New ${\cal N}{=}\,2$ superspace Calogero models

Starting from the Hamiltonian formulation of ${\cal N}{=}\,2$ supersymmetric Calogero models associated with the classical $A_n, B_n, C_n$ and $D_n$ series and their hyperbolic/trigonometric cousins, we provide their superspace description. The key ingredients include $n$ bosonic and $2n(n{-}1)$ fermionic ${\cal N}{=}\,2$ superfields, the latter being subject to a nonlinear chirality constraint. This constraint has a universal form valid for all Calogero models. With its help we find more general supercharges (and a superspace Lagrangian), which provide the ${\cal N}{=}\,2$ supersymmetrization for bosonic potentials with arbitrary repulsive two-body interactions.


Introduction
Recent progress in the construction of supersymmetric extensions of Calogero models [1,2] (see [3] for a review) was achieved by adding to the system more fermions as compared to the standard supersymmetrization [4,5,6]. It was inspired by supersymmetric extensions of the matrix models which, upon reduction or gauge fixing, give rise to the familiar bosonic systems. The approach developed in [7,8,9] for the rational spin-Calogero models with N = 2, 4 supersymmetry was recently extended to N = 2, 4 supersymmetric hyperbolic Calogero models [10,11]. However, it suffers from an unclear structure of the bosonic matrix model one has to start from.
In the series of papers [12,13,14] we developed a different approach. Mainly working in the Hamiltonian formulation, we worked out an ansatz for the supercharges which accommodates all Calogero models associated with the classical A n , B n , C n and D n Lie algebras and their trigonometric/hyperbolic extensions [14]. Having at hands the Hamiltonian description of N -extended supersymmetric Calogero models, it is of interest to gain also its superfield formulation, at least in the simplest case of N = 2 supersymmetry. 1 The superspace picture will help in understanding the general supersymmetry structure and clarify the role played by the additional matrix fermions. This is the main goal of this work.
The plan of the paper is as follows. In Section 2 we review the Hamiltonian description of the supersymmetric Calogero models. Their superfield treatment is performed in Section 3 (A 1 ⊕ A n−1 models) and in Section 4 (B n , C n , D n models). The most remarkable result here is a universal nonlinear fermionic chiral supermultiplet which collects all matrix fermions occurring in all super-extended Calogero models. In Section 5 we present more general supercharges (and a superspace Lagrangian), which provide an N = 2 supersymmetrization for a bosonic potential 2 with an arbitrary function f . We conclude with a short summary and possible extensions.

Hamiltonian description of N = supersymmetric Calogero models
In the Hamiltonian approach the n-particle supersymmetric Calogero model with N = 2-extended supersymmetry [14,12,13] features the following degrees of freedom: • n bosonic coordinates x i and momenta p i with i = 1, . . . , n, Their non-vanishing Poisson brackets are A central role of our construction take the composite objects Π ij and Π ij defined as One may easily check that Π ij and Π ij together form an s(u(n) ⊕ u(n)) algebra, 2 Our N = 2 supersymmetric Calogero models of A-type [14,13] are defined by a generic form of their supercharges, with some function f , to be specified in a moment. Note that Π ij does not appear here. These supercharges form an N = 2 super-Poincaré algebra, together with the Hamiltonian Here, we abbreviated 8) and the constant parameter α and the function f are given as follows, . (2.9) For the B, C and D-type models, the supercharges take a more complicated generic form (including Π ij ), Here, 11) and the function f is the same as in (2.9). The supercharges (2.10) form the same N = 2 super-Poincaré algebra (2.6) together with the Hamiltonian (2.12)

