On Intermediate Exceptional Series

The Freudenthal--Tits magic square m (A₁, A₂) for A=R, C, H, O of semi-simple Lie algebras can be extended by including the sextonions S. A series of non-reductive Lie algebras naturally appear in the new row associated with the sextonions, which we will call the intermediate exceptional series, with the largest one as the intermediate Lie algebra E₇+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 gI belonging to the intermediate exceptional series, the intermediate VOA L₁ (gI) has characters of irreducible modules coinciding with those of the simple rational C₂-cofinite W-algebra W-₇^/₆ (g, f_) studied by Kawasetsu, with g belonging to the Cvitanovi\'c--Deligne exceptional series. We propose some new intermediate VOA Lₖ (gI) with integer level k and investigate their properties. For example, for the intermediate Lie algebra D₆+₁/₂ between D₆ and E₇ in the subexceptional series and also in Vogel's projective plane, we find that the intermediate VOA L₂ (D₆+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 L₂ (E₇) /L₂ (D₆+₁/₂) and L₁ (D₆+₁/₂) ^2\!/L₂ (D₆+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.