Vertex algebras and TKK algebras

TL;DR

This paper establishes a deep connection between TKK algebras and vertex algebras by classifying modules and twisted modules, revealing new structures and classifications in the theory of vertex algebras associated with Lie algebras.

Contribution

It constructs an isomorphism between categories of modules for TKK algebras and twisted modules for certain vertex algebras, and classifies irreducible twisted modules.

Findings

Category of restricted TKK modules is isomorphic to twisted modules of a vertex algebra.

Integrable restricted TKK modules correspond exactly to twisted modules of a quotient vertex algebra.

Complete classification of irreducible twisted modules for the vertex algebra.

Abstract

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.