Excited States of Calogero-Sutherland Model and Singular Vectors of the $W_N$ Algebra

TL;DR

This paper establishes a connection between excited states of the Calogero-Sutherland model, represented by Jack polynomials, and singular vectors of the $W_N$ algebra, providing new integral representations and algebraic methods.

Contribution

It introduces a novel relation between Jack polynomials and $W_N$ algebra singular vectors, along with integral representations and an algebraic derivation method.

Findings

Derived integral representations of Jack polynomials and skew-Jack polynomials.

Established a direct relation between Calogero-Sutherland excited states and $W_N$ singular vectors.

Provided an algebraic method confirming the integral representations.

Abstract

Using the collective field method, we find a relation between the Jack symmetric polynomials, which describe the excited states of the Calogero-Sutherland model, and the singular vectors of the $W_N$ algebra. Based on this relation, we obtain their integral representations. We also give a direct algebraic method which leads to the same result, and integral representations of the skew-Jack polynomials.