A step to the perfect classification of 24dimensional meromorphic vertex operator algebras

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A concept of vertex operator algebras (VOA shortly) has originated from the moonshine vertex operator algebra, which was constructed in order to explain a mysterious relation (the moonshine conjecture) between Monster simple finite group (the largest sporadic finite simple group) and the classical elliptic modular function. It is now understand to be a rigorous and mathematical concept of 2 dimensional conformal field theory in physics. Namely, it offers axioms on such a theory. Although 24 dimensional meromorphic conformal field theory have a special important value in physics, the examples we know at present are only Moonshine VOA, VOAs constructed from the Niemeier lattices and their orbifold VOAs. Recently, Dong and Mason found that all of them contain a tensor product of 48 Ising modules.With the support of this Grant, M.Miyamoto (Head of this research) has been studying VOAs containing a tensor product of Ising modules, he found new VOAs called "code VOAs", which are easy to hand … More le compared with the other VOAs in 1997. We also determined their representations (all modules) and introduced a concept of "induced modules." In 1998, we found a special property of Hamming code VOAs (constructed from an extended [8, 4, 4]-Hamming code) and determined its fusion rules among its irreducible modules. Using this special property, we found a new construction of the famous moonshine VOA and then a new construction of Monster simple group. It is very easier than the original construction and obtained a lot of properties of Monster simple group. In 2000, we applied the new method of construction to the known VOAs (lattice VOAs, etc.) and found that it is possible to construct all known 24 dimensional holomorphic VOAs by this way. We also succeed to construct twisted modules of code VOAs. For twisted modules, the existence was proved theoretically, but we don't know examples except very easy one. So we hope that this new construction have many applications, especially we expect to construct twisted modules for the moonshine VOAs, which are the essential parts of the moonshine conjectures. Less