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The construction of the moonshine module as $$\mathbb{Z}_ p$$-orbifold. (English) Zbl 0808.17014
Sally, Paul J. jun. (ed.) et al., Mathematical aspects of conformal and topological field theories and quantum groups. AMS-IMS-SIAM summer research conference, June 13-19, 1992, Mount Holyoke College, South Hadley, MA, USA. Providence, RI: American Mathematical Society. Contemp. Math. 175, 37-52 (1994).
Summary: We discuss our recent work (currently being prepared for publication) on the monstrous moonshine module first constructed by Frenkel-Lepowsky- Meurman. As is well-known, the work of FLM can be loosely described as the construction of the conformal field theory $$V_{\Lambda/ \mathbb{Z}_ 2}$$ based on a $$\mathbb{Z}_ 2$$-orbifold $$T^ \Lambda/ \mathbb{Z}_ 2$$ where $$\Lambda$$ is the Leech lattice and $$T^ \Lambda= \mathbb{R}^{24}/ \Lambda$$. We consider:
(i) Conformal field theory $$V_{\Lambda/ \mathbb{Z}_ p}$$ based on certain $$\mathbb{Z}_ p$$-orbifolds of $$T^ \Lambda$$, including a completely rigorous construction for $$p=3$$.
(ii) The identification $$V_{\Lambda/ \mathbb{Z}_ 2} \simeq V_{\Lambda/ \mathbb{Z}_ 3}$$ and characterization of the moonshine module as an irreducible module for the affine Griess algebra.
The detailed version of this paper will be submitted for publication elsewhere.
For the entire collection see [Zbl 0801.00049].

##### MSC:
 17B65 Infinite-dimensional Lie (super)algebras 17B68 Virasoro and related algebras 20D08 Simple groups: sporadic groups