Collective Field Theory, Calogero-Sutherland Model and Generalized Matrix Models

On the basis of the collective field method, we analyze the Calogero– Sutherland model (CSM) and the Selberg–Aomoto integral, which defines, in particular case, the partition function of the matrix models. Vertex operator realizations for some of the eigenstates (the Jack polynomials) of the CSM Hamiltonian are obtained. We derive Virasoro constraint for the generalized matrix models and indicate relations with the CSM operators. Similar results are presented for the q–deformed case (the Macdonald operator and polynomials), which gives the generating functional of infinitely many conserved charges in the CSM. ∗JSPS fellow Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606, Japan; E–mail:[email protected] Uji research center, Yukawa Institute for Theoretical Physics, Kyoto University, Uji 611, Japan; E–mail: [email protected] Department of Physics, Faculty of Liberal Arts, Shinshu University, Matsumoto 390, Japan; E–mail: [email protected] Department of Physics, University of Tokyo, Tokyo 113, Japan; E–mail: [email protected]