Positive Energy Representations of Affine Vertex Algebras
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Communications in
Mathematical Physics
Positive Energy Representations of Affine Vertex Algebras Vyacheslav Futorny1,2 , Libor Kˇrižka1 1 Instituto de Matemática e Estatística, Universidade de São Paulo, São Paulo, Brazil.
E-mail: [email protected]; [email protected]; [email protected]
2 International Center for Mathematics, SUSTech, Shenzhen, China
Received: 24 March 2020 / Accepted: 28 July 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
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 Tα on the category of modules over the universal affine vertex algebra Vκ (g) of level κ for any positive root α of g, and the Wakimoto functor from a certain category of g-modules to the category of Vκ (g)-modules. These two functors commute (taking a proper restriction of Tα on g-modules) and the image of the Wakimoto functor consists of relaxed Wakimoto gκ -modules. In particular, applying the twisting functor Tα to the relaxed Wakimoto gκ -module whose top degree component g is isomorphic to the Verma g-module Mb (λ), we obtain the relaxed Wakimoto gκ module whose top degree component is isomorphic to the α-Gelfand–Tsetlin g-module g Wb (λ, α). We show that the relaxed Verma module and relaxed Wakimoto module whose top degree components are such α-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 Lκ (g) of admissible level κ by means of the twisting functor applied on generalized Verma modules for the parabolic subalgebra p. Contents 1. 2.
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Preliminaries . . . . . . . . . . . . . . . . . . . . . . Affine Kac–Moody Algebras and Weyl Algebras . . . 2.1 Affine Kac–Moody algebras . . . . . . . . . . . 2.2 Smooth modules over affine Kac–Moody algebras 2.3 Weyl algebras . . . . . . . . . . . . . . . . . . . Representations of Vertex Algebras . . . . . . . . . . 3.1 Vertex algebras . . . . . . . . . . . . . . . . . . 3.2 Positive energy representations . . . . . . . . . .
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V. Futorny, L. Kˇrižka
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3.3 Affine vertex algebras . . . . . . . . . . . . . . . 3.4 Weyl vertex algebras . . . . . . . . . . . . . . . Twisting Functors and Gelfand–Tsetlin Modules . . . 4.1 Twisting functors for semisimple Lie algebras . . 4.2 Affine α-Gelfand–Tsetlin modules . . . . . . . . 4.3 Twisting functors for affine Kac–Moody algebras 4.4 Tensoring with Weyl modules . . . . . . . . . . . 4.5 Relaxed Verma modules . . . . . . . . . . . . . . Relaxed Wakimoto Modules . . . . . . . . . . . . . . 5.1 Feigin–Frenkel homomor
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