Vertex algebra

It has been suggested that this article or section be merged with vertex operator algebra. (Discuss)

In mathematics, a vertex algebra is a (complex) vector space $V$ with a linear map

$\begin{alignat}{2} V\otimes V & \to V((x)) \\ a\otimes b & \mapsto a(x)b=\sum_{n\in \mathbb{Z}}a_nbx^{-n-1} \\ \end{alignat}$

and a vacuum vector $\mathbf{1}\in V$ satisfying:

$\mathbf{1}(x)b=b,$

$a(x)\mathbf{1}\in a+V[[x]]x,$

$\sum_{n\in \mathbb{Z}} (x-y)^nz^{-n-1}a(x)b(y)-\sum_{n\in \mathbb{Z}}(-y+x)^nz^{-n-1}b(y)a(x)=\sum_{n\in \mathbb{Z}} (x-z)^ny^{-n-1}(a(z)b)(y)$

where the binomials above are expanded as formal power series using the binomial expansion convention:

$(x+y)^n=\sum_{k\in\mathbb{N}}\binom{n}{k}x^{n-k}y^k,$

where

$\binom{n}{k}=\frac{1}{k!}\prod_{i=0}^{k-1}(n-i)=\frac{1}{k!}n(n-1)\dots(n-k+1)$

Equivalently:

$(a,b)\mapsto a_nb\text{ is linear in }a\text{ and }b,$

$a_nb=0\text{ for all but finitely many }n\ge 0,$

$\mathbf{1}_{-1}a=a=a_{-1}\mathbf{1},$

$\mathbf{1}_nb=0\text{ if }n\ne -1,$

$a_n\mathbf{1}=0\text{ if }n\ge 0,$

$(a_nb)_mc=\sum_{k\in\N}\binom{n}{k}\big((-1)^ka_{n-k}(b_{m+k}c)-(-1)^{n-k}b_{n+m-k}(a_kc)\big).$

[edit] See also

James Lepowsky, Haisheng Li, "Introduction to Vertex Operator Algebras and Their Representations". Progress in mathematics, v. 227. Birkhauser, Boston, MA, 2004.