Bosonizations of $\widehat{\mathfrak{sl}}_2$ and Integrable Hierarchies

TL;DR

This paper develops new bosonization techniques for the affine Lie algebra fsl_2, leading to novel integrable hierarchies of differential equations with level as a parameter.

Contribution

It introduces a method combining Wakimoto realization and Friedan-Martinec-Shenker bosonization to embed fsl_2 in lattice vertex algebras, producing new integrable hierarchies.

Findings

Constructed embeddings of fsl_2 in lattice vertex algebras.

Derived two new integrable hierarchies of PDEs.

Hierarchies depend on the level parameter of fsl_2.

Abstract

We construct embeddings of $\widehat{\mathfrak{sl}}_2$ in lattice vertex algebras by composing the Wakimoto realization with the Friedan-Martinec-Shenker bosonization. The Kac-Wakimoto hierarchy then gives rise to two new hierarchies of integrable, non-autonomous, non-linear partial differential equations. A new feature of our construction is that it works for any value of the central element of $\widehat{\mathfrak{sl}}_2$; that is, the level becomes a parameter in the equations.