"Affine sl(2)"의 두 판 사이의 차이

2010년 9월 26일 (일) 01:15 판

Gannon 190p, 193p, 196p,371p

$$A^{(1)}_1$$

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 $\Phi$
- $\alpha_0=-\alpha$
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
$\begin{pmatrix} 2 & -2 & 1\\ -2 & 2 &0 \\ 1 &0 & 0 \end{pmatrix}$

$\Phi=\{\alpha+n\delta|\alpha\in\Phi^{0},n\in\mathbb{Z}\}\cup \{n\delta|n\in\mathbb{Z},n\neq 0\}$
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
- $\Phi^{+}=\{\alpha+n\delta|\alpha\in\Phi^{0},n>0\}\cup (\Phi^{0})^{+}\cup \{n\delta|n\in\mathbb{Z},n> 0\}$

invariant symmetric non-deg bilinear forms
$<h_i,h_j>=A_{ij}$
$<h_0,d>=1$
$<h_1,d>=0$
$<d,d>=0$

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

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
$m_{\Lambda}=\frac{(\Lambda+\rho)^2}{2(k+h^{\vee})}-\frac{\rho^2}{2h^{\vee}}=h_{\lambda}-\frac{c(\lambda)}{24}$

@@ 90번째 줄: / 90번째 줄: @@
 <h5>central charge</h5>
-*  central charge<br><math>c_{\lambda}=\frac{k}{k+h^{\vee}}\text{dim }\mathfrak{\bar{g}}</math><br>
+*  central charge (depends on the level only)<br><math>c_{\lambda}=\frac{k}{k+h^{\vee}}\text{dim }\mathfrak{\bar{g}}</math><br>
 *  conformal weight<br><math>h_{\lambda}=\frac{(\lambda|\lambda+2\rho)}{2(k+h^{\vee})}</math><br>
 *  definition of conformal anomaly<br><math>m_{\Lambda}=\frac{(\Lambda+\rho)^2}{2(k+h^{\vee})}-\frac{\rho^2}{2h^{\vee}}</math><br>