The authors analyze the category of discrete modules over affine Kac-Moody algebras \(\mathfrak{\widehat{g}}\) associated to complex Lie algebras \(\mathfrak{g}\) endowed with an invariant inner product (called \(k\)), as well as the category of \(k\)-twisted right \(D\)-modules on the Grassmanian associated to the algebraic group of adjoint type of the Lie algebra \(\mathfrak{g}\). It is shown that the functor of global sections from the category of \(k\)-twisted modules onto the category of discrete \(\mathfrak{\widehat{g}}\)-modules is exact when \(k\) equals \(\frac{-1}{2}\kappa\) (called the critical level), \(\kappa\) being the Killing form. To this extent, chiral algebras of differential operators are used. This result extends previous work of Drinfel’d and Beilinson on the exactness of this functor for irrational or less than critical levels [A. Beilinson and V. Drinfeld, Chiral algebras, Am. Math. Soc. Colloq. Publ. 51 (2004; Zbl 1138.17300)].