Level one Weyl modules for toroidal Lie algebras

TL;DR

This paper establishes a correspondence between level one global Weyl modules for toroidal Lie algebras and certain twisted modules constructed via vertex operators, providing explicit character formulas.

Contribution

It identifies level one Weyl modules with twisted modules from prior constructions, extending understanding of their structure and characters in the context of toroidal Lie algebras.

Findings

Derived a formula for the character of level one local Weyl modules.

Obtained the graded character of level one graded local Weyl modules.

Extended vertex operator constructions to toroidal Lie algebra modules.

Abstract

We identify level one global Weyl modules for toroidal Lie algebras with certain twists of modules constructed by Moody-Eswara Rao-Yokonuma via vertex operators for type ADE and by Iohara-Saito-Wakimoto and Eswara Rao for general type. The twist is given by an action of $\mathrm{SL}_{2}(\mathbb{Z})$ on the toroidal Lie algebra. As a byproduct, we obtain a formula for the character of the level one local Weyl module over the toroidal Lie algebra and that for the graded character of the level one graded local Weyl module over an affine analog of the current Lie algebra.