Algebras of twisted chiral differential operators and affine localization of {\mathfrak {g}} -modules

We propose a notion of algebra of twisted chiral differential operators over algebraic manifolds with vanishing 1st Pontrjagin class. We show that such algebras possess families of modules depending on infinitely many complex parameters, which we classify in terms of the corresponding algebra of twisted differential operators. If the underlying manifold is a flag manifold, our construction recovers modules over an affine Lie algebra parameterized by opers over the Langlands dual Lie algebra. The spaces of global sections of “smallest” such modules are irreducible ^(\mathfrakg)]{{\hat{{\mathfrak{g}}}}} -modules, and all irreducible \mathfrakg{{\mathfrak{g}}} -integrable ^(\mathfrakg)]{{\hat{{\mathfrak{g}}}}} -modules at the critical level arise in this way.