From Jack polynomials to minimal model spectra

TL;DR

This paper explores the connection between Jack symmetric polynomials and the representation theory of minimal model vertex operator algebras, offering simplified proofs and new insights into their classification.

Contribution

It introduces a novel approach combining free field realizations and symmetric polynomials to classify minimal model representations more efficiently.

Findings

Simplified proofs of minimal model representation classification

New application of Jack polynomials in conformal field theory

Enhanced understanding of free field realizations

Abstract

In this note, a deep connection between free field realisations of conformal field theories and symmetric polynomials is presented. We give a brief introduction into the necessary prerequisites of both free field realisations and symmetric polynomials, in particular Jack symmetric polynomials. Then we combine these two fields to classify the irreducible representations of the minimal model vertex operator algebras as an illuminating example of the power of these methods. While these results on the representation theory of the minimal models are all known, this note exploits the full power of Jack polynomials to present significant simplifications of the original proofs in the literature.