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

2012년 7월 17일 (화) 13:17 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

\(V=V(c,0)\) : a highest weight module for Virasoro algebra
- \(\mathbf{1}\in V\) is the highest weight vector (vacuum)
- central charge \(c\in \mathbb{C}\)
- \(L_{-n}\mathbf{1}=0\) for \(n\geq 1\)
- \(V(c,0)=M(c,0)/\langle L_{-1}\mathbf{1} \rangle\)
conformal vector
\(\omega=L_{-2}.\mathbf{1}\)
\(Y(\omega,z)=L(z)=\sum L(n)z^{-n-2}\)

@@ 17번째 줄: / 17번째 줄: @@
 * <math>Y(\mathbf{1},z)=\operatorname{id}_{V}</math>
 *  (creation property)<br><math>Y(v,z).\mathbf{1}=v+\cdots</math><br>
-*  conformal vector<br><math>Y(\omega,z)=L(z)=\sum L(n)z^{-n-2}</math><br>
+*  conformal vector<br><math>Y(\omega,z)=L(z)=\sum L(n)z^{-n-2}</math> satisfies<br><math>[L_m,L_n]=(m-n)L_{m+n}+\frac{c}{12}(m^3-m)\delta_{m+n}</math><br>
-*  locality<br><math>(z_1-z_2)^N[Y(v_1,z_1),Y(v_2,z_2)]=0</math> for some positive integer N<br>
+* <math>L(0)v=nv</math> for <math>n\in\mathbb{Z}</math> and <math>v\in V_{(n)}</math>
-* creativity
-*  derivation with<br><math>D.\mathbf{1}=0</math><br>
 *  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>