Monstrous moonshine MOC

Vertex algebra

A vertex algebra 𝑉 is a ℤ-graded vector space (by weight¹)

𝑉=⨁𝑛∈ℤ𝑉(𝑛);dim⁡𝑉(𝑛)<∞

truncated from below such that 𝑉(𝑛) =0 sufficiently small 𝑛, equipped with a linear map into formal sums over endomorphisms called vertex operators

𝑉→(End⁡𝑉)[[𝑧,𝑧−1]]𝑣↦𝑌(𝑣,𝑧)=∑𝑛∈ℤ𝑣𝑛𝑧−𝑛−1

with a distinguished vacuum element 𝟙 ∈𝑉 such that the following conditions holds for 𝑢,𝑣 ∈𝑉² #m/def/voa

1. 𝑢𝑛𝑣 =0 for sufficiently large 𝑛;
2. 𝑌(𝟙,𝑣) =1;
3. 𝑌(𝑣,𝑧)𝟙 ∈𝑉[[𝑧]] and lim𝑧→0𝑌(𝑣,𝑧)𝟙 =𝑣; and
4. the generalized Jacobi identity holds

𝑧−10𝛿(𝑧1−𝑧2𝑧0)𝑌(𝑢,𝑧1)𝑌(𝑣,𝑧2)−𝑧−10𝛿(𝑧2−𝑧1−𝑧0)𝑌(𝑣,𝑧2)𝑌(𝑢,𝑧1)=𝑧−12𝛿(𝑧1−𝑧0𝑧2)𝑌(𝑌(𝑢,𝑧0)𝑣,𝑧2)

where 𝛿(𝑧) is the formal delta and all terms are well-defined acting on 𝑣 ∈𝑉 from the left.

Most vertex algebras appearing “in nature” carry a representation of the Virasoro algebra and are hence vertex operator algebras.

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develop | en | SemBr

Footnotes

1. i.e. the grade of a homogenous element 𝑣 ∈𝑉(𝑛) is called its weight and denoted wt⁡𝑣. ↩

2. 1988. Vertex operator algebras and the Monster. §8.10, pp. 244–245 ↩