Correspondence between conformal field theory and Calogero-Sutherland model

We use the Jack symmetric functions as a basis of the Fock space, and study the action of the Virasoro generators $L_n$. We calculate explicitly the matrix elements of $L_n$ with respect to the Jack-basis. A combinatorial procedure which produces these matrix elements is conjectured. As a limiting case of the formula, we obtain a Pieri-type formula which represents a product of a power sum and a Jack symmetric function as a sum of Jack symmetric functions. Also, a similar expansion was found for the case when we differentiate the Jack symmetric functions with respect to power sums. As an application of our Jack-basis representation, a new diagrammatic interpretation is presented, why the singular vectors of the Virasoro algebra are proportional to the Jack symmetric functions with rectangular diagrams. We also propose a natural normalization of the singular vectors in the Verma module, and determine the coefficients which appear after bosonization in front of the Jack symmetric functions.Comment: 23 pages, references adde

Multidimensional Calogero systems from matrix models

We show that a particular many-matrix model gives rise, upon hamiltonian reduction, to a multidimensional version of the Calogero-Sutherland model and its spin generalizations. Some simple solutions of these models are demonstrated by solving the corresponding matrix equations. A connection of this model to the dimensional reduction of Yang-Mills theories to (0+1)-dimensions is pointed out. In particular, it is shown that the low-energy dynamics of D0-branes in sectors with nontrivial fermion content is that of spin-Calogero particles.Comment: 12 pages, no figures, plain tex, phyzzx macr

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.Comment: 14 pages, LaTeX, no figure

Generalized Calogero-Sutherland systems from many-matrix models

We construct generalizations of the Calogero-Sutherland-Moser system by appropriately reducing a model involving many unitary matrices. The resulting systems consist of particles on the circle with internal degrees of freedom, coupled through modifications of the inverse-square potential. The coupling involves SU(M) non-invariant (anti)ferromagnetic interactions of the internal degrees of freedom. The systems are shown to be integrable and the spectrum and wavefunctions of the quantum version are derived.Comment: 8 pages, LaTeX, no figure

Exact Spectrum of SU(n) Spin Chain with Inverse-Square Exchange

The spectrum and partition function of a model consisting of SU(n) spins positioned at the equilibrium positions of a classical Calogero model and interacting through inverse-square exchange are derived. The energy levels are equidistant and have a high degree of degeneracy, with several SU(n) multiplets belonging to the same energy eigenspace. The partition function takes the form of a q-deformed polynomial. This leads to a description of the system by means of an effective parafermionic hamiltonian, and to a classification of the states in terms of "modules" consisting of base-n strings of integers.Comment: 12 pages, CERN-TH-7040/9

Yangian-invariant field theory of matrix-vector models

We extend our study of the field-theoretic description of matrix-vector models and the associated many-body problems of one dimensional particles with spin. We construct their Yangian-su(R) invariant Hamiltonian. It describes an interacting theory of a c=1 collective boson and a k=1 su(R) current algebra. When $R \geq 3$ cubic-current terms arise. Their coupling is determined by the requirement of the Yangian symmetry. The Hamiltonian can be consistently reduced to finite-dimensional subspaces of states, enabling an explicit computation of the spectrum which we illustrate in the simplest case.Comment: Extensive modifications in the construction of the spinon basis and the subsequent computations of eigenvalues and eigenvectors. Title and references modified. To appear in Nuclear Physics B. 21 pages, Late

Applications of the Collective Field Theory for the Calogero-Sutherland Model

We use the collective field theory known for the Calogero-Sutherland model to study a variety of low-energy properties. These include the ground state energy in a confining potential upto the two leading orders in the particle number, the dispersion relation of sound modes with a comparison to the two leading terms in the low temperature specific heat, large amplitude waves, and single soliton solutions. The two-point correlation function derived from the dispersion relation of the sound mode only gives its nonoscillatory asymptotic behavior correctly, demonstrating that the theory is applicable only for the low-energy and long wavelength excitations of the system.Comment: LaTeX, 31 page

Invariants of the Haldane-Shastry SU(N)SU(N) Chain

Using a formalism developed by Polychronakos, we explicitly construct a set of invariants of the motion for the Haldane-Shastry $SU(N)$ chain.Comment: 11 pages, UVA-92-0

On the solutions of multicomponent generalizations of the Lam{\'e} equation

We describe a class of the singular solutions to the multicomponent analogs of the Lam{\'e} equation, arising as equations of motion of the elliptic Calogero--Moser systems of particles carrying spin 1/2. At special value of the coupling constant we propose the ansatz which allows one to get meromorphic solutions with two arbitrary parameters. They are quantized upon the requirement of the regularity of the wave function on the hyperplanes at which particles meet and imposing periodic boundary conditions. We find also the extra integrals of motion for three-particle systems which commute with the Hamiltonian for arbitrary values of the coupling constant.Comment: 8 pages, 1 table, no figures, revtex