W-algebras and chiral differential operators at the critical level

Let A(crit) be the chiral algebra corresponding to the affine Kac-Moody algebra at the critical level (g) over cap (crit). Let 3(crit) be the center of A(crit). The commutative chiral algebra 3crit admits a canonical deformation into a non-commutative chiral algebra W-(h) over bar. In this paper we will express the resulting first order deformation via the chiral algebra D-crit of chiral differential operators on G(crit) at the critical level. (C) 2013 Elsevier Inc. All rights reserved.