Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Relaxed and logarithmic modules of $\widehat{\mathfrak{sl}_3}$   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.

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The categories of this paper are: atom-ph
9
The categories of this paper are: nlin
6
The categories of this paper are: astro-ph
4
The categories of this paper are: cond-mat
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The categories of this paper are: dg-ga
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The categories of this paper are: q-fin
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The categories of this paper are: comp-gas
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The categories of this paper are: chem-ph
5
The categories of this paper are: gr-qc
15
The categories of this paper are: nucl-ex
0
The categories of this paper are: hep-ph
13
The categories of this paper are: quant-ph
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The categories of this paper are: econ
14
The categories of this paper are: math-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: eess