Sutherland models

Starting from the Hamiltonian formulation of supersymmetric Calogero models associated with the classical An, Bn, Cn and Dn series we construct the N=2 and N=4 supersymmetric extensions of the their hyperbolic/trigonometric Calogero–Sutherland cousins. The bosonic core of these models are the standard Calogero– Sutherland hyperbolic/trigonometric systems. PACS numbers: 11.30.Pb, 11.30.-j, 02.30.lk


Introduction
There is a lot of confirmation that N = 4 supersymmetric extensions of Calogero-Moser systems must include a large number of fermions -far more than the 4n fermions expected within the standard (but not very successful) approach [1,2,3,4,5,6]. The source of these fermions is the supersymmetrization of the matrix models from which, in the purely bosonic cases, the Calogero-Moser systems can be obtained by a reduction (see e.g. [7]).
A suitable approach to supersymmetric Calogero-like models has been proposed in [1,2,3]. Starting from a supersymmetrization of the Hermitian matrix model, the resulting matrix fermionic degrees of freedom are packaged in N = 4 superfields. In a recent paper [2], N = 2 and N = 4 supersymmetric extensions of the multiparticle hyperbolic Calogero-Sutherland system were constructed by applying a gauging procedure [8] to one-dimensional matrix superfield systems. However, their bosonic part does not reproduce the ordinary Calogero systems but only spin-Calogero ones. 1 In a series of papers [4,5,6] 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. Here, the supercharges contain the fermion-cube terms only through the combination fermion × fermion bilinear, where the fermion bilinears span an s(u(n) ⊕ u(n)) algebra.
In this paper we use this ansatz to construct N = 4 supersymmetric extensions of the Calogero-Sutherland models associated with the classical A n , B n , C n and D n series (Section 2). As a separate application, we also find the N = 4 supersymmetric extensions of the trigonometric/hyperbolic cousins of the Euler-Calogero-Moser system (Section 3).

Basic ingredients
The starting point of our construction is the same set of the fields as in the N -extended supersymmetric Calogero-Moser model [4] which is nothing but a supersymmetric extension of the Hermitian matrix model [9,10,7]. This set of fields includes the following ones • n bosonic coordinates x i , which come from the diagonal elements of the Hermitian matrix X, and the corresponding momenta p i for i, j = 1, . . . , n which obey the standard brackets

1)
• fermionic matrices containing N n 2 elements ξ a ij ,ξ ij a for a = 1, . . . , N /2 with (ξ a ij ) † =ξ ji a and brackets Using these ingredients one may construct the fermionic bilinears ξ a ikξ kj a +ξ ik a ξ a kj , which form an s(u(n) ⊕ u(n)) algebra, Using these ingredients in [6] the supercharges and Hamiltonian have been constructed for arbitrary even-N supersymmetric extensions of the A n , B n , C n and D n rational Calogero models. In what follows we will use the same ingredients to construct N = 2, 4 trigonometric/hyperbolic Calogero-Sutherland models with the supercharges and the Hamiltonian obeying the N = 2, 4-extended super-Poincaré algebra.
In this simplest case the supercharges have a quite simple structure 2 Note that Π ij does not appear here and the function f will be specified in a moment. These supercharges form an N = 2 super-Poincaré algebra together with the Hamiltonian Here, we abbreviated 9) and the constant parameter β and the function f are given as follows, . (2.10) Thus, the supercharges (2.6) and the Hamiltonian (2.8) describe an N = 2-extended supersymmetric Calogero-Sutherland models of type A n−1 ⊕ A 1 .
It should be noted that when checking that the supercharges form the superalgebra (2.7) it is not enough to know the brackets between Π ij and the fermions ξ ij ,ξ ij . Instead, the explicit expressions for Π ij (2.3) have to be substitute in the (2.6). This makes the calculations slightly more complicated as comparing to those ones discussed in [6].

N = 4 supersymmetric A n−1 ⊕ A 1 Calogero-Sutherland models
Due to the absence of any guiding rules for construction of N = 4 supercharges, the reasonable starting point is the straightforward generalization of the N = 2 supercharges (2.6) to the N = 4 supersymmetry reads (2.11) Unfortunately, this guess is not correct and the supercharges (2.11) do not form the N = 4 superalgebra in contrast with their N = 2 cousins (2.6) . The possible modification of the supercharges looks as follows which has the same form as N = 2 Hamiltonian (2.8).
It should be mentioned that in the N = 2 case the additional, β-dependent terms in the supercharges (2.13) are automatically nullified in virtue of the structure of Π ij (2.3) and thus, the supercharges (2.13) reduced to the supercharges (2.6) in the limit a, b = 1. Finally, all we said above is valid only for the functions f from the list (2.10).

N = 4 supersymmetric B n , C n and D n Calogero-Sutherland models
It is strange but for the B, C and D-type models the N = 4 supercharges take the same form as N = 2 ones. Indeed, one may check that the following supercharges (including Π ij ), (2.17) Here, and the function f is the same as in (2.10). The bosonic sector of the Hamiltonian (2.17) 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. Finally, let us noted that in the N = 2 supersymmetric case the last term in the Hamiltonian (2.17) nullified automatically due to structure of the Π ij and Π ij (2.3) (2.20)

Towards higher supersymmetries
It is interesting to note that the supercharges (2.13) obey the relations for arbitrary range of the indices a, b running from one to N /2. However, the anti-commutators between these supercharges have more complicated structure 3 Euler-Calogero-Moser models

Basic ingredients
The construction of the supersymmetric extension [5] of the Euler-Calogero-Moser systems [9] is a more economical as comparing to the supersymmetric Calogero-Sutherland systems we considered in the previous Sections. Indeed, to construct the corresponding N supercharges one needs to introduce "only" N 1 2 n(n + 1) fermions ρ a ij ,ρ ij a symmetric over indices i, j ρ a ij = ρ a ji ,ρ a ij =ρ a ji and obeying the brackets The internal degrees of freedom of ECM models are encoded in the angular momenta ℓ ij = −ℓ ji with the Poisson brackets forming the so(n) algebra Similarly to the construction of the supersymmetric Calogero-Sutherland systems, our anzats for the supercharges include the following fermionic bilinears 3 ρ a ikρ kj a + ρ a jkρ ki a , Finally, one may check that the N supercharges Q a , Q b form N -extended super Poincaré algebra (2.12) [5] together with the Hamiltonian (3.5)

N = 4 supersymmetry
The construction of the supersymmetric extension of the hyperbolic/trigonometric ECM model is very similar to the case of the Calogero model. Again, we succeeded in the construction of the N = 4 supersymmetric extensions, only. The main idea of our construction is to maximally preserve the anzatz for the supercharges (3.4), i.e. we admit the appearance of the three-linear fermionic terms in the supercharges only through the bilinears (3.3). Thus, our anzatz for the supercharges reads