Quantum affine vertex algebras associated to untwisted quantum affinization algebras

Let $\mathcal U_\hbar(\hat{\mathfrak g})$ be the untwisted quantum affinization of a symmetrizable quantum Kac-Moody algebra $\mathcal U_\hbar({\mathfrak g})$. For $\ell\in\mathbb C$, we construct an $\hbar$-adic quantum vertex algebra $V_{\hat{\mathfrak g},\hbar}(\ell,0)$, and establish a one-to-one correspondence between $\phi$-coordinated $V_{\hat{\mathfrak g},\hbar}(\ell,0)$-modules and restricted $\mathcal U_\hbar(\hat{\mathfrak g})$-modules of level $\ell$. Suppose that $\ell$ is a positive integer. We construct a quotient $\hbar$-adic quantum vertex algebra $L_{\hat{\mathfrak g},\hbar}(\ell,0)$ of $V_{\hat{\mathfrak g},\hbar}(\ell,0)$, and establish a one-to-one correspondence between certain $\phi$-coordinated $L_{\hat{\mathfrak g},\hbar}(\ell,0)$-modules and restricted integrable $\mathcal U_\hbar(\hat{\mathfrak g})$-modules of level $\ell$. Suppose further that ${\mathfrak g}$ is of finite type. We prove that $L_{\hat{\mathfrak g},\hbar}(\ell,0)/\hbar L_{\hat{\mathfrak g},\hbar}(\ell,0)$ is isomorphic to the simple affine vertex algebra $L_{\hat{\mathfrak g}}(\ell,0)$.