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

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 $V_P = V^{\Lamba_0} \oplus V^{\Lamba_1} \oplus V^{\Lamba_6}$\mathrm{.}TheDynkindiagramoftype$E_6$ 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 $\widehat{\tau }$ 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 $V^{{\text{Lamba}}_i}$. 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 $Y(\text{omg},z)$ by explicitly determining the generator $\text{omg}$. We also give the explicit highest weight vectors for each $Vir\otimes F_4^{(}1)$-module in the decomposition of each $V^{{\text{Lamba}}_i}$. This explicit work on determining highest weight vectors gives some insight into the relationship to the Zamolodchikov $W_3$-algebra.