Admissible representations of simple affine vertex algebras

TL;DR

This paper characterizes the highest weights of simple modules over affine vertex algebras of type A at admissible levels, using combinatorial and Gelfand-Tsetlin methods, and classifies certain modules explicitly.

Contribution

It provides an explicit combinatorial description of highest weights and realizations for admissible modules, including a full classification of admissible sl(2)-induced modules for the principal orbit.

Findings

Explicit highest weight descriptions for admissible modules.

Realizations of modules via Gelfand-Tsetlin theory.

Complete classification of admissible sl(2)-induced modules for the principal orbit.

Abstract

We provide an explicit combinatorial description of highest weights of simple highest weight modules over the simple affine vertex algebra of type A of admissible level k. For admissible simple highest weight modules corresponding to the principal, subregular and maximal parabolic nilpotent orbits we give a realization using the Gelfand-Tsetlin theory, which also allows us to obtain a realization of certain classes of simple admissible sl(2)-induced modules in these orbits. In particular, simple admissible sl(2)-induced modules are fully classified for the principal nilpotent orbit.