Abstract
We give a new construction of the moonshine module vertex operator algebra $V^{\natural}$, which was originally constructed in [FLM2]. We construct it as a framed VOA over the real number field ${\mathrm{R}}$. We also offer ways to transform a structure of framed VOA into another framed VOA. As applications, we study the five framed VOA structures on $V_{E_8}$ and construct many framed VOAs including $V^{\natural}$ from a small VOA. One of the advantages of our construction is that we are able to construct $V^{\natural}$ as a framed VOA with a positive definite invariant bilinear form and we can easily prove that ${\mathrm{Aut}}(V^{\natural})$ is the Monster simple group. By similar ways, we also construct an infinite series of holomorphic framed VOAs with finite full automorphism groups. At the end of the paper, we calculate the character of a $3C$ element of the Monster simple group.
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