Selecta Mathematica

, Volume 17, Issue 1, pp 1-46

First online:

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

  • Tomoyuki ArakawaAffiliated withResearch Institute for Mathematical Sciences, Kyoto University
  • , Dmytro ChebotarovAffiliated withDepartment of Mathematics, University of Southern California
  • , Fyodor MalikovAffiliated withDepartment of Mathematics, University of Southern California Email author 

Rent the article at a discount

Rent now

* Final gross prices may vary according to local VAT.

Get Access


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 \({{\hat{{\mathfrak{g}}}}}\) -modules, and all irreducible \({{\mathfrak{g}}}\) -integrable \({{\hat{{\mathfrak{g}}}}}\) -modules at the critical level arise in this way.


Rings of differential operators Chiral differential operators Representations

Mathematics Subject Classification (2010)

Primary 17B69 Secondary 14F10 17B67