Double Yangian and reflection algebras of the Lie superalgebra ${\mathfrak{g}\mathfrak{l}}_{M\mathrm{\mid }N}$, II: Quantum currents

In this paper, we continue our research on the double Yangian and reflection algebras of the Lie superalgebra ${\mathfrak{g}\mathfrak{l}}_{M\mathrm{\mid }N}$. We construct families of central elements in the double Yangian for ${\mathfrak{g}\mathfrak{l}}_{M\mathrm{\mid }N}$ at the critical level. Next, extending the Etingof-Kazhdan construction, we introduce the structure of $h$-adic quantum vertex superalgebra on the vacuum module $V^c({\mathfrak{g}\mathfrak{l}}_{M\mathrm{\mid }N})$ over the double Yangian of level $c\in \mathbb{C}$. Finally, we show that restricted modules over the reflection algebra of ${\mathfrak{g}\mathfrak{l}}_{M\mathrm{\mid }N}$ are naturally equipped with the structure of quasi $V^c({\mathfrak{g}\mathfrak{l}}_{M\mathrm{\mid }N})$-module.