Lectures on Wakimoto modules, opers and the center at the critical level

TL;DR

This paper explores the construction of Wakimoto modules for affine Kac-Moody algebras, linking them to the center at the critical level and the space of opers, within the framework of vertex algebra theory.

Contribution

It provides a vertex algebra perspective on Wakimoto modules and their role in identifying the center at the critical level with functions on opers.

Findings

Wakimoto modules constructed via vertex algebra theory.

Center at critical level identified with functions on opers.

Connection established between representation theory and geometric Langlands program.

Abstract

Wakimoto modules are representations of affine Kac-Moody algebras in Fock modules over infinite-dimensional Heisenberg algebras. In these lectures we present the construction of the Wakimoto modules from the point of view of the vertex algebra theory. We then use Wakimoto modules to identify the center of the completed universal enveloping algebra of an affine Kac-Moody algebra at the critical level with the algebra of functions on the space of opers for the Langlands dual group on the punctured disc. These results were originally obtained by B. Feigin and the author.