Positive Energy Representations of Affine Vertex Algebras

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 the universal affine vertex algebra V-kappa (g) 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 V-kappa (g)-modules. These two functors commute (taking a proper restriction of T-alpha on g-modules) and the image of the Wakimoto functor consists of relaxed Wakimoto (g) over cap (kappa)-modules. In particular, applying the twisting functor T-alpha to the relaxed Wakimoto (g) over cap (kappa)-module whose top degree component is isomorphic to the Verma g-module M-b(g)(lambda), we obtain the relaxed Wakimoto (g) over cap kappa-module whose top degree component is isomorphic to the alpha-Gelfand-Tsetlin g-module W-b(g)(lambda, alpha). We show that the relaxed Verma module and relaxed Wakimoto module whose top degree components are such a-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 new large family of positive energy representations of the simple affine vertex algebra L-kappa(g) of admissible level kappa by means of the twisting functor applied on generalized Verma modules for the parabolic subalgebra p.