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

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<h5>definition</h5>
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==definition==
  
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* vertex operator algbera is a quadruple <math>(V,Y,\mathbf{1},\omega)</math> with the following axioms
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* 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>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>
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*  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>
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*  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>
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* two distinguished vectors <math>\mathbf{1}\in V_{(0)}</math> and  <math>\omega\in V_{(2)}</math>
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* two distinguished vectors <math>\mathbf{1}\in V_{(0)}</math> and  <math>\omega\in V_{(2)}</math>
  
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<h5>vertex algebra vs VOA</h5>
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==vertex algebra vs VOA==
  
 
* grading on V
 
* grading on V
  
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<h5>axioms</h5>
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==axioms==
  
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*  <br><math>u_{n}v=0</math> for <math>n>>0</math><br>
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*   <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>
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*  (creation property)<br><math>Y(v,z).\mathbf{1}=v+\cdots</math><br>
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*  (creation property)<math>Y(v,z).\mathbf{1}=v+\cdots</math>
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*  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>
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*  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>
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*  translation covariance  <br><math>[D, Y(v,z)]=\sum_{n}[D,V_n]z^{-n-1}=\partial Y(v,z)</math><br>
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*  translation covariance  <math>[D, Y(v,z)]=\sum_{n}[D,V_n]z^{-n-1}=\partial Y(v,z)</math>
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*  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>
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*  Jacobi identity
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:<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>
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==remark on Jacobi identity==
  
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<h5>remark on Jacobi identity</h5>
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*  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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* Jacobi identity for Lie algebra says<br><math>(\operatorname{ad} u)(\operatorname{ad} v)-(\operatorname{ad} v)-(\operatorname{ad} u)=(\operatorname{ad}(\operatorname{ad} u) v)</math><br>
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==related items==
  
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<h5>history</h5>
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* http://www.google.com/search?hl=en&tbs=tl:1&q=
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==expositions==
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* Mason, [http://ws15.amsi.org.au/wp-content/uploads/sites/9/2015/05/Vanderbilt.pdf Vertex Operator Algebras]
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* Mason, [http://ws15.amsi.org.au/wp-content/uploads/sites/9/2015/05/Heidelberg.pdf Vertex Operator Algebras, Modular Forms and Moonshine]
  
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==articles==
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* 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
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* 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.
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* 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.
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* 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.
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* 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.
  
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<h5>related items</h5>
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[[분류:개인노트]]
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[[분류:migrate]]
  
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==메타데이터==
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===위키데이터===
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* ID : [https://www.wikidata.org/wiki/Q28509 Q28509]
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===Spacy 패턴 목록===
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<h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">encyclopedia</h5>
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* [{'LOWER': 'vertex'}, {'LOWER': 'operator'}, {'LEMMA': 'algebra'}]
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* http://en.wikipedia.org/wiki/
 
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* http://www.scholarpedia.org/
 
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* http://eom.springer.de
 
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* http://www.proofwiki.org/wiki/
 
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* Princeton companion to mathematics([[2910610/attachments/2250873|Companion_to_Mathematics.pdf]])
 
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<h5>books</h5>
 
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* [[2011년 books and articles]]
 
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* http://library.nu/search?q=
 
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* http://library.nu/search?q=
 
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<h5>expositions</h5>
 
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<h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">articles</h5>
 
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* http://www.ams.org/mathscinet
 
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* http://www.zentralblatt-math.org/zmath/en/
 
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* http://arxiv.org/
 
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* http://www.pdf-search.org/
 
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* http://pythagoras0.springnote.com/
 
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* [http://math.berkeley.edu/%7Ereb/papers/index.html http://math.berkeley.edu/~reb/papers/index.html]
 
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* http://dx.doi.org/
 
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<h5>question and answers(Math Overflow)</h5>
 
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* http://mathoverflow.net/search?q=
 
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* http://math.stackexchange.com/search?q=
 
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* http://physics.stackexchange.com/search?q=
 
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<h5>blogs</h5>
 
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*  구글 블로그 검색<br>
 
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**  http://blogsearch.google.com/blogsearch?q=<br>
 
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** http://blogsearch.google.com/blogsearch?q=
 
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* http://ncatlab.org/nlab/show/HomePage
 
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<h5>experts on the field</h5>
 
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* http://arxiv.org/
 
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<h5>links</h5>
 
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* [http://detexify.kirelabs.org/classify.html Detexify2 - LaTeX symbol classifier]
 
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* [http://pythagoras0.springnote.com/pages/1947378 수식표현 안내]
 

2021년 2월 17일 (수) 02:15 기준 최신판

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\((\operatorname{ad} u)(\operatorname{ad} v)-(\operatorname{ad} v)-(\operatorname{ad} u)=(\operatorname{ad}(\operatorname{ad} u) v)\)



related items

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.

메타데이터

위키데이터

Spacy 패턴 목록

  • [{'LOWER': 'vertex'}, {'LOWER': 'operator'}, {'LEMMA': 'algebra'}]
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