Whittaker modules for $\widehat{\mathfrak{g}l}$ and ${\mathcal{W}}_{1+\infty }$-modules which are not tensor products

Abstract

We consider the Whittaker modules $M_1(\lambda ,\mu )$ for the Weyl vertex algebra $M$, constructed in arXiv:1811.04649, where it was proved that these modules are irreducible for each finite cyclic orbifold $M^{{\mathbb{Z}}_n}$. In this paper, we consider the modules $M_1(\lambda ,\mu )$ as modules for the $\mathbb{Z}$-orbifold of $M$, denoted by $M^0$. $M^0$ is isomorphic to the vertex algebra ${\mathcal{W}}_{1+\infty ,c=-1}=\mathcal{M}(2)\otimes M_1(1)$ which is the tensor product of the Heisenberg vertex algebra $M_1(1)$ and the singlet algebra $\mathcal{M}(2)$. Furthermore, these modules are also modules of the Lie algebra $\widehat{\mathfrak{g}l}$ with central charge $c=-1$. We prove they are reducible as $\widehat{\mathfrak{g}l}$-modules (and therefore also as $M^0$-modules), and we completely describe their irreducible quotients $L(d,\lambda ,\mu )$. We show that $L(d,\lambda ,\mu )$ in most cases are not tensor product modules for the vertex algebra $\mathcal{M}(2)\otimes M_1(1)$. Moreover, we show that all constructed modules are typical in the sense that they are irreducible for the Heisenberg-Virasoro vertex subalgebra of ${\mathcal{W}}_{1+\infty ,c=-1}$.

25 pages