A note on the affine vertex algebra associated to $\frak{gl}(1 \vert 1)$ at the critical level and its generalizations

Abstract: 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.