Vertex algebras and TKK algebras

In this paper, we associate the TKK algebra $\widehat{\mathcal{G}}(\mathcal{J})$ with vertex algebras through twisted modules. Firstly, we prove that for any complex number $\mathrm{\ell }$, the category of restricted $\widehat{\mathcal{G}}(\mathcal{J})$-modules of level $\mathrm{\ell }$ is canonically isomorphic to the category of $\sigma$-twisted $V_{\widehat{{\mathcal{C}}_{\mathfrak{g}}}}(\mathrm{\ell },0)$-modules, where $V_{\widehat{{\mathcal{C}}_{\mathfrak{g}}}}(\mathrm{\ell },0)$ is a vertex algebra arising from the toroidal Lie algebra of type $C_2$ and $\sigma$ is an isomorphism of $V_{\widehat{{\mathcal{C}}_{\mathfrak{g}}}}(\mathrm{\ell },0)$ induced from the involution of this toroidal Lie algebra. Secondly, we prove that for any nonnegative integer $\mathrm{\ell }$, the integrable restricted $\widehat{\mathcal{G}}(\mathcal{J})$-modules of level $\mathrm{\ell }$ are exactly the $\sigma$-twisted modules for the quotient vertex algebra $L_{\widehat{{\mathcal{C}}_{\mathfrak{g}}}}(\mathrm{\ell },0)$ of $V_{\widehat{{\mathcal{C}}_{\mathfrak{g}}}}(\mathrm{\ell },0)$. Finally, we classify the irreducible $\frac{1}{2}\mathbb{N}$-graded $\sigma$-twisted $L_{\widehat{{\mathcal{C}}_{\mathfrak{g}}}}(\mathrm{\ell },0)$-modules.