Our models belong to same class which was proposed in [16,17]. Its main features are • a huge number of fermionic coordinates, namely N n 2 in number rather than the N n to be expected • the supercharges and the Hamiltonian contain terms which a fermionic power much larger than three.
Clearly, these features merit a more careful and detailed analysis.
The following further developments come to mind: • a superspace description of the constructed models, at least for N = 2 and N = 4 supersymmetry, presumably with nonlinear chiral supermultiplets • an extension to the Calogero-Sutherland inverse-sine-square model • an extension to the Euler-Calogero-Moser system [11] and its reduction to the goldfish system [13], yielding a supersymmetric goldfish model upon reduction, to be compared with recent results from [20].