Editing Vertex operator algebra (section)

=== Operator product expansion ===
In vertex algebra theory, due to associativity, we can abuse notation to write, for <math>A, B, C \in V,</math>
<math display = block>Y(A, z)Y(B,w)C = \sum_{n \in \mathbb{Z}}\frac{Y(A_{(n)}\cdot B, w)}{(z-w)^{n+1}}C.</math>
This is the '''operator product expansion'''. Equivalently,
<math display = block>Y(A, z)Y(B,w) = \sum_{n \geq 0}\frac{Y(A_{(n)}\cdot B, w)}{(z-w)^{n+1}} + :Y(A,z)Y(B,w):.</math>
Since the normal ordered part is regular in <math>z</math> and <math>w</math>, this can be written more in line with physics conventions as
<math display = block>Y(A, z)Y(B,w) \sim \sum_{n \geq 0}\frac{Y(A_{(n)}\cdot B, w)}{(z-w)^{n+1}},</math>
where the [[equivalence relation]] <math>\sim</math> denotes equivalence up to regular terms.

==== Commonly used OPEs ====
Here some OPEs frequently found in conformal field theory are recorded.{{sfn|Kac|1998|p=38}}

{| class="wikitable"
|+ OPEs
|-
! 1st distribution !! 2nd distribution !! Commutation relations !! OPE !! Name !! Notes
|-
| <math>a(z) = \sum a_m z^{-m-1}</math> || <math>b(w) = \sum b_n z^{-n-1}</math> || <math>[a_m, b_n] = c_{m+n}</math> || <math>a(z)b(w) \sim \frac{\sum c_nw^{-n-1}}{z-w}</math> || Generic OPE ||
|-
| <math>a(z) = \sum a_m z^{-m-1}</math> || <math>b(w) = \sum b_n z^{-n-1}</math> || <math>[a_m, b_n] = m \delta_{m+n,0}</math> || <math>a(z)b(w) \sim \frac{1}{(z-w)^2}</math> || Free boson OPE || Invariance under <math>z \leftrightarrow w</math> shows 'bosonic' nature of this OPE.
|-
| <math>L(z) = \sum L_m z^{-m-2}</math> || <math>a(w) = \sum a_n w^{-n-\Delta}</math> || <math>[L_m, a_n] = ((\Delta - 1)m - n)a_{m+n}</math> || <math>L(z)a(w) \sim \frac{\Delta a(w)}{(z-w)^2} + \frac{\partial a(w)}{z-w}</math> || Primary field OPE || [[Primary field]]s are defined to be fields a(z) satisfying this OPE when multiplied with the Virasoro field. These are important as they are the fields which transform 'like tensors' under coordinate transformations of the [[worldsheet]] in [[string theory]].
|-
| <math>L(z) = \sum L_m z^{-m-2}</math> || <math>L(w) = \sum L_n z^{-n-2}</math> || <math>[L_m, L_n] = (m-n)L_{m+n} + \frac{m^3 - m}{12}\delta_{m+n,0}c</math> || <math>L(z)L(w) \sim \frac{c/2}{(z-w)^4} + \frac{2L(w)}{(z-w)^2} + \frac{\partial L(w)}{z-w}</math> || TT OPE || In physics, the Virasoro field is often identified with the [[stress-energy tensor]] and labelled T(z) rather than L(z).
|}