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