Recursions in Calogero-Sutherland Model Based on Virasoro Singular Vectors

TL;DR

This paper explores the Virasoro algebra's structure within the Calogero-Sutherland model to develop a recursive construction of Jack symmetric functions, offering new integral representations and insights into integrable systems.

Contribution

It introduces a novel recursive method for constructing Jack symmetric functions using Virasoro singular vectors and their skew hierarchy, both operatorially and via integrals.

Findings

Virasoro singular vectors form a skew hierarchy in the CS model

Recursive construction of Jack symmetric functions is achieved

New integral representation for Jack functions is proposed

Abstract

The present work is much motivated by finding an explicit way in the construction of the Jack symmetric function, which is the spectrum generating function for the Calogero-Sutherland(CS) model. To accomplish this work, the hidden Virasoro structure in the CS model is much explored. In particular, we found that the Virasoro singular vectors form a skew hierarchy in the CS model. Literally, skew is analogous to coset, but here specifically refer to the operation on the Young tableaux. In fact, based on the construction of the Virasoro singular vectors, this hierarchical structure can be used to give a complete construction of the CS states, i.e. the Jack symmetric functions, recursively. The construction is given both in operator formalism as well as in integral representation. This new integral representation for the Jack symmetric functions may shed some insights on the spectrum constructions for the other integrable systems.