On Intermediate Exceptional Series

Abstract

The Freudenthal--Tits magic square $\mathfrak{m}({\mathbb{A}}_1,{\mathbb{A}}_2)$ for $\mathbb{A}=\mathbb{R},\mathbb{C},\mathbb{H},\mathbb{O}$ of semi-simple Lie algebras can be extended by including the sextonions $\mathbb{S}$. A series of non-reductive Lie algebras naturally appear in the new row associated with the sextonions, which we will call the \textit{intermediate exceptional series}, with the largest one as the intermediate Lie algebra $E_{7+1/2}$ constructed by Landsberg--Manivel. We study various aspects of the intermediate vertex operator (super)algebras associated with the intermediate exceptional series, including rationality, coset constructions, irreducible modules, (super)characters and modular linear differential equations. For all ${\mathfrak{g}}_I$ belonging to the intermediate exceptional series, the intermediate VOA $L_1({\mathfrak{g}}_I)$ has characters of irreducible modules coinciding with those of the simple rational $C_2$-cofinite $W$-algebra $W_{-h^{\vee }/6}(\mathfrak{g},f_{\theta })$ studied by Kawasetsu, with $\mathfrak{g}$ belonging to the Cvitanović--Deligne exceptional series. We propose some new intermediate VOA $L_k({\mathfrak{g}}_I)$ with integer level $k$ and investigate their properties. For example, for the intermediate Lie algebra $D_{6+1/2}$ between $D_6$ and $E_7$ in the subexceptional series and also in Vogel's projective plane, we find that the intermediate VOA $L_2(D_{6+1/2})$ has a simple current extension to a SVOA with four irreducible Neveu--Schwarz modules. We also provide some (super) coset constructions such as $L_2(E_7)/L_2(D_{6+1/2})$ and $L_1(D_{6+1/2}{)}^{\otimes 2}/L_2(D_{6+1/2})$. In the end, we find that the theta blocks associated with the intermediate exceptional series produce some new holomorphic Jacobi forms of critical weight and lattice index.

46 pages