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

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