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

Abstract

Abstract. In this note we present an explicit realization of the affine vertex algebra V cri (gl(1|1)) inside of the tensor product F ⊗ 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 (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 (gl(1|1)). Moreover, we construct a family of irreducible V cri (gl(1|1))-modules realized on F and parameterized by χ + , χ − ∈ C((z)). We propose a generalization of V cri (gl(1|1)) as a critical level version of the super W 1+∞ vertex algebra.

On fusion rules and intertwining operators for the Weyl vertex algebra

“…satisfy the commutation relations for the affine Lie algebra g = gl(1|1), so that M ⊗ F is a g-module of level 1. (See also [5] for a realization of gl(1|1) at the critical level).…”