TAILIEUCHUNG - Đề tài " A new construction of the moonshine vertex operator algebra over the real number field "

We give a new construction of the moonshine module vertex operator algebra V , which was originally constructed in [FLM2]. We construct it as a framed VOA over the real number field 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 VE8 and construct many framed VOAs including V from a small VOA. One of the advantages of our construction is that we are able to construct V as a framed VOA with a positive definite invariant bilinear form and we. | Annals of Mathematics A new construction of the moonshine vertex operator algebra over the real number field By Masahiko Miyamoto Annals of Mathematics 159 2004 535 596 A new construction of the moonshine vertex operator algebra over the real number field By Masahiko MiyAMOTO Abstract We give a new construction of the moonshine module vertex operator algebra Vt which was originally constructed in FLM2 . We construct it as a framed VOA over the real number field 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 VE8 and construct many framed VOAs including Vfrom a small VOA. One of the advantages of our construction is that we are able to construct Vas a framed VOA with a positive definite invariant bilinear form and we can easily prove that Aut Vb 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. 1. Introduction All vertex operator algebras VOAs V Y 1 w in this paper are simple VOAs defined over the real number field R and satisfy V 0V and dim V0 1. CV denotes the complexification C Or V of V. Throughout this paper V m denotes a coefficient of vertex operator Y v z 2mỊỉ V m Z m 1 of v at z m-1 and Y w z 52rnez L m z m 2 where w is the Virasoro element of V. VOAs conformal field theories are usually considered over C but VOAs over R are extremely important for finite group theory. The most interesting example of VOAs is the moonshine module VOA V 52i 0 Vi over R constructed in FLM2 whose second primary space V2 coincides with the Griess algebra and the full automorphism group is the Monster simple group M. Although it has many interesting properties the original construction essentially depends on the actions of the centralizer CM ff 21 of a 2B-involution ff of M and it is hard

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