Editing Vertex operator algebra (section)

===Vertex algebra===
A '''vertex algebra''' is a collection of data that satisfy certain axioms.

==== Data ====
* a [[vector space]] <math>V</math>, called the space of states.  The underlying [[field (mathematics)|field]] is typically taken to be the [[complex number]]s, although Borcherds's original formulation allowed for an arbitrary [[commutative ring]].
* an identity element <math>1\in V</math>, sometimes written <math>|0\rangle</math> or <math>\Omega</math> to indicate a vacuum state.
* an [[endomorphism]] <math>T:V\rightarrow V</math>, called "translation". (Borcherds's original formulation included a system of divided powers of <math>T</math>, because he did not assume the ground ring was divisible.)
* a linear multiplication map <math>Y:V\otimes V\rightarrow V((z))</math>, where <math>V((z))</math> is the space of all [[formal Laurent series]] with coefficients in <math>V</math>. This structure has some alternative presentations:
** as an infinite collection of bilinear products <math> \cdot_n : u \otimes v \mapsto u_n v</math> where <math>n \in \mathbb{Z}</math> and <math> u_n \in \mathrm{End}(V)</math>, so that for each <math>v</math>, there is an <math>N</math> such that <math>u_n v = 0</math> for <math>n < N</math>.
** as a left-multiplication map <math>V\rightarrow \mathrm{End}(V)[[z^{\pm 1}]]</math>. This is the 'state-to-field' map of the so-called state-field correspondence. For each <math>u\in V</math>, the endomorphism-valued [[formal distribution]] <math>Y(u,z)</math> is called a vertex operator or a field, and the coefficient of <math>z^{-n-1}</math> is the operator <math>u_{n}</math>. In the context of vertex algebras, a '''field''' is more precisely an element of <math>\mathrm{End}(V)[[z^{\pm 1}]]</math>, which can be written <math>A(z) = \sum_{n \in \mathbb{Z}}A_n z^n, A_n \in \mathrm{End}(V)</math> such that for any <math>v \in V, A_n v = 0</math> for sufficiently small <math>n</math> (which may depend on <math>v</math>). The standard notation for the multiplication is
::<math display = block>u \otimes v \mapsto Y(u,z)v = \sum_{n \in \mathbf{Z}} u_n v z^{-n-1}.</math>

==== Axioms ====
These data are required to satisfy the following axioms:

* '''Identity.''' For any <math>u\in V\,,\,Y(1,z)u=u</math> and <math>\,Y(u,z)1\in u+zV[[z]]</math>.{{efn|This last axiom can be used to provide a 'field-to-state' map for the state-field correspondence}}
* '''Translation.''' <math>T(1)=0</math>, and for any <math>u,v\in V</math>,
::<math>[T,Y(u,z)]v = TY(u,z)v - Y(u,z)Tv = \frac{d}{dz}Y(u,z)v</math>

* '''Locality (Jacobi identity, or Borcherds identity).''' For any <math>u,v\in V</math>, there exists a positive [[integer]] {{mvar|N}} such that:
::<math> (z-x)^N Y(u, z) Y(v, x) = (z-x)^N Y(v, x) Y(u, z).</math>

===== Equivalent formulations of locality axiom =====
The locality axiom has several equivalent formulations in the literature, e.g., Frenkel–Lepowsky–Meurman introduced the Jacobi identity: <math>\forall u,v,w\in V</math>,

:<math> \begin{aligned}&z^{-1}\delta\left(\frac{x-y}{z}\right)Y(u,x)Y(v,y)w - z^{-1}\delta\left(\frac{-y+x}{z}\right)Y(v,y)Y(u,x)w \\&= y^{-1}\delta\left(\frac{x-z}{y}\right)Y(Y(u,z)v,y)w\end{aligned},</math>

where we define the formal delta series by:

:<math>\delta\left(\frac{x-y}{z}\right) := \sum_{s \geq 0, r \in \mathbf{Z}} \binom{r}{s} (-1)^s y^{r-s}x^s z^{-r}.</math>

Borcherds{{sfn|Borcherds|1986}} initially used the following two identities: for any <math>u,v,w\in V</math> and integers <math> m,n</math> we have

:<math>(u_m (v))_n (w) = \sum_{i \geq 0} (-1)^i \binom{m}{i} \left (u_{m-i} (v_{n+i} (w)) - (-1)^m v_{m+n-i} (u_i (w)) \right)</math>
and
:<math> u_m v=\sum_{i\geq 0}(-1)^{m+i+1}\frac{T^{i}}{i!}v_{m+i}u </math>.

He later gave a more expansive version that is equivalent but easier to use: for any <math>u,v,w\in V</math> and integers <math> m,n,q</math> we have

:<math>\sum_{i \in \mathbf{Z}} \binom{m}{i} \left(u_{q+i} (v) \right )_{m+n-i} (w) = \sum_{i\in \mathbf{Z}} (-1)^i \binom{q}{i} \left (u_{m+q-i} \left(v_{n+i} (w) \right ) - (-1)^q v_{n+q-i} \left (u_{m+i} (w) \right ) \right)</math>

This identity is the same as the Jacobi identity by expanding both sides in all formal variables. Finally, there is a formal function version of locality: For any <math>u,v,w\in V</math>, there is an element

:<math>X(u,v,w;z,x) \in V[[z,x]] \left[z^{-1}, x^{-1}, (z-x)^{-1} \right]</math>

such that <math>Y(u,z)Y(v,x)w</math> and <math>Y(v,x)Y(u,z)w</math> are the corresponding expansions of <math>X(u,v,w;z,x)</math> in <math>V((z))((x))</math> and <math>V((x))((z))</math>.