Generalized Calogero models through reductions by discrete symmetries

We construct generalizations of the Calogero-Sutherland-Moser system by appropriately reducing a classical Calogero model by a subset of its discrete symmetries. Such reductions reproduce all known variants of these systems, including some recently obtained generalizations of the spin-Sutherland model, and lead to further generalizations of the elliptic model involving spins with SU(n) non-invariant couplings. PACS: 03.65.Fd, 71.10.Pm, 11.10.Lm, 03.20.+i E-mail: [email protected] The inverse-square interacting particle system [1, 2, 3] and its spin generalizations [4, 5, 6, 7, 8, 9] are important models of many-body systems, due to their exact solvability and intimate connection to spin chain systems [10, 11, 12, 13], 2-dimensional Yang-Mills theories [14, 15, 16] etc. Most of the variants of these systems can be though of as appropriate ‘foldings’ of the basic Calogero model with an augmented number of particles. Versions of this idea have appeared in the early literature, and have been been used, e.g., to motivate the Sutherland [2] and elliptic (Weierstrass) [17] versions of these systems. In this paper, we use this approach in the case of spin-generalized systems to give a more intuitive derivation and interpretation to some recently produced models and to derive new models. We begin with a brief review of known cases. We consider the Calogero model in connection with some of its discrete symmetries D. The equations of motion remain invariant under the phase space mapping φ → D(φ), where φ are phase space variables. Then the reduction to the invariant subspace φ = D(φ) is kinematically preserved, that is, the equations of motion do not move the system out of this subspace. Therefore, reducing the initial value data to this subspace trivially produces a system as solvable as the original one. The motion will be generated by the original hamiltonian on the reduced space. The starting point will be the inverse-square scattering particle system H = N ∑