Positive energy representations of affine vertex algebras

Abstract

We construct new families of positive energy representations of affine vertex algebras together with their free field realizations by using localization technique. We introduce the twisting functor T_\alpha on the category of modules over affine Kac--Moody algebra \widehat{g}_\kappa of level \kappa for any positive root \alpha of g, and the Wakimoto functor from a certain category of g-modules to the category of smooth \widehat{g}_\kappa-modules. These two functors commute and the image of the Wakimoto functor consists of relaxed Wakimoto \widehat{g}_\kappa-modules. In particular, applying the twisting functor T_\alpha to the relaxed Wakimoto \widehat{g}_\kappa-module whose top degree component is isomorphic to the Verma g-module M^g_b(\lambda), we obtain the relaxed Wakimoto \widehat{g}_\kappa-module whose top degree component is isomorphic to the \alpha-Gelfand--Tsetlin g-module W^g_b(\lambda, \alpha). We show that the relaxed Verma module and relaxed Wakimoto module whose top degree components are such \alpha-Gelfand--Tsetlin modules, are isomorphic generically. This is an analogue of the result of E.Frenkel for Wakimoto modules both for critical and non-critical level. For a parabolic subalgebra p of g we construct a large family of admissible g-modules as images under the twisting functor of generalized Verma modules induced from p. In this way, we obtain new simple positive energy representations of simple affine vertex algebras.