CFT Zoo

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minimal_models [2021/09/17 19:15]
Brian McPeak 		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==
 +
 +==Derivation using BPZ equation==
 +
 +Primaries in degenerate representations have vanishing descendents, by definition. The $\sigma$ operator in the Ising model is the field corresponding to the doubly degenerate representation 
 +
 +$$\mathcal{R} = \mathcal{R}_{2,1} = \mathcal{R}_{2,2} \, .$$
 +
 +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)
 +
  
 ====More Examples==== ====More Examples====
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 Like most of the 2d CFT content, this page follows the excellent notes of Sylvain Ribault. The most complete form of those notes is given [[ https://arxiv.org/abs/1406.4290 | here]] Like most of the 2d CFT content, this page follows the excellent notes of Sylvain Ribault. The most complete form of those notes is given [[ https://arxiv.org/abs/1406.4290 | here]]
 +
 +The derivation of the 2d Ising model correlation function using the BPZ equation is given in [[https://pos.sissa.it/305/003/pdf | Yin's lecture notes]]
Last modified: 2021/09/17 19:15
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