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

TL;DR

This paper introduces twisted chiral differential operator algebras on algebraic manifolds, classifies their modules, and connects them to affine Lie algebra modules, especially over flag manifolds, revealing new insights into representation theory.

Contribution

It defines twisted chiral differential operator algebras, classifies their modules, and relates these modules to affine Lie algebra representations, especially at the critical level.

Findings

Modules depend on infinitely many parameters and are classified by twisted differential operators.

Over flag manifolds, modules correspond to affine Lie algebra modules parameterized by opers.

All irreducible g-integrable affine Lie algebra modules at the critical level are obtained.

Abstract

We propose a notion of algebra of {\it 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 $\ghat$-modules and all irreducible $\frak{g}$-integrable $\ghat$-modules at the critical level arise in this way.