A classification of irreducible Wakimoto modules for the affine Lie algebra $A_1 ^{(1)}$

TL;DR

This paper classifies all Wakimoto modules for the affine Lie algebra $A_1^{(1)}$ at the critical level that are irreducible, revealing the role of Schur polynomial zeros in this classification.

Contribution

It provides a complete classification of irreducible Wakimoto modules for $A_1^{(1)}$ at the critical level based on the analysis of certain functions $ ext{chi}$.

Findings

Zeros of Schur polynomials are key to irreducibility.

All $ ext{chi} otin$ zeros of Schur polynomials yield irreducible modules.

The classification is explicit and complete for the affine Lie algebra $A_1^{(1)}$.

Abstract

By using methods developed in arXiv:math/0602181 we study the irreducibility of certain Wakimoto modules for $\widehat{sl_2}$ at the critical level. We classify all $\chi \in {\Bbb C}((z))$ such that the corresponding Wakimoto module $W_{\chi}$ is irreducible. It turns out that zeros of Schur polynomials play important rule in the classification result.