Modules for a sheaf of Lie algebras on loop manifolds

TL;DR

This paper constructs sheaves of modules for a centrally extended sheaf of Lie algebras on loop manifolds using vertex algebra techniques, extending prior work on the chiral de Rham complex.

Contribution

It introduces a new construction of sheaves of modules for a sheaf of Lie algebras on loop manifolds via vertex algebra methods, expanding existing theories.

Findings

Constructed sheaves of modules for the extended Lie algebra sheaf.

Extended Malikov-Schechtman-Vaintrob's work on the chiral de Rham complex.

Provided a new framework connecting Lie algebra sheaves and vertex algebras.

Abstract

We consider a central extension of the sheaf of Lie algebras of maps from a manifold into a finite-dimensional simple Lie algebra, together with the sheaf of vector fields. Using vertex algebra methods we construct sheaves of modules for this sheaf of Lie algebras. Our results extend the work of Malikov-Schechtman-Vaintrob on the chiral de Rham complex.