Vertex algebras and TKK algebras

Abstract

In this paper, we associate the TKK algebra $\hat{\mathcal{G}}(\mathcal{J})$ with vertex algebras through twisted modules. Firstly, we prove that for any complex number $\ell$, the category of restricted $\hat{\mathcal{G}}(\mathcal{J})$-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 $\hat{\mathcal{G}}(\mathcal{J})$-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.

arXiv comment19 pages