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On Intermediate Exceptional Series

Kimyeong Lee, Kaiwen Sun, Haowu Wang

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 $W_{-h^\vee/6}(\mathfrak{g},f_θ)$ 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 $L_1(D_{6+1/2})^{\otimes2}\!/L_2(D_{6+1/2})$. 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.

On Intermediate Exceptional Series

Abstract

The Freudenthal--Tits magic square for of semi-simple Lie algebras can be extended by including the sextonions . 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 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 belonging to the intermediate exceptional series, the intermediate VOA has characters of irreducible modules coinciding with those of the simple rational -cofinite -algebra studied by Kawasetsu, with belonging to the Cvitanović--Deligne exceptional series. We propose some new intermediate VOA with integer level and investigate their properties. For example, for the intermediate Lie algebra between and in the subexceptional series and also in Vogel's projective plane, we find that the intermediate VOA has a simple current extension to a SVOA with four irreducible Neveu--Schwarz modules. We also provide some (super) coset constructions such as and . 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.
Paper Structure (38 sections, 197 equations, 3 figures, 14 tables)

This paper contains 38 sections, 197 equations, 3 figures, 14 tables.

Table of Contents

  1. Introduction
  2. Magic square, exceptional series and Vogel's projective plane
  3. Universal properties of intermediate exceptional series
  4. Algebraic properties
  5. Affine VOA of level $1$
  6. Intermediate Lie algebra $D_{6+1/2}$
  7. Weyl vector and quadratic Casimir invariants
  8. Irreducible modules of $D_{6+1/2}$
  9. Weyl dimension formula
  10. Decomposition $D_6\subset D_{6+1/2} \subset E_7$
  11. Tensor product decomposition
  12. Affine VOA $L_k(D_{6+1/2})$
  13. Affine VOA $D_{6+1/2}$ at level $1$
  14. Affine VOA $D_{6+1/2}$ at level $2$
  15. Affine VOA $D_{6+1/2}$ at level $3$
  16. ...and 23 more sections

Figures (3)

  • Figure 1: The analogy of the Dynkin diagram for $D_{6+1/2}$ and irreducible modules associated with fundamental weights. The three white circle nodes are fermionic fundamental weights.
  • Figure 2: The analogy of Dynkin diagram for $A_{5+1/2}$ and irreducible modules associated with fundamental weights. The three circled nodes are fermionic fundamental weights.
  • Figure 3: The analogy of the Dynkin diagram for $C_{3+1/2}$ and irreducible representations associated with fundamental weights.

Theorems & Definitions (4)

  • Conjecture 1.1
  • Conjecture 1.2
  • Conjecture 1.3
  • Conjecture 7.1