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The calculation of the character of moonshine VOA. (English) Zbl 1017.17025
Bannai, Eiichi (ed.) et al., Groups and combinatorics - in memory of Michio Suzuki. Tokyo: Mathematical Society of Japan. Adv. Stud. Pure Math. 32, 325-336 (2001).
The author computes the character of the moonshine vertex operator algebra $$V^\sharp$$ and the Thompson series for $$2A$$ and $$2B$$ involutions of the Monster simple group [J. H. Conway and S. P. Norton, Bull. Lond. Math. Soc. 11, 308-339 (1979; Zbl 0424.20010)] explicitly in terms of the character of Ising models $$L(1/2,h)$$, $$h = 0,1/2,1/16$$. The moonshine vertex operator algebra $$V^\sharp$$ is decomposed into a direct sum of irreducible $$T$$-modules, where $$T$$ is a vertex operator subalgebra of $$V^\sharp$$ isomorphic to $$L(1/2,0)^{\otimes 48}$$. The decomposition is completely described by M. Miyamoto [Ann. Math. (to appear), see also C. Dong, R. L. Griess and G. Höhn [Commun. Math. Phys. 193, 407-448 (1998; Zbl 0908.17018)]. In fact, Miyamoto constructed $$V^\sharp$$ from Ising models and two binary even codes. Using Miyamoto’s description of $$V^\sharp$$ and some basic formulas, the author obtains the results. Some vertex operator algebras constructed in a similar way as Miyamoto’s construction of $$V^\sharp$$ are also discussed.
For the entire collection see [Zbl 0983.00069].
##### MSC:
 17B69 Vertex operators; vertex operator algebras and related structures 20D08 Simple groups: sporadic groups 17B68 Virasoro and related algebras