Lectures on Wakimoto modules, opers and the center at the critical level

  title={Lectures on Wakimoto modules, opers and the center at the critical level},
  author={Edward Frenkel},
  journal={arXiv: Quantum Algebra},
  • E. Frenkel
  • Published 2 October 2002
  • Mathematics
  • arXiv: Quantum Algebra

Geometric realizations of Wakimoto modules at the critical level

We study the Wakimoto modules over the affine Kac-Moody algebras at the critical level from the point of view of the equivalences of categories proposed in our previous works, relating categories of

Self-extensions of Verma modules and differential forms on opers

We compute the algebras of self-extensions of the vacuum module and the Verma modules over an affine Kac–Moody algebra $\hat{\mathfrak g}$ in suitable categories of Harish-Chandra modules. We show

On the Endomorphisms of Weyl Modules over Affine Kac–Moody Algebras at the Critical Level

We present an independent short proof of the recently established result that the algebra of endomorphisms of a Weyl module of critical level is isomorphic to the algebra of functions on the space of

Fusion and convolution: applications to affine Kac-Moody algebras at the critical level

Let g be a semi-simple Lie algebra, and let g^ be the corresponding affine Kac-Moody algebra. Consider the category of g^-modules at the critical level, on which the action of the Iwahori subalgebra

On Higher-Order Sugawara Operators

The higher Sugawara operators acting on the Verma modules over the affine Kac-Moody algebra at the critical level are related to the higher Hamiltonians of the Gaudin model due to the work of Feigin,


We apply the technique of localization for vertex algebras to the Segal- Sugawara construction of an "internal" action of the Virasoro algebra on affine Kac- Moody algebras. The result is a lifting

Jantzen sum formula for restricted Verma modules over affine Kac-Moody algebras at the critical level

For a restricted Verma module of an affine Kac-Moody algebra at the critical level we describe the Jantzen filtration and give an alternating sum formula which corresponds to the Jantzen sum formula

Affine opers and conformal affine Toda

  • C. Young
  • Mathematics
    Journal of the London Mathematical Society
  • 2021
For g a Kac–Moody algebra of affine type, we show that there is an AutO ‐equivariant identification between FunOpg(D) , the algebra of functions on the space of g ‐opers on the disc, and W⊂π0 , the

The linkage principle for restricted critical level representations of affine Kac–Moody algebras

Abstract We study the restricted category 𝒪 for an affine Kac–Moody algebra at the critical level. In particular, we prove the first part of the Feigin–Frenkel conjecture: the linkage principle for

Localization of g^-modules on the affine Grassmannian

We consider the category of modules over the affine Kac-Moody algebra g^ of critical level with regular central character. In our previous paper math.RT/0508382 we conjectured that this category is




We prove Drinfeld's conjecture that the center of a certain completion of the universal enveloping algebra of an affine Kac-Moody algebra at the critical level is isomorphic to the Gelfand-Dikii

Wakimoto modules for twisted affine lie algebras

We construct Wakimoto modules for twisted a!ne Lie algebras , and interpret this construction in terms of vertex algebras and their twisted modules. Using the Wakimoto construction, we prove the

Affine Kac-Moody algebras and semi-infinite flag manifolds

We study representations of affine Kac-Moody algebras from a geometric point of view. It is shown that Wakimoto modules introduced in [18], which are important in conformal field theory, correspond

Representations of affine Kac-Moody algebras, bosonization and resolutions

We study boson representations of the affine Kac-Moody algebras and give an explicit description of primary fields and intertwining operators, using vertex operators. We establish the resolution of

Gaudin model, Bethe Ansatz and critical level

We propose a new method of diagonalization oif hamiltonians of the Gaudin model associated to an arbitrary simple Lie algebra, which is based on the Wakimoto modules over affine algebras at the

Integrable Hierarchies and Wakimoto Modules

In our earlier papers we proposed a new approach to integrable hierarchies of soliton equations and their quantum deformations. We have applied this approach to the Toda field theories and the

Representations of Affine Kac-Moody Algebras

In the first chapter we explained how simple finite-dimensional Lie algebras can be completely characterized in terms of their Cartan matrices or Dynkin diagrams. The same holds for an arbitrary

Semi-infinite Weil complex and the Virasoro algebra

We define a semi-infinite analogue of the Weil algebra associated an infinite-dimensional Lie algebra. It can be used for the definition of semi-infinite characteristic classes by analogy with the

Fock representations of the affine Lie algebraA1(1)

The aim of this note is to show that the affine Lie algebraA1(1) has a natural family πμ, υ,v of Fock representations on the spaceC[xi,yj;i ∈ ℤ andj ∈ ℕ], parametrized by (μ,v) ∈C2. By corresponding

Wakimoto Realizations of Current Algebras: An Explicit Construction

Abstract:A generalized Wakimoto realization of can be associated with each parabolic subalgebra of a simple Lie algebra according to an earlier proposal by Feigin and Frenkel. In this paper the