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

Abstract

In this note we present an explicit realization of the affine vertex algebra $V^{cri}(\mathfrak{gl}(1|1))$ 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 $V^{cri}(\mathfrak{gl}(1|1)))$ as a subalgebra $M_0$ of $M$. We reconstruct the Molev-Mukhin formula for the Hilbert-Poincare series of the center of $V^{cri}(\mathfrak{gl}(1|1))$. Moreover, we construct a family of irreducible $V^{cri}(\mathfrak{gl}(1|1))$ -modules realized on $F$ and parameterized by ${\chi }^{+},{\chi }^{-}\in \mathbb{C}((z)).$ We propose a generalization of $V^{cri}(\mathfrak{gl}(1|1))$ as a critical level version of the super ${\mathcal{W}}_{1+\infty }$ vertex algebra.

12 pages

References in corpus (1)

• Lectures on Wakimoto modules, opers and the center at the critical level