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.