Differential equations in vertex algebras and simple modules for the Lie algebra of vector fields on a torus

TL;DR

This paper investigates irreducible representations of the Lie algebra of vector fields on a 2-torus, utilizing vertex algebra techniques and differential equations to analyze module irreducibility under certain conditions.

Contribution

It introduces a method to determine irreducibility of modules for the Lie algebra of vector fields on a torus using vertex algebra and differential equations.

Findings

Certain representations remain irreducible when restricted to loop subalgebras.

Vertex algebra techniques effectively analyze module structure.

Differential equations derived from vertex algebra facilitate the study of irreducibility.

Abstract

We study irreducible representations for the Lie algebra of vector fields on a 2-dimensional torus constructed using the generalized Verma modules. We show that for a certain choice of parameters these representations remain irreducible when restricted to a loop subalgebra in the Lie algebra of vector fields. We prove this result by studying vertex algebras associated with the Lie algebra of vector fields on a torus and solving non-commutative differential equations that we derive using the vertex algebra technique.