Integrable hierarchy for homogeneous realization of the toroidal Lie algebra $\mathcal{L}^{\rm tor}_{r+1}(\mathfrak{sl}_\ell)$

TL;DR

This paper constructs an integrable hierarchy associated with the toroidal Lie algebra using lattice vertex algebras, extending the $ ext{KP}$ hierarchy through explicit Lax and Hirota bilinear equations.

Contribution

It introduces a novel integrable hierarchy for the toroidal Lie algebra based on a homogeneous lattice vertex algebra realization, connecting it to the $ ext{KP}$ hierarchy.

Findings

Derived a hierarchy of Hirota bilinear equations

Represented the hierarchy via Lax equations

Extended a reduction of the $ ext{KP}$ hierarchy

Abstract

Starting from a fairly explicit homogeneous realization of the toroidal Lie algebra $\mathcal{L}^{\tor}_{r+1}(\fsl_\ell)$ via a lattice vertex algebra, we derive an integrable hierarchy of Hirota bilinear equations. Moreover, we represent this hierarchy in the form of Lax equations, and show that it is an extension of a certain reduction of the $\ell$-component KP hierarchy.