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  \(\omega\in V_{(2)}\)

 

 

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

 

 

history

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

 

 

 

expositions

• Mason, Vertex Operator Algebras
• Mason, Vertex Operator Algebras, Modular Forms and Moonshine