Vertex operator algebra (VOA)

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 \(n\geq 1\)
    
  • \(V(c,0)=M(c,0)/\langle L_{-1}\mathbf{1} \rangle\)
    
• conformal vector
  \(\omega=L_{-2}.\mathbf{1}\)
  \(Y(\omega,z)=L(z)=\sum L(n)z^{-n-2}\)
  

 

 

history

• http://www.google.com/search?hl=en&tbs=tl:1&q=

 

 

 

 

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

 

• 2011년 books and articles
• http://library.nu/search?q=
• http://library.nu/search?q=

 

 

 

 

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=
• http://physics.stackexchange.com/search?q=

 

 

 

blogs

• 구글 블로그 검색
  
  • http://blogsearch.google.com/blogsearch?q=
    
  • http://blogsearch.google.com/blogsearch?q=
• http://ncatlab.org/nlab/show/HomePage

 

 

experts on the field

• http://arxiv.org/

 

 

links

• Detexify2 - LaTeX symbol classifier
• 수식표현 안내