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

TL;DR

This paper provides an explicit realization of the affine vertex algebra for rak{gl}(1|1) at the critical level, describes its center, reconstructs its Hilbert-Poincare9 series, and proposes a generalization related to super al W_{1+e9} algebras.

Contribution

It offers an explicit realization of the affine vertex algebra at the critical level and introduces a generalization linked to super al W_{1+e9} vertex algebras.

Findings

Explicit realization of the affine vertex algebra inside a tensor product of fermionic and commutative vertex algebras.

Reconstruction of the Molev-Mukhin formula for the center's Hilbert-Poincare9 series.

Construction of irreducible modules parameterized by complex functions.

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.