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minimal_models [2021/09/17 20:32]
Brian McPeak [Example: Critical Ising Model] 		minimal_models [2021/09/17 21:26] (current)
Brian McPeak [Example: Critical Ising Model]
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 $$ \begin{array}{c}\begin{array}{c|ccc}  & r = 1 & r = 2 & r = 3  \\ \hline  s =  1 & \Delta =  0 & \Delta = \frac{1}{16} & \Delta = \frac{1}{2}  \\ s = 2 & \Delta =  \frac{1}{2} & \Delta =  \frac{1}{16} & \Delta =  0 \end{array} \end{array} $$ $$ \begin{array}{c}\begin{array}{c|ccc}  & r = 1 & r = 2 & r = 3  \\ \hline  s =  1 & \Delta =  0 & \Delta = \frac{1}{16} & \Delta = \frac{1}{2}  \\ s = 2 & \Delta =  \frac{1}{2} & \Delta =  \frac{1}{16} & \Delta =  0 \end{array} \end{array} $$
  
-This example is the 2d critical Ising model, as we will see in more detail below. There are only three unique representations, as e.g. $\Delta_{\langle 1, 1\rangle} = \Delta_{\langle 3,2 \rangle}$.+This example is the 2d critical Ising model, as we will see in more detail below. There are only three unique representations, as e.g. $\Delta_{\langle 1, 1\rangle} = \Delta_{\langle 3,2 \rangle}$. 
  
 ==Correlation Functions== ==Correlation Functions==
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 $$\mathcal{R} = \mathcal{R}_{2,1} = \mathcal{R}_{2,2} \, .$$ $$\mathcal{R} = \mathcal{R}_{2,1} = \mathcal{R}_{2,2} \, .$$
  
-Let's determine the level-two descendent which vanishes. This must take the general form+Let's determine the level-two descendent which vanishes. This is the same as the primary of the  subrepresentation we mod out by, defined by 
 + 
 +$$ | \chi_{2,1} \rangle =  L_{2,1} | \Delta_{2,1} \rangle = (a L_{-1}^2 + L_{-2} ) | \Delta_{2,1} \rangle \, .$$ 
 + 
 +This will be a primary if $L_1  | \chi_{2,1} \rangle = 0$ and if $L_2  | \chi_{2,1} \rangle = 0$.  Using the $c = 1/2$ Virasoro algebra, and solving these constraints, we find 
 + 
 +$$ L_{2,1} =  -\frac{4}{3} L_{-1}^2 + L_{-2}  \, .$$ 
 + 
 +Therefore the degenerate field $\sigma$ must satisfy 
 + 
 +$$ L_{2,1} \sigma(z) = 0 \, .$$ 
 + 
 +This equation will allow us to compute the four-point function 
 + 
 +(To Be Finished Later) 
  
-$$ (a L_{-1}^2 + L_{-2} ) | \chi_{2,1} \rangle = 0 $$ 
 ====More Examples==== ====More Examples====
  
Last modified: 2021/09/17 20:32
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