Monstrous moonshine MOC

Vertex algebra

A vertex algebra is a -graded vector space (by weight[^wt])

truncated from below such that sufficiently small , equipped with a linear map into formal sums over endomorphisms called vertex operators

with a distinguished vacuum element such that the following conditions holds for [^1988] #m/def/voa

1. for sufficiently large ;
2. ;
3. and ; and
4. the generalized Jacobi identity holds

\begin{align*} z_{0}^{-1} \delta\left( \frac{z_{1}-z_{2}}{z_{0}} \right) Y(u,z_{1}) Y(v,z_{2}) - z_{0}^{-1} \delta\left( \frac{z_{2}-z_{1}}{-z_{0}} \right)Y(v,z_{2})Y(u,z_{1}) \ = z_{2}^{-1}\delta\left( \frac{z_{1}-z_{0}}{z_{2}} \right)Y(Y(u,z_{0})v,z_{2}) \end{align*}