On Vertex Operator Algebras of Affine Type at Admissible Levels

Abstract: The main purpose of this thesis is the study of the structure and representation theory of simple vertex operator algebras (VOAs) of affine type at admissible levels. To do this, it is crucial to obtain knowledge of the singular vectors which generate the maximal submodules, with respect to which these VOAs appear as irreducible quotients, and therefore a substantial part of the text is devoted to this matter. We study in particular the simple VOAs associated to affine sl(3, C) with half-integer admissible levels, and especially the one with the minimal admissible level -3/2. We tackle the problem of describing singular vectors in Verma modules for affine Lie algebras by providing a novel way of realizing the ideas presented in an article by F. G. Malikov, B. L. Feigin and D. B. Fuchs. Our approach is based on the rigorous construction of a broader algebraic framework by means of Ore localization in the universal enveloping algebra and via the introduction of certain conjugation automorphisms. We are able to express operators corresponding to those of Malikov et al. and to partially extend to our setting their main result regarding whether or not these operators represent elements of the enveloping algebra. Using this knowledge about singular vectors we deal with the problem of finding the irreducible modules in the category O for VOAs of affine type, when the level is admissible. Applying the theory of Zhu's algebra, the highest weights of these modules are characterized as the zeros of a polynomial ideal determined by the single singular vector generating the maximal proper submodule of the generalized Verma module. For the VOA associated to affine sl(3, C) at level -3/2, we prove that these highest weights are precisely the four admissible weights of level -3/2, and moreover that any module in the category O for this VOA is completely reducible. We also show that there are no nontrivial intertwining operators between these irreducible modules, except those deriving from the module structures. Furthermore, we demonstrate how the Sapovalov form can be employed to gain insight into the polynomial ideal, if merely the weight of the corresponding singular vector is known.