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 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, and the supercharacters can be realized by a fermionic Hecke operator on the \(N=1\) Virasoro minimal model \(L(c_{16,2},0)\). 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.