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minimal_models [2021/09/17 18:55]
Brian McPeak [Construction] 		minimal_models [2021/09/17 21:26] (current)
Brian McPeak [Example: Critical Ising Model]
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 We see that both $r_1 + r_2$, and $r_1' + r_2'$ are now given constraints on upper range for $r_3$ (and the same for $s$).  We see that both $r_1 + r_2$, and $r_1' + r_2'$ are now given constraints on upper range for $r_3$ (and the same for $s$). 
  
-Now, for a given central charge, which defines $p$ and $q$, there is an upper bound in the possible values of $r_3$ and $s_3. So only a finite number of representations can possibly be generated by fusion. At a given central charge, the set allowed representations are called **Kac table**. +Now, for a given central charge, which defines $p$ and $q$, there is an upper bound in the possible values of $r_3$ and $s_3. So only a finite number of representations can possibly be generated by fusion. At a given central charge, the set allowed representations are called **Kac table**.  See below for an example.
  
-**Example**: consider $c = 1 / 2$, which implies $p = 4, \, q = 3$. The Kac table of doubly degenerate representations is given by+==Unitarity== 
 + 
 +The minimal models we've just constructed are only unitarity when $q = p+ 1$. Non-unitarity models are consistent but cannot be given a quantum-mechanical interpretation. Nonetheless, they are interesting objects to study and often have applications in statistical systems. 
 + 
 +(I'll have to give a thorough discussion of unitarity later) 
 + 
 +==== Example: Critical Ising Model==== 
 + 
 +Consider $c = 1 / 2$, which implies $p = 4, \, q = 3$. The Kac table of doubly degenerate representations is given by
  
 $$ \begin{array}{c}\begin{array}{c|ccc}  & r = 1 & r = 2 & r = 3  \\ \hline  s =  1 & \Delta =  0 & \Delta = \frac{1}{16} & \Delta = \frac{1}{2}  \\ s = 2 & \Delta =  \frac{1}{2} & \Delta =  \frac{1}{16} & \Delta =  0 \end{array} \end{array} $$ $$ \begin{array}{c}\begin{array}{c|ccc}  & r = 1 & r = 2 & r = 3  \\ \hline  s =  1 & \Delta =  0 & \Delta = \frac{1}{16} & \Delta = \frac{1}{2}  \\ s = 2 & \Delta =  \frac{1}{2} & \Delta =  \frac{1}{16} & \Delta =  0 \end{array} \end{array} $$
  
-This example is the 2d critical Ising model, as we will see in more detail below.+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. $\Delta_{\langle 1, 1\rangle} = \Delta_{\langle 3,2 \rangle}$. 
  
-==Unitarity==+==Correlation Functions==
  
-The minimal models we've just constructed are only unitarity when $q = p+ 1$. Non-unitarity models are consistent but cannot be given a quantum-mechanical interpretation. Nonetheless, they are interesting objects to study and often have applications in statistical systems.+==Derivation using BPZ equation==
  
-(I'll have to give a thorough discussion of unitarity later)+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 
  
-=====Related Theories=====+$$\mathcal{R} = \mathcal{R}_{2,1} = \mathcal{R}_{2,2} \, .$$ 
 + 
 +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==== 
 + 
 +====Related Theories====
  
   * [[Generalized Minimal Models]]   * [[Generalized Minimal Models]]
   * [[Runkel-Watts theory]]   * [[Runkel-Watts theory]]
 +
 +====Sources====
 +
 +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 18:55
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