Quantum vertex algebra associated to quantum toroidal ${\mathfrak{g}\mathfrak{l}}_N$

In this paper, we associate the quantum toroidal algebra ${\mathcal{E}}_N$ of type ${\mathfrak{g}\mathfrak{l}}_N$ with quantum vertex algebra through equivariant $\phi$-coordinated quasi modules. More precisely, for every $\mathrm{\ell }\in \mathbb{C}$, by deforming the universal affine vertex algebra of ${\mathfrak{s}\mathfrak{l}}_{\mathrm{\infty }}$, we construct an $\mathrm{\hslash }$-adic quantum $\mathbb{Z}$-vertex algebra $V_{{\widehat{\mathfrak{s}\mathfrak{l}}}_{\mathrm{\infty }},\mathrm{\hslash }}(\mathrm{\ell },0)$. Then we prove that the category of restricted ${\mathcal{E}}_N$-modules of level $\mathrm{\ell }$ is canonically isomorphic to that of equivariant $\phi$-coordinated quasi $V_{{\widehat{\mathfrak{s}\mathfrak{l}}}_{\mathrm{\infty }},\mathrm{\hslash }}(\mathrm{\ell },0)$-modules.