Wakimoto Modules Research Articles

Semi-infinite induction and Wakimoto modules

The purpose of this paper is to suggest the construction and study properties of semi-infinite induction, which relates to semi-infinite cohomology the same way induction relates to homology and coinduction to cohomology. We prove a version of the Shapiro Lemma, showing that the semi-infinite cohomology of a module is isomorphic to that of the semi-infinitely induced module. A practical outcome of our construction is a simple construction of the Wakimoto modules, highest-weight modules used in double-sided BGG resolutions of irreducible modules.

Bosonization and Integral Representation of Solutions¶of the Knizhnik-Zamolodchikov-Bernard Equations

We construct an integral representation of solutions of the Knizhnik-Zamolodchikov-Bernard equations, using the Wakimoto modules.

Wakimoto Modules and Knizhnik-Zamolodchikov-Bernard Equations

This article is a short review on the construction of solutions of the KnizhnikZamolodchikov (KZ) equation and the Knizhnik-Zamolodchikov-Bernard (KZB) equation in integral form with the help of the Wakimoto realization of affine Lie algebras. Correlation functions of the (chiral) Wess-Zumino-Witten models on a Riemann sphere satisfy the well-known Knizhnik-Zamolodchikov (KZ) equations. 1), 2) There are many works on integral representation of solutions of the KZ equations. The most general solutions were found in Ref. 3) from the viewpoint of the theory of hypergeometric type integrals. There is another approach which makes use of free field theories on Riemann surfaces and the Wakimoto realization of affine Lie algebras; e.g., Refs. 4) 8). We review this result in §3. Bernard 9) found a system of equations for the genus one case, which is now called the Knizhnik-Zamolodchikov-Bernard (KZB) equations. Felder and Varchenko found an integral representation of solutions of this system in Ref. 10) by the same method as in Ref. 3). In our paper 11) we applied the free field theory to the KZB equations. We will review this result in §4. The key lemma is the screening current Ward identity used in Refs. 7) and 8) (with necessary modification). For the sl(2) case Bernard and Felder found the same result in Ref. 6) by using Wakimoto modules.

Free Field Representation of Level-k Yangian Double DY(sl2) k and deformation of Wakimoto Modules

A free field representation of level-k (≠0,–2) Yangian double DY(sl2) k and a corresponding deformation of Wakimoto modules are presented. We also realize two Types of vertex operators intertwining these modules.

On bosonization of 2d conformal field theories

We show how bosonic (free field) representations for so-called degenerate conformal theories are built by singular vectors in Verma modules. Based on this construction, general expressions of conformal blocks are proposed. As an example, we describe new modules for the SL(2) Wess-Zumino-Witten model. They are, in fact, the simplest nontrivial modules in a full set of bosonized highest weight representations of the ŝl 2 algebra. The Verma and Wakimoto modules appear as boundary modules of this set. Our construction also yields a new kind of bosonization in 2d conformal field theories.

Gaudin model, Bethe Ansatz and critical level

We propose a new method of diagonalization of hamiltonians of the Gaudin model associated to an arbitrary simple Lie algebra, which is based on Wakimoto modules over affine algebras at the critical level. We construct eigenvectors of these hamiltonians by restricting certain invariant functionals on tensor products of Wakimoto modules. In conformal field theory language, the eigenvectors are given by certain bosonic correlation functions. Analogues of Bethe ansatz equations naturally appear as Kac-Kazhdan type equations on the existence of certain singular vectors in Wakimoto modules. We use this construction to expalain a connection between Gaudin's model and correlation functions of WZNW models.

Aq-deformation of Wakimoto modules, primary fields and screening operators

Theq-vertex operators of Frenkel and Reshetikhin are studied by means of aq-deformation of the Wakimoto module for the quantum affine algebraU q \((\widehat{\mathfrak{s}\mathfrak{l}}_2 )\) at an arbitrary levelk≠0, −2. A Fock-module version of theq-deformed primary field of spinj is introduced, as well as the screening operators which (anti-)commute with the action ofU q \((\widehat{\mathfrak{s}\mathfrak{l}}_2 )\) up to a total difference of a field. A proof of the intertwining property is given for theq-vertex operators corresponding to the primary fields of spinj∉1/2Z≦0. A sample calculation of the correlation function is also given.

Determinant formulas for the free field representations of the Virasoro and Kac-Moody algebras

The determinant formulas are given for the Feigin-Fuchs modules over the Virasoro algebra and for the Wakimoto modules over the affine Kac-Moody algebras. These formulas imply that some of the modules are isomorphic to the Verma modules or their duals. In particular, this is the case for all Feigin-Fuchs modules over the Virasoro algebra with central charge c⩾25. Seiberg's condition is a simple criterion as to whether such a module is isomorphic to a dual Verma module.

SL( N) Kac-Moody algebras and Wess-Zumino-Witten models

Wess-Zumino-Witten models are represented by Wakimoto modules. Representation theory of SL( N) is used to reduce arbitrary three point functions to generalized Dotsenko-Fateev integrals.

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 to certain sheaves on a semi-infinite flag manifold with support on its Schhubert cells. This manifold is equipped with a remarkable semi-infinite structure, which is discussed; in particular, the semi-infinite homology of this manifold is computed. The Cousin-Grothendieck resolution of an invertible sheaf on a semi-infinite flag manifold gives a two-sided resolution of an irreducible representation of an affine algebras, consisting of Wakimoto modules. This is just the BRST complex. As a byproduct we compute the homology of an algebra of currents on the real line with values in a nilpotent Lie algebra.