Monstrous moonshine MOC

Vertex operator algebra

A vertex operator algebra 𝑉 is a Vertex algebra which carries a representation of the Virasoro algebra, voa specifically there is a distinguished homogenous conformal element πœ” βˆˆπ‘‰ with vertex operator π‘Œ(πœ”,𝑧) =βˆ‘π‘›βˆˆβ„€πΏ(𝑛)π‘§βˆ’π‘›βˆ’2 such that1

[𝐿(π‘š),𝐿(𝑛)]=(π‘šβˆ’π‘›)𝐿(π‘š+𝑛)+rank⁑𝑉12(π‘š3βˆ’π‘š)π›Ώπ‘š+𝑛

where

  1. rank⁑𝑉 βˆˆβ„š;
  2. 𝐿(0)𝑣 =(wt⁑𝑣)𝑣 for homogenous 𝑣 βˆˆπ‘‰(𝑛); and
  3. π‘‘π‘‘π‘§π‘Œ(𝑣,𝑧) =π‘Œ(𝐿( βˆ’1)𝑣,𝑧).

Properties

[𝐿(βˆ’1),π‘Œ(𝑣,𝑧)]=π‘Œ(𝐿(βˆ’1)𝑣,𝑧)
[𝐿(0),π‘Œ(𝑣,𝑧)]=π‘Œ(𝐿(0)𝑣,𝑧)+π‘§π‘Œ(𝐿(βˆ’1)𝑣,𝑧)
𝑛β‰₯βˆ’1⟹𝐿(𝑛)πŸ™=0
𝐿(βˆ’2)πŸ™=πœ”
𝐿(0)πœ”=2πœ”

develop | en | SemBr

Footnotes

  1. 1988. Vertex operator algebras and the Monster. Β§8.10, pp. 244–245 ↩

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