W-algebras and chiral differential operators at the critical Level

TL;DR

This paper explores the deformation of the center of a critical level chiral algebra into a non-commutative algebra, connecting it with chiral differential operators on loop groups.

Contribution

It provides a new expression for the first order deformation of the center of the critical level chiral algebra using chiral differential operators.

Findings

Deformation of the center into a non-commutative algebra is explicitly described.

Connection established between W-algebras and chiral differential operators.

First order deformation characterized via chiral algebra of differential operators.

Abstract

Let $\mathcal{A}_{crit}$ be the chiral algebra corresponding to the affine Kac-Moody algebra at the critical level $\hat{\mathfrak{g}}_{crit}$. Let $\mathfrak{Z}_{crit}$ be the center of $\mathcal{A}_{crit}$. The commutative chiral algebra $\mathfrak{Z}_{crit}$ admits a canonical deformation into a non-commutative chiral algebra $\mathcl{W}_{h}$. In this paper we will express the resulting first order deformation via the chiral algebra $\mathcal{D}_{crit}$ of chiral differential operators of $G((t))$ at the critical level.