Branching Rule Decomposition of Irreducible Level-1 E_6^(1)-modules with respect to F_4^(1)

TL;DR

This paper explores the decomposition of level-1 irreducible modules of the affine Lie algebra E6^(1) into modules of the subalgebra F4^(1), using vertex operator algebra automorphisms, coset constructions, and character theory, revealing connections to Ramanujan identities.

Contribution

It provides an explicit decomposition of E6^(1) modules into F4^(1) modules, constructs coset Virasoro operators, and identifies highest weight vectors, linking to Ramanujan identities and W3-algebra insights.

Findings

Explicit highest weight vectors for each module

Decomposition rules for E6^(1) modules into F4^(1) modules

Connection to Ramanujan identities and W3-algebra

Abstract

It is well known that using the weight lattice of type $E_6$, $P$, and the lattice construction for vertex operator algebras one can obtain all three level 1 irreducible $\tilde{g}$-modules with $V_P = V^{\Lamba_0} \oplus V^{\Lamba_1} \oplus V^{\Lamba_6}$. The Dynkin diagram of type $E_6$ has an order 2 automorphism, $\tau$, which can be lifted to $\tilde{{\tau}}$, a Lie algebra automorphism of $\tilde{g}$ of type $E_6^(1)$. The fixed points of $\tilde{{\tau}}$ are a subalgebra $\tilde{a}$ of type $F_4^(1)$. The automorphism $\tau$ lifts further to $\hat{{\tau}}$ a vertex operator algebra automorphism of $V_P$. We investigate the branching rules, how these three modules for the affine Lie algebra $\tilde{g}$ decompose as a direct sum of irreducible $\tilde{a}$-modules. To complete the decomposition we use the Godard-Kent-Olive coset construction which gives a $c = \frac{4}{5}$ module for the Virasoro algebra on $V_P$ which commutes with $\tilde{a}$. We use the irreducible modules for this coset Virasoro to give the space of highest weight vectors for $\tilde{a}$ in each $V^{\Lamba_i}$. The character theory related to this decomposition is examined and we make a connection to one of the famous Ramanujan identities. This dissertation constructs coset Virasoro operators $Y(\omg,z)$ by explicitly determining the generator $\omg$. We also give the explicit highest weight vectors for each $Vir \otimes F_4^(1)$-module in the decomposition of each $V^{\Lamba_i}$. This explicit work on determining highest weight vectors gives some insight into the relationship to the Zamolodchikov $W_3$-algebra.