Relaxed singular vectors, Jack symmetric functions and fractional level $\widehat{\mathfrak{sl}}(2)$ models

TL;DR

This paper combines Wakimoto's free field realization with Jack symmetric functions to analyze fractional level fsl(2) models, providing explicit formulas for singular vectors and classifying modules in these logarithmic conformal field theories.

Contribution

It introduces explicit formulas for singular vectors in relaxed Wakimoto modules and classifies simple relaxed highest weight modules, advancing the understanding of fractional level fsl(2) models.

Findings

Explicit formulas for singular vectors in relaxed Wakimoto modules

Classification of simple relaxed highest weight modules

Presentation of Zhu's algebra and new proof of module classification

Abstract

The fractional level models are (logarithmic) conformal field theories associated with affine Kac-Moody (super)algebras at certain levels $k \in \mathbb{Q}$. They are particularly noteworthy because of several longstanding difficulties that have only recently been resolved. Here, Wakimoto's free field realisation is combined with the theory of Jack symmetric functions to analyse the fractional level $\widehat{\mathfrak{sl}}(2)$ models. The first main results are explicit formulae for the singular vectors of minimal grade in relaxed Wakimoto modules. These are closely related to the minimal grade singular vectors in relaxed (parabolic) Verma modules. Further results include an explicit presentation of Zhu's algebra and an elegant new proof of the classification of simple relaxed highest weight modules over the corresponding vertex operator algebra. These results suggest that generalisations to higher rank fractional level models are now within reach.