Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A note on the affine vertex algebra associated to $\frak{gl}(1 \vert 1)$
  at the critical level and its generalizations   In this note we present an explicit realization of the affine vertex algebra
$V^{cri}(\frak{gl}(1 \vert 1)) $ inside of the tensor product $F\otimes M$
where $F$ is a fermionic verex algebra and $M$ is a commutative vertex algebra.
This immediately gives an alternative description of the center of
$V^{cri}(\frak{gl}(1 \vert 1) ) )$ as a subalgebra $M _ 0$ of $M$. We
reconstruct the Molev-Mukhin formula for the Hilbert-Poincare series of the
center of $V^ {cri}(\frak{gl}(1 \vert 1) )$. Moreover, we construct a family of
irreducible $V^{cri}(\frak{gl}(1 \vert 1))$ -modules realized on $F$ and
parameterized by $\chi^+, \chi ^- \in {\Bbb C}((z)). $ We propose a
generalization of $V^ {cri}(\frak{gl}(1 \vert 1))$ as a critical level version
of the super $\mathcal W_{1+\infty}$ vertex algebra.

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The categories of this paper are: cmp-lg
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The categories of this paper are: solv-int
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The categories of this paper are: chao-dyn
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The categories of this paper are: plasm-ph
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The categories of this paper are: nucl-th
4
The categories of this paper are: cond-mat
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The categories of this paper are: q-alg
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The categories of this paper are: nlin
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The categories of this paper are: astro-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: alg-geom
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The categories of this paper are: quant-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: eess
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The categories of this paper are: hep-lat
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The categories of this paper are: stat