"Vertex operator algebra (VOA)"의 두 판 사이의 차이

2012년 7월 17일 (화) 13:30 판

vertex operator algbera is a quadruple $(V,Y,\mathbf{1},\omega)$ with the following axioms
$V=\bigoplus_{n\in\mathbb{Z}}V_{(n)}$ vector space
$\dim V_{(n)} <\infty$ for $n\in \mathbb{Z}$
$\dim V_{(n)}=0$ for $n<<0$
vertex operator
$V\to (\operatorname{End})[[x,x^{-1}]]$
$v\mapsto Y(v,x)=\sum_{n\in \mathbb{Z}}v_{n}x^{-n-1}$
two distinguished vectors $\mathbf{1}\in V_{(0)}$ and $\omega\in V_{(2)}$

$u_{n}v=0$ for $n>>0$
$Y(\mathbf{1},z)=\operatorname{id}_{V}$
(creation property)
$Y(v,z).\mathbf{1}=v+\cdots$
conformal vector
$Y(\omega,z)=L(z)=\sum L(n)z^{-n-2}$ satisfies
$[L_m,L_n]=(m-n)L_{m+n}+\frac{c}{12}(m^3-m)\delta_{m+n}$
$L(0)v=nv$ for $n\in\mathbb{Z}$ and $v\in V_{(n)}$
translation covariance
$[D, Y(v,z)]=\sum_{n}[D,V_n]z^{-n-1}=\partial Y(v,z)$
Jacobi identity
$ $z_0^{-1}\delta(\frac {z_1-z_2}{z_0})Y(u,z_1)Y(v,z_2)-z_0^{-1}\delta(\frac {z_2-z_1}{-z_0})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)$$

@@ 20번째 줄: / 20번째 줄: @@
 * <math>L(0)v=nv</math> for <math>n\in\mathbb{Z}</math> and <math>v\in V_{(n)}</math>
 *  translation covariance  <br><math>[D, Y(v,z)]=\sum_{n}[D,V_n]z^{-n-1}=\partial Y(v,z)</math><br>
-*  Jacobi identity<br>  <br>
+*  Jacobi identity<br><math> $z_0^{-1}\delta(\frac   {z_1-z_2}{z_0})Y(u,z_1)Y(v,z_2)-z_0^{-1}\delta(\frac   {z_2-z_1}{-z_0})Y(v,z_2)Y(u,z_1)=z_2^{-1}\delta\left(\frac   {z_1-z_0}{z_2}\right)Y(Y(u,z_0)v,z_2)$</math><br>
@@ 26번째 줄: / 26번째 줄: @@
-<h5 style="line-height: 2em; margin: 0px;">Virasoro VOA</h5>
+<h5>remark on Jacobi identity</h5>
-* <math>V=V(c,0)</math> : a highest weight module for [[Virasoro algebra]]<br>
-** <math>\mathbf{1}\in V</math> is the highest weight vector (vacuum)<br>
-**  central charge <math>c\in \mathbb{C}</math><br>
-** <math>L_{-n}\mathbf{1}=0</math> for <math>n\geq 1</math><br>
-** <math>V(c,0)=M(c,0)/\langle L_{-1}\mathbf{1} \rangle</math><br>
-*  conformal vector<br><math>\omega=L_{-2}.\mathbf{1}</math><br><math>Y(\omega,z)=L(z)=\sum L(n)z^{-n-2}</math><br>