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

By E. Frenkel. 2002

183 citations recorded by the source corpus.

Wakimoto modules are representations of affine Kac-Moody algebras in Fock modules over infinite-dimensional Heisenberg algebras. In these lectures we present the construction of the Wakimoto modules from the point of view of the vertex algebra theory. We then use Wakimoto modules to identify the center of the completed universal enveloping algebra of an affine Kac-Moody algebra at the critical level with the algebra of functions on the space of opers for the Langlands dual group on the punctured disc. These results were originally obtained by B. Feigin and the author.

Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/7126ddaadb4f39a13ea26ff6b0c6448465f69f1a

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In Geometry and Topology

1. J. Teschner
2. Carlos Simpson
3. Alexander I. Molev
4. É. Ragoucy
5. E. Frenkel
6. Y. Nakayama
7. A. Okounkov
8. Davesh Maulik

In Mathematical Physics

1. J. Teschner
2. Carlos Simpson
3. Alexander I. Molev
4. É. Ragoucy
5. E. Frenkel
6. E. Witten
7. Sergei Gukov
8. D. Gaiotto

In Statistical and Nonlinear Physics

1. J. Teschner
2. Carlos Simpson
3. Alexander I. Molev
4. É. Ragoucy
5. E. Frenkel
6. D. Gaiotto
7. E. Witten
8. T. Arakawa

Citations in the corpus

• Lectures on the Langlands program and conformal field theory (2005, methodology): “GRAM AND CONFORMAL FIELD THEORY 95 dual Lie algebra is closely related to the T-duality in free bosonic conformal field theory compactified on a torus. 8.5. Free field realization. The idea of the proof [11, 12] of Theorem 9 is to realize the center z(g) inside the Poisson version of the chiral algebra”
• Quantization of the Hitchin moduli spaces, Liouville theory, and the geometric Langlands correspondence I (2010, background): “terizes a choice of a parabolic subgroup at z0. In order to eliminate this dependence and in order to strengthen analogies with the definitions of the Hecke operators in othe r circumstances (see e.g. [Fr05] for a discussion) let us consider ξ(w) := ξ(0|w), ξ(u|w) := (Ik 2 Φk 2)(u|w), (8.29) where Ij”
• Iterated destabilizing modifications for vector bundles with connection (2008, background): “The classical moduli space of GL(r)-opers [4] [18] is the subset Gα defined in Proposition 5.”
• The MacMahon Master Theorem for right quantum superalgebras and higher Sugawara operators for \hat gl(m|n) (2009, background): “glmjn) of the vertex algebra Vn m(glmjn); cf. [ 11 ].”
• Dynkin Automorphism Actions on Gaudin Algebras (2023, unknown): “The isomorphism Spec z ( g ) ≃ Op g ∨ ( D ) was originally constructed using the language of vertex algebras and Wakimoto modules, as described in [Fre05b], [Fre07]. b We believe that it is possible to proceed through all the constructions made there and to convince oneself that they are all σ…”
• Recent advances in the Langlands Program (2003, methodology): “It is interesting that their work uses results from representation theory of ane Kac-Moody algebras [FF], [ Fr ], which now play the role of the reductive groups over local non-archimedian elds.”