Editing Goddard–Thorn theorem (section)

==Applications==
The bosonic string quantization functors described here can be applied to any conformal vertex algebra of central charge 26, and the output naturally has a Lie algebra structure.  The Goddard–Thorn theorem can then be applied to concretely describe the Lie algebra in terms of the input vertex algebra.

Perhaps the most spectacular case of this application is [[Richard Borcherds]]'s proof of the [[monstrous moonshine]] conjecture, where the unitarizable Virasoro representation is the [[monster vertex algebra]] (also called "moonshine module") constructed by [[Igor Frenkel|Frenkel]], [[James Lepowsky|Lepowsky]], and [[Arne Meurman|Meurman]].  By taking a tensor product with the vertex algebra attached to a rank-2 hyperbolic lattice, and applying quantization, one obtains the [[monster Lie algebra]], which is a [[generalized Kac–Moody algebra]] graded by the lattice.  By using the Goddard–Thorn theorem, Borcherds showed that the homogeneous pieces of the Lie algebra are naturally isomorphic to graded pieces of the moonshine module, as representations of the [[monster simple group]].

Earlier applications include Frenkel's determination of upper bounds on the root multiplicities of the Kac–Moody Lie algebra whose Dynkin diagram is the [[Leech lattice]], and Borcherds's construction of a generalized Kac–Moody Lie algebra that contains Frenkel's Lie algebra and saturates Frenkel's 1/∆ bound.