"Affine sl(2)"의 두 판 사이의 차이

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*  strange formula<br><math>\frac{<\rho,\rho>}{2h^{\vee}}=\frac{\operatorname{dim}\mathfrak{g}}{24}</math><br>
 
*  strange formula<br><math>\frac{<\rho,\rho>}{2h^{\vee}}=\frac{\operatorname{dim}\mathfrak{g}}{24}</math><br>
 
*  very strange formula<br>
 
*  very strange formula<br>
−
*  conformal anomaly <br><math>m_{\Lambda}=\frac{(\Lambda+\rho)^2}{2(k+h^{\vee})}-\frac{\rho^2}{2h^{\vee}}=h_{\lambda}-\frac{c(\lambda)}{24}</math><br>
+
*  conformal anomaly <br><math>m_{\Lambda}=\frac{(\Lambda+\rho)^2}{2(k+h^{\vee})}-\frac{\rho^2}{2h^{\vee}}=h_{\lambda}-\frac{c_{\lambda}}{24}</math><br>
  
 
 
 
 

2013년 1월 6일 (일) 15:28 판

introduction

  • affine sl(2) \(A^{(1)}_1\)

 

construction from semisimple Lie algebra

  • this is borrowed from affine Kac-Moody algebra entry
  • Let \(\mathfrak{g}\) be a semisimple Lie algebra with root system \(\Phi\) and the invariant form \(<\cdot,\cdot>\)
  • say \(\mathfrak{g}=A_1\),  \(\Phi=\{\alpha,-\alpha\}\)
  • Cartan matrix
    \(\mathbf{A} = \begin{pmatrix} 2 \end{pmatrix}\)
  • Find the highest root  \(\alpha\)
  • Add another simple root \(\alpha_0\) to the root system \(\Phi\) which is \(\alpha_0=-\alpha\), but we regard this as an independent one now.
  • Construct a new Cartan matrix
    \(A' = \begin{pmatrix} 2 & -2 \\ -2 & 2 \end{pmatrix}\)
  • Note that this matrix has rank 1 since \((1,1)\) belongs to the null space
  • construct a Lie algebra from the new Cartan matrix \(A'\)
  • Add a outer derivation\(d=-l_0\) to compensate the degeneracy of the Cartan matrix
    \(\begin{pmatrix} 2 & -2 & 1\\ -2 & 2 &0 \\ 1 &0 & 0 \end{pmatrix}\)

 

 

basic quantities

  • a_i=1
  • c_i=a_i^{\vee}=1
  • a_{ij}
  • coxeter number 2
  • dual Coxeter number 2
  • Weyl vector

 

 

root systems

  • \(\Phi=\{\alpha+n\delta|\alpha\in\Phi^{0},n\in\mathbb{Z}\}\cup \{n\delta|n\in\mathbb{Z},n\neq 0\}\)
  • real roots
    • \(\{\alpha+n\delta|\alpha\in\Phi^{0},n\in\mathbb{Z}\}\)
  • imaginary roots   
    • \(\{n\delta|n\in\mathbb{Z},n\neq 0\}\)
    • \(\delta=\alpha_0+\alpha_1\)
  • simple roots
    • \(\alpha_0,\alpha_1\)
  • positive roots
    • \(\Phi^{+}=\{\alpha+n\delta|\alpha\in\Phi^{0},n>0\}\cup (\Phi^{0})^{+}\cup \{n\delta|n\in\mathbb{Z},n> 0\}\)

 

 

fixing a Cartan subalgebra and its dual

  • H is a 3-dimensional space
  • basis of the Cartan subalgebra H (this defines C and l_0 also)
    \(h_0=C-h_1\)
    \(h_1\)
    \(d=-l_0\)
  • basis of dual Cartan algebra
    \(\omega_0,\alpha_0,\alpha_1\)
  • dual basis for H
    \(\omega_0,\omega_1=\omega_0+\frac{1}{2}\alpha_1,\delta=\alpha_0+\alpha_1\)
  • Weyl vector
    \(\rho=\omega_0+\omega_1\)
  • pairing
    \(\alpha_0(h_0)=2\)
    \(\alpha_0(h_1)=-2\)
    \(\alpha_0(d)=1\)
    \(\alpha_1(h_0)=-2\)
    \(\alpha_1(h_1)=2\)
    \(\alpha_1(d)=0\)
    \(\omega_0(h_0)=1\)
    \(\omega_0(h_1)=0\)
    \(\omega_0(d)=0\)

 

 

killing form

  • invariant symmetric non-deg bilinear forms
    \(<h_i,h_j>=A_{ij}\)
    \(<h_0,d>=1\)
    \(<h_1,d>=0\)
    \(<d,d>=0\)
  • with centers (note that C=h_0+h_1)
    \(<C,h_0>=0\)
    \(<C,h_1>=0\)
    \(<C,d>=1\)

 

 

explicit construction

  • start with a semisimple Lie algebra \(\mathfrak{g}\) with invariant form \(<\cdot,\cdot>\)
  • make a vector space from it
  • Construct a Loop algbera
    \(\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}]\)
    \(\alpha(m)=\alpha\otimes t^m\)
  • Add a central element to get a central extension and give a bracket
    \(\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}]\oplus\mathbb{C}c\)
    \([E(m),F(n)]=H\otimes t^{m+n}+m\delta_{m,-n}c\)
    \([H(m),E(n)]=2E\otimes t^{m+n}\)
    \([H(m),F(n)]=-2F\otimes t^{m+n}\)
    \([E(m),E(n)]=[F(m),F(n)]=0\)
    \(<c,\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}]\oplus\mathbb{C}c>=0\)
  • Add a derivation \(d\)
    \(d=t\frac{d}{dt}\)
    \(\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}]\oplus\mathbb{C}c \oplus\mathbb{C}d\)
    \(d(\alpha(n))=n\alpha(n)\)
    \(d(c)=0\)
    \(<c,d>=0\)
  • Define a Lie bracket
    \([d,x]=d(x)\)

 

 

denominator formula

 

 

 

level k highest weight representation

  • integrable highest weight
    \(\lambda=\lambda_{0}\omega_0+\lambda_{1}\omega_1\), \(\lambda_{i}\in\mathbb{N}\)
  • level
    \(k=\lambda_{0}+\lambda_{1}\in\mathbb{N}\)
  • therefore \(\lambda_{0}\in\{0,1,\cdots,k\}\)

 

 

central charge

  • central charge (depends on the level only)
    \(c_{\lambda}=\frac{k}{k+h^{\vee}}\text{dim }\mathfrak{\bar{g}}\)
  • conformal weight
    \(h_{\lambda}=\frac{(\lambda|\lambda+2\rho)}{2(k+h^{\vee})}\)
  • definition of conformal anomaly
    \(m_{\Lambda}=\frac{(\Lambda+\rho)^2}{2(k+h^{\vee})}-\frac{\rho^2}{2h^{\vee}}\)
  • strange formula
    \(\frac{<\rho,\rho>}{2h^{\vee}}=\frac{\operatorname{dim}\mathfrak{g}}{24}\)
  • very strange formula
  • conformal anomaly 
    \(m_{\Lambda}=\frac{(\Lambda+\rho)^2}{2(k+h^{\vee})}-\frac{\rho^2}{2h^{\vee}}=h_{\lambda}-\frac{c_{\lambda}}{24}\)

 

 

 

vertex operator construction

 

 

 

related items

 

computational resource

encyclopedia


 

 

books

 

 

articles

 

 

question and answers(Math Overflow)

 

 

blogs

 

 

experts on the field

 

 

links

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