Collective fields, Calogero-Sutherland model and generalized matrix models

Abstract

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.