Non-unitary c(2,k+2) minimal models

introduction

• important

 

non-unitary \(c(2,k+2)\)'minimal models'

• central charge
  \(c(2,k+2)=1-\frac{3k^2}{k+2}\)
  \(k \geq 3\), odd
  
• primary fields have conformal dimensions
  \(h_j=-\frac{j(k-j)}{2(k+2)}\), \(j\in \{0,1,\cdots,[k/2]\}\)
  
• effective central charge
  \(c_{eff}=\frac{k-1}{k+2}\)
  
• dilogarithm identity
  \(\sum_{i=1}^{[k/2]}L(\frac{\sin^2\frac{\pi}{k+2}}{\sin^2\frac{(i+1)\pi}{k+2}})=\frac{k-1}{k+2}\cdot \frac{\pi^2}{6}\)
  
• character functions
  \(\chi_j(\tau)=q^{h_j-c/24}\prod_{n\neq 0,\pm(j+1)}(1-q^n)^{-1}\)
  
• to understand the factor \(q^{h-c/24}\), look at the finite size effect page also
  
• quantum dimension and there recurrence relation
  \(d_i=\frac{\sin \frac{(i+1)\pi}{k+2}}{\sin \frac{\pi}{k+2}}\) satisfies
  \(d_i^2=1+d_{i-1}d_{i+1}\) where \(d_0=1\), \(d_k=1\)
  

 

1. (*choose k for c (2,k+2) minimal model*)k := 11
   (*define Rogers dilogarithm*)
   L[x_] := PolyLog[2, x] + 1/2 Log[x] Log[1 - x]
   (*quantum dimension for minimal models*)
   f[k_, i_] := (Sin[Pi/(k + 2)]/Sin[(i + 1) Pi/(k + 2)])^2
   (*effective central charge*)
   g[k_] := (k*Pi^2)/(2 (k + 2))
   (*compare the results*)
   N[Sum[L[f[k, i]], {i, 1, k - 1}] + Pi^2/6, 10]
   N[g[k], 10]
   d[k_, i_] := Sin[(i + 1) Pi/(k + 2)]/Sin[Pi/(k + 2)]
   Table[{i, d[k, i]}, {i, 1, k}] // TableForm
   Table[{i, N[(d[k, i])^2 - (1 + d[k, i - 1]*d[k, i + 1]), 10]}, {i, 1,
      k}] // TableForm
   

 

related items

• Andrews-Gordon identity

 

computational resource

• https://docs.google.com/file/d/0B8XXo8Tve1cxdXRlbU40OExkeW8/edit