ar X iv : h ep - t h / 96 01 17 0 v 1 3 1 Ja n 19 96 Semi - infinite wedges and the conformal limit of the fermionic Calogero - Sutherland Model with spin

The conformal limit over an anti-ferromagnetic vacuum of the fermionic spin 1 2 Calogero-Sutherland Model is derived by using the wedge product formalism. The space of states in the conformal limit is identified with the Fock space of two complex fermions, or, equivalently, with a tensor product of an irreducible level-1 module of ŝl2 and a Fock space module of the Heisenberg algebra. The Hamiltonian and the Yangian generators of the CalogeroSutherland Model are represented in terms of ŝl2 currents and bosons. At special values of the coupling constant they give rise to the Hamiltonian and the Yangian generators of the conformal limit of the Haldane-Shastry Model acting in an irreducible level-1 module of ŝl2. At generic values of the coupling constant the space of states is decomposed into irreducible representations of the Yangian. 0 Introduction Solvable models with long-range interaction have become a rapidly developing area of research over the course of the past few years. One important source of these new developments was the paper [3] where the original spinless Calogero-Sutherland Model [6] had been generalized so as to include particles with spin, and where a connection between these generalized Calogero-SutherlandModels and the HaldaneShastry long-range interacting spin chain [10, 20] had been established. An essential feature of both generalized Calogero-Sutherland and the Haldane-Shastry Models, as emphasized in [3], is the presence of the Yangian symmetry algebra which commutes with the Hamiltonian. This infinite-dimensional algebra of symmetries can be efficiently used to compute spectra of excitations and, to some extent, dynamical correlation functions [11]. Remarkably, the Yangian was found to be an exact symmetry of the generalized Calogero-Sutherland and the (trigonometric) HaldaneShastry Models even when the last two are considered in finite volume and at fixed finite number of particles. This distinguishes the long-range interacting models from the class of solvable models with point interactions, such as the Heisenberg XXX spin chain and the Hubbard Model where Yangian symmetry is exact only in thermodynamic limit taken over anti-ferromagnetic vacuum. The work on conformal limit of the spin12 Haldane-Shastry Model, which was initiated in [12] and continued in [4] and [5], brought equally remarkable results. It was found, that in the conformal limit taken in the vicinity of the anti-ferromagnetic ground state, the space of states of the Haldane-Shastry Model can be identified with the direct sum of the two integrable, irreducible level-1 representations of the algebra ŝl2, so that the generators of the Yangian symmetry and the Hamiltonian are expressed in terms of ŝl2 currents. The decomposition of the space of states into irreducible representations of the Yangian provides a new basis of the level-1 representations of ŝl2 and, therefore, new character formulas for these representations. This new basis is written in terms of the Vertex Operators associated with the level-1 ŝl2-representations, which, in the context of the Haldane-Shastry Model are interpreted as creation operators of spinon excitations – particles with spin 12 and half-integer statistics [11]. These results were further generalized to include sln Haldane-Shastry Model , and the q-deformed situation [13]. In the case of finite number of particles several authors observed [22],[19] , that the Haldane-Shastry spin chain can be considered a special subcase of the generalized Calogero-Sutherland Model where coordinates of the particles are frozen so that only the spin degrees of freedom remain relevant. This suggests, that a similar situation may be encountered in the conformal limit as well. In fact, this is a point of view adopted in [4]. Thus the natural problem arises – to find what is the conformal limit of the Calogero-Sutherland Model with spin and how to derive the Hamiltonian and the Yangian symmetry of the Haldane-Shastry Model from the corresponding Calogero-Sutherland objects. This is the main issue with which we deal in this paper. Methodologically, a new feature of the present work is an extensive use of the wedge product formalism. About this formalism a reader can consult the works [21],[14] where it is introduced in the general, q-deformed setting. The approach which we use can be summarized as follows. First we reformulate the finite-particle fermionic Calogero-Sutherland Model with spin 12 in terms of the finite wedge product of infinite-dimensional spaces V (z) = C[z, z] ⊗ C. In the space V (z) and in the wedge product one naturally defines a level-0 action of ŝl2 . In the wedge product language anti-ferromagnetic vacua of the Model have simple and explicit form. Taking advantage of this one can implement a conformal limit in the vicinity of any of these vacua by going from finite to semi-infinite wedge product. The wedge product formulation makes the transition to the conformal limit especially transparent. The expressions for the spin Calogero-Sutherland Hamilto-