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

 

 

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
  \(\chi_j(\tau)=q^{h_j-c/24}\prod_{n\neq 0,\pm(j+1)}(1-q^n)^{-1}\)
  
• 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
   

 

 

 

history

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related items

• Andrews-Gordon identity

 

 

encyclopedia

• http://en.wikipedia.org/wiki/
• http://www.scholarpedia.org/
• http://www.proofwiki.org/wiki/
• Princeton companion to mathematics(Companion_to_Mathematics.pdf)

 

 

books

 

• 2010년 books and articles
  
• http://gigapedia.info/1/
• http://gigapedia.info/1/
• http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=

 

 

 

 

 

articles

 

• http://www.ams.org/mathscinet
• http://www.zentralblatt-math.org/zmath/en/
• http://arxiv.org/
• http://www.pdf-search.org/
• http://pythagoras0.springnote.com/
• http://math.berkeley.edu/~reb/papers/index.html
• http://dx.doi.org/

 

 

question and answers(Math Overflow)

• http://mathoverflow.net/search?q=
• http://mathoverflow.net/search?q=

 

 

blogs

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• http://ncatlab.org/nlab/show/HomePage

 

 

experts on the field

• http://arxiv.org/

 

 

links

• Detexify2 - LaTeX symbol classifier
• 수식표현 안내
• The On-Line Encyclopedia of Integer Sequences
• http://functions.wolfram.com/