"Vertex operator algebra (VOA)"의 두 판 사이의 차이
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| 5번째 줄: | 5번째 줄: | ||
* <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 | + | * vertex operator<math>V\to (\operatorname{End})[[x,x^{-1}]]</math><math>v\mapsto Y(v,x)=\sum_{n\in \mathbb{Z}}v_{n}x^{-n-1}</math> |
* two distinguished vectors <math>\mathbf{1}\in V_{(0)}</math> and <math>\omega\in V_{(2)}</math> | * two distinguished vectors <math>\mathbf{1}\in V_{(0)}</math> and <math>\omega\in V_{(2)}</math> | ||
| 22번째 줄: | 22번째 줄: | ||
==axioms== | ==axioms== | ||
| − | * | + | * <math>u_{n}v=0</math> for <math>n>>0</math> |
* <math>Y(\mathbf{1},z)=\operatorname{id}_{V}</math> | * <math>Y(\mathbf{1},z)=\operatorname{id}_{V}</math> | ||
| − | * (creation property) | + | * (creation property)<math>Y(v,z).\mathbf{1}=v+\cdots</math> |
| − | * conformal vector | + | * conformal vector<math>Y(\omega,z)=L(z)=\sum L(n)z^{-n-2}</math> satisfies<math>[L_m,L_n]=(m-n)L_{m+n}+\frac{c}{12}(m^3-m)\delta_{m+n}</math> |
* <math>L(0)v=nv</math> for <math>n\in\mathbb{Z}</math> and <math>v\in V_{(n)}</math> | * <math>L(0)v=nv</math> for <math>n\in\mathbb{Z}</math> and <math>v\in V_{(n)}</math> | ||
| − | * translation covariance | + | * translation covariance <math>[D, Y(v,z)]=\sum_{n}[D,V_n]z^{-n-1}=\partial Y(v,z)</math> |
* Jacobi identity | * Jacobi identity | ||
:<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> | :<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> | ||
| 36번째 줄: | 36번째 줄: | ||
==remark on Jacobi identity== | ==remark on Jacobi identity== | ||
| − | * Jacobi identity for Lie algebra says | + | * Jacobi identity for Lie algebra says<math>(\operatorname{ad} u)(\operatorname{ad} v)-(\operatorname{ad} v)-(\operatorname{ad} u)=(\operatorname{ad}(\operatorname{ad} u) v)</math> |
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2020년 11월 16일 (월) 11:06 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 \(\omega\in V_{(2)}\)
vertex algebra vs VOA
- grading on V
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\(Y(\omega,z)=L(z)=\sum L(n)z^{-n-2}\) 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
\[z_0^{-1}\delta(\frac{z_1-z_2}{z_0})Y(u,z_1)Y(v,z_2)-z_0^{-1}\delta(\frac{z_2-z_1}{-z_0})Y(v,z_2)Y(u,z_1)=z_2^{-1}\delta\left(\frac {z_1-z_0}{z_2}\right)Y(Y(u,z_0)v,z_2)\]
remark on Jacobi identity
- Jacobi identity for Lie algebra says\((\operatorname{ad} u)(\operatorname{ad} v)-(\operatorname{ad} v)-(\operatorname{ad} u)=(\operatorname{ad}(\operatorname{ad} u) v)\)
expositions
articles
- Yi-Zhi Huang, Jinwei Yang, Associative algebras for (logarithmic) twisted modules for a vertex operator algebra, arXiv:1603.04367 [math.QA], March 14 2016, http://arxiv.org/abs/1603.04367
- Gilroy, Thomas, and Michael P. Tuite. “Genus Two Zhu Theory for Vertex Operator Algebras.” arXiv:1511.07664 [hep-Th], November 24, 2015. http://arxiv.org/abs/1511.07664.
- Ai, Chunrui, and Xingjun Lin. “On the Unitary Structures of Vertex Operator Superalgebras.” arXiv:1510.08609 [math], October 29, 2015. http://arxiv.org/abs/1510.08609.
- Ding, Lu, Wei Jiang, and Wei Zhang. “Zhu’s Algebra of a C1-Cofinite Vertex Algebra.” arXiv:1508.06351 [math], August 25, 2015. http://arxiv.org/abs/1508.06351.
- van Ekeren, Jethro, Sven Möller, and Nils R. Scheithauer. “Construction and Classification of Holomorphic Vertex Operator Algebras.” arXiv:1507.08142 [math], July 29, 2015. http://arxiv.org/abs/1507.08142.