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Wakimoto modules, opers and the center at the critical level. (English) Zbl 1129.17014

Wakimoto modules are representations of affine Kac-Moody algebras in Fock spaces over infinite-dimensional Heisenberg algebras. These modules have useful applications in representation theory and conformal field theory, in particular they provide a bridge between representation theory of affine algebras and the geometry of the semi-infinite flag manifolds. In the paper under review the author constructs Wakimoto modules using vertex operator algebras, in particular, making the connection between Wakimoto modules and vertex algebras very explicit. It turns out that the construction of Wakimoto modules amounts to constructing a homomorphism from a certain vertex operator algebra, associated to an affine Kac-Moody algebra, to another vertex operator algebra, associated to an infinite-dimensional Heisenberg algebra. The existence of such homomorphism is proved by homological methods. The author describes the cohomology class responsible for the obstruction and shows that it is equal to the class defining the affine Kac-Moody algebra. The author then extends the construction of Wakimoto modules to a more general context in which the Heisenberg subalgebra is replaced by a central extension of the loop algebra of the Levi subalgebra of an arbitrary parabolic Lie subalgebra of the original Lie algebra. This establishes a semi-infinite parabolic induction pattern for representations of affine Kac-Moody algebras. Next, the author gives a uniform construction for the screening operators (the intertwiners between Wakimoto modules). Finally, using the screening operators the author identifies 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, giving another proof of the earlier theorem by B. Feigin and the authors.

MSC:

17B69 Vertex operators; vertex operator algebras and related structures
17B70 Graded Lie (super)algebras
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras

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