We show using Borcherds products that for any fixed-point free automorphism of the Leech lattice satisfying a "no massless states" condition, the corresponding cyclic orbifold of the Leech lattice vertex operator algebra is isomorphic to the Monster vertex operator algebra. This induces an "orbifold duality" bijection between algebraic conjugacy classes of fixed-point free automorphisms of the Leech lattice satisfying this condition and algebraic conjugacy classes of non-Fricke elements in the Monster. We use the duality to show that non-Fricke Monstrous Lie algebras are Borcherds-Kac-Moody Lie algebras, and prove a refinement of Norton's Generalized Moonshine conjecture: the ambiguous constants relating generalized moonshine Hauptmoduln under conjugation and modular transformations are necessarily roots of unity. We also describe a class of rank 2 Borcherds-Kac-Moody Lie algebras attached to the Conway group.

中文翻译：

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Moonshine 模块的 51 种结构

我们使用 Borcherds 产品表明，对于满足“无质量状态”条件的 Leech 格的任何不动点自由自同构，Leech 格顶点算子代数的相应循环 orbifold 与 Monster 顶点算子代数同构。这在满足此条件的 Leech 格子的不动点自由自同构的代数共轭类与 Monster 中非 Fricke 元素的代数共轭类之间引发了“轨道对偶性”双射。我们使用对偶性证明非 Fricke Monstrous Lie 代数是 Borcherds-Kac-Moody Lie 代数，并证明了 Norton 广义 Moonshine 猜想的改进：在共轭和模变换下与广义 Moonshine Hauptmoduln 相关的模糊常数必然是统一的根。