Relaxed and logarithmic modules of $\widehat{\mathfrak{sl}_3}$

TL;DR

This paper generalizes the realization of affine vertex algebras from $rak{sl}_2$ to $rak{sl}_3$, constructing and analyzing various modules, including logarithmic ones, using a combination of vertex algebra techniques.

Contribution

It introduces a new realization of $L_k(rak{sl}_3)$ as a subalgebra of a tensor product of vertex algebras, extending prior $rak{sl}_2$ results and studying module irreducibility.

Findings

Constructed $L_k(rak{sl}_3)$ as a subalgebra of $ ext{W}_k imes eta ext{-} ext{gamma} imes ext{Pi}(0)$.

Proved irreducibility of relaxed highest weight modules with finite-dimensional weight spaces.

Developed logarithmic modules of rank three for $L_k(rak{sl}_3)$.

Abstract

In [8], the affine vertex algebra $L_k(\mathfrak{sl}_2)$ is realized as a subalgebra of the vertex algebra $Vir_c \otimes \Pi(0)$, where $Vir_c$ is a simple Virasoro vertex algebra and $\Pi(0)$ is a half-lattice vertex algebra. Moreover, all $L_k(\mathfrak{sl}_2)$--modules (including, modules in the category $KL_k$, relaxed highest weight modules and logarithmic modules) are realized as $Vir_c \otimes \Pi(0)$--modules. A natural question is the generalization of this construction in higher rank. In the current paper, we study the case $\mathfrak{g}= \mathfrak{sl}_3$ and present realization of the VOA $L_k(\mathfrak g)$ for $k \notin {\mathbb Z}_{\ge 0}$ as a vertex subalgebra of $\mathcal{W}_k \otimes \mathcal S \otimes \Pi(0)$, where $\mathcal{W}_k$ is a simple Breshadsky Polykov vertex algebra and $\mathcal S$ is the $\beta \gamma$ vertex algebra. We use this realization to study ordinary modules, relaxed highest weight modules and logarithmic modules. We prove the irreducibility of all our relaxed highest weight modules having finite-dimensional weight spaces (whose top components are Gelfand-Tsetlin modules). The irreducibility of relaxed highest weight modules with infinite-dimensional weight spaces is proved up to a conjecture on the irreducibility of certain $\mathfrak g$--modules which are not Gelfand-Tsetlin modules. The next problem that we consider is the realization of logarithmic modules. We first analyse the free-field realization of $\mathcal{W}_k$ from [11] and obtain a realization of logarithmic modules for $\mathcal{W}_k$ of nilpotent rank two at most admissible levels. Using logarithmic modules for the $\beta \gamma$ VOA, we are able to construct logarithmic $L_k(\mathfrak g)$--modules of rank three.