"Vertex operator algebra (VOA)"의 두 판 사이의 차이
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| − | <h5> | + | <h5>definition</h5> |
| − | * <math>V=\bigoplus_{n\in\mathbb{Z}}V_{(n)}</math> | + | * vertex operator algbera is a quadruple <math>(V,Y,\mathbf{1},\omega)</math> with the following axioms |
| + | * <math>V=\bigoplus_{n\in\mathbb{Z}}V_{(n)}</math> vector space | ||
* <math>\dim V_{(n)} <\infty</math> for <math>n\in \mathbb{Z}</math> | * <math>\dim V_{(n)} <\infty</math> for <math>n\in \mathbb{Z}</math> | ||
* <math>\dim V_{(n)}=0</math> for <math>n<<0</math> | * <math>\dim V_{(n)}=0</math> for <math>n<<0</math> | ||
| − | * vertex operator<br><math>V\to (\operatorname{End})[[x,x^{-1}]]</math><br><math>v\mapsto Y(v,x)=\ | + | * vertex operator<br><math>V\to (\operatorname{End})[[x,x^{-1}]]</math><br><math>v\mapsto Y(v,x)=\sum_{n\in \mathbb{Z}}v_{n}x^{-n-1}</math><br> |
| − | * | + | * two distinguished vectors <math>\mathbf{1}\in V_{(0)}</math> and <math>\omega\in V_{(2)}</math> |
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| + | <h5>axioms</h5> | ||
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| + | * <br><math>u_{n}v=0</math> for <math>n>>0</math><br> | ||
| + | * <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> | ||
* 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> | * 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> | ||
| − | * | + | * creativity |
* derivation with<br><math>D.\mathbf{1}=0</math><br> | * 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> | * translation covariance <br><math>[D, Y(v,z)]=\sum_{n}[D,V_n]z^{-n-1}=\partial Y(v,z)</math><br> | ||
2012년 7월 17일 (화) 13:12 판
definition
- 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 \(\omega\in V_{(2)}\)
axioms
-
\(u_{n}v=0\) for \(n>>0\) - \(Y(\mathbf{1},z)=\operatorname{id}_{V}\)
- (creation property)
\(Y(v,z).\mathbf{1}=v+\cdots\) - conformal vector
\(Y(\omega,z)=L(z)=\sum L(n)z^{-n-2}\) - locality
\((z_1-z_2)^N[Y(v_1,z_1),Y(v_2,z_2)]=0\) for some positive integer N - creativity
- derivation with
\(D.\mathbf{1}=0\) - translation covariance
\([D, Y(v,z)]=\sum_{n}[D,V_n]z^{-n-1}=\partial Y(v,z)\)
Virasoro VOA
- \(V=V(c,0)\) : 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 \(n\geq 1\)
- \(V(c,0)=M(c,0)/\langle L_{-1}\mathbf{1} \rangle\)
- \(\mathbf{1}\in V\) is the highest weight vector (vacuum)
- conformal vector
\(\omega=L_{-2}.\mathbf{1}\)
\(Y(\omega,z)=L(z)=\sum L(n)z^{-n-2}\)
history
encyclopedia
- http://en.wikipedia.org/wiki/
- http://www.scholarpedia.org/
- http://eom.springer.de
- http://www.proofwiki.org/wiki/
- Princeton companion to mathematics(Companion_to_Mathematics.pdf)
books
expositions
articles
- http://www.ams.org/mathscinet
- http://www.zentralblatt-math.org/zmath/en/
- http://arxiv.org/
- http://www.pdf-search.org/
- http://pythagoras0.springnote.com/
- http://math.berkeley.edu/~reb/papers/index.html
- http://dx.doi.org/
question and answers(Math Overflow)
- http://mathoverflow.net/search?q=
- http://math.stackexchange.com/search?q=
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blogs
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experts on the field