zbMATH — the first resource for mathematics

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].
17B69 Vertex operators; vertex operator algebras and related structures
20D08 Simple groups: sporadic groups
17B68 Virasoro and related algebras