Its bosonic sector reads
Due to the presence of only two coupling constants, g and g ′ , we may describe B, C and D-type models in the rational case and C and D (but not B)-type models in the hyperbolic/trigonometric case.
To provide a superspace description of N = 2 supersymmetric Calogero models one has, firstly, to assemble the physical components x i , ψ i ,ψ i , ξ ij andξ ij into N = 2 superfields. It immediately follows from the structure of the supercharges Q and Q (2.5) that under N = 2 supersymmetry the coordinates x i transform into the fermions ψ i andψ i : Thus, one is let to n bosonic N = 2 superfields x i with the components 3 Concerning the fermionic components ξ ij ,ξ ij , we have no other possibility than to put them into 2n(n−1) new fermionic superfields ξ ij andξ ij with vanishing diagonal parts, i.e.
As N = 2 superfields the ξ ij andξ ij contain a lot of components. Hence, they have to be constrained somehow.
The appropriate constraints derive from the explicit form of the supercharges Q and Q (2.5), which leads to the following supersymmetry transformations of the leading components ξ ij andξ ij of these superfields, (3.4) To realize this transformation property we are forced to impose a nonlinear chirality condition on the superfields ξ ij andξ ij , (3.5) This condition leaves in the superfields ξ ij andξ ij only the components Finally, to obtain the correct brackets (2.1) for (ψ i ,ψ i ) and (ξ ij ,ξ ij ) after passing to the Hamiltonian formalism, the kinetic terms for these fermionic components must have the form In N = 2 superspace, this amounts to the free action (g = 0) More interesting is the construction of the interaction terms. Again, some hints come from the transformation properties of the fermions ξ ij andξ ij under Q and Q supersymmetry, respectively, To reproduce such terms in superspoace, the unique possibility is to add to the action S 0 (3.8) a term To be supersymmetrically invariant, the integrands in (3.10) must be chiral and antichiral, respectively. It is not too hard to check that this is indeed so: the nonlinear chirality constraint (3.5) implies that Combining all these facts together, we conclude that the superfield action reads where the superfields ξ ij andξ ij are subject to the nonlinear chirality constraint (3.5).
It is important to note that, after passing to new fermionic superfields the nonlinear constraint (3.5) is slightly simplified to (3.14) In this form, the constraint has lost any f -dependence, which however will reappear in the action, Also, the component Lagrangian, Hamiltonian and Poisson brackets will be more complicated in terms of the composite superfields λ ij andλ ij . Despite the extremely simple form of the superfield action (3.12), its component version looks quite complicated due to the constraint (3.5). Indeed, after integration over θ in (3.12) we get the off-shell Lagrangian (3.17) To eliminate the auxiliary fields A i and B ij one firstly has to evaluate the terms in the second line of (3.17) by using the constraint (3.5). This is a straightforward but rather tedious calculation. After employing the equations of motion we finally obtain the desired result, 4 N = 2 superspace B n , C n and D n Calogero models The supercharges of the N = 2 supersymmetric B, C and D-type Calogero models (2.10) have a more complicated structure than those in (2.5). Therefore, it is expected that the nonlinear chirality constraint for the superfields ξ ij andξ ij are more intricate as well. Indeed, the explicit structure of the supercharges (2.10) uniquely fixes this constraint to be (4.1) The complicated form of this constraint disappears after passing to the composite superfields in which it acquires its familiar form (3.14), Finally, the superfield action reads where h ′ (y ii ) = f (y ii ). (4.4) Compared to the action of the A 1 ⊕ A n−1 Calogero models (3.15), only the term 1 2 g ′ dt d 2 θ h(y ii ) carrying the new coupling constant g ′ appears in the action (4.3). All other terms just mimic those in (3.15).
It is a matter of straightforward but tedious calculations to check that, after excluding the auxiliary fields by their equations of motion, the final Lagrangian acquires the expected form

Conclusions
In this paper we have provided a superspace description of the N = 2 supersymmetric Calogero models, rational as well as trigonometric/hyperbolic, associated with the classical A n , B n , C n and D n Lie algebras. We presented a minimal superfield content accomodating the 2n 2 fermions for the N = 2 supersymmetric n-particle model. As 2n fermions accompany the n bosonic coordinates in general bosonic N = 2 superfields, the remaining 2n(n−1) fermions must be put into additional fermionic N = 2 superfields, which have to be constrained such as to describe those fermions alone. The nonlinear chirality condition (3.14) written in terms of commposite superfields solves this task. These composite fermionic N = 2 superfields make the constraint look simple and universal but complicate the Lagrangian. In terms of the fundamental fermions it is the other way around. We finally presented more general supercharges (and the superspace Lagrangian) which provides an N = 2 supersymmetrization of bosonic n-particle systems with an arbitrary repulsive two-body interaction. One might criticize that the approach presented here is unnecessarily complicated, because all N = 2 supersymmetric Calogero models can be more or less straightforwardly formulated in the standard fashion employing the minimum of 2n fermions [4]. This, however, is no longer the case with N > 2 supersymmetric Calogero models, where our treatment with additional fermions becomes essential. Hence, we consider our results here as a preparation for attacking Calogero systems with more supersymmetry in a superspace setting. The main excuse for presenting of our N = 2 results is the universal form of the nonlinear chirality constraint together with the almost trivial generalization of the supercharges to N = 4 supersymmetry [14] which will further the superspace construction of N ≥ 4 Calogero models.
The fourth and sixth terms cancel each other out. Then, replacing the indices in the first and second terms as k ↔ n makes them cancel with the third and seventh terms. Thus, after replacing the indices in the first term as k ↔ n, we remain with n =i,j k =i,n λ ik λ kn λ nj − k =i,j n =k,j λ ik λ kn λ nj . (A.5) A sum over k in the first term can be split into two pieces, corresponding to k = j or k = j, yielding n =i,j k =i,n λ ik λ kn λ nj = n =i,j λ ij λ jn λ nj + n =i,j k =i,j,n λ ik λ kn λ nj . (A.6) Performing the analogous splitting for the index n in the second term of (A.5), n = i or n = i, one finds k =i,j n =k,j λ ik λ kn λ nj = k =i,j λ ik λ ki λ ij + k =i,j n =i,j,k λ ik λ kn λ nj . (A.7) After relabeling k ↔ n, the second terms on the r.h.s. of (A.6) and (A.7) become identical. Hence, their contribution to (A.5) cancels, It remains to show that these twelve terms combine to zero. The sum of the first five term cancels with the eighth term. Rewriting the sixth term as a sum of two pieces, separating n = i and n = i, one may observe it to cancel with the seventh and the twelveth term. Analogously, the ninth, tenth and eleventh term add to zero. Nothing remains. This completes the proof.