Non-unitary c(2,2k+1) minimal models

 

 

 

central charge and conformal dimensions

• central charge
  \(c(2,2k+1)=1-\frac{3(2k-1)^2}{2k+1}\)
  
• primary fields have conformal dimensions
  \(h_j=-\frac{j(2k-1-j)}{2(2k+1)}\), \(j\in \{0,1,\cdots,k-1\}\) or by setting i=j+1
  \(h_i=-\frac{(i-1)(2k-i)}{2(2k+1)}\) \(i\in \{1,2, \cdots,k\}\) (this is An's notation in his paper)
  
• effective central charge
  \(c_{eff}=c-24h_{min}\)
  \(c_{eff}=\frac{2k-2}{2k+1}\)
  

 

 

character formula and Andrew-Gordon identity

• Andrews-Gordon identity

• character functions
  \(\chi_i(\tau)=q^{h_i-c/24}\prod_{n\neq 0,\pm i\pmod {2k+1}}(1-q^n)^{-1}\)
  
• to understand the factor \(q^{h-c/24}\), look at the finite size effect page also
  
• bosonic characters of Virasoro minimal models(Rocha-Caridi formula)
  

 

 

different expressions for central charge

• from above
  \(h_i-c(2,2k+1)/24\)
  \(c(2,2k+1)=1-\frac{3(2k-1)^2}{2k+1}\)
  \(h_i=-\frac{(i-1)(2k-i)}{2(2k+1)}\), \(i\in \{1,2, \cdots,k\}\)
  
• L-values
  \(\frac{k}{12}+\frac{2k+1}{12}-\frac{j(N-j)}{2N}\)
  

 

 

Dirichlet L-function

• 디리클레 L-함수

\(L(-1, \chi) = \frac{1}{2f}\sum_{n=1}^{\infty}{\chi(n)}{n}\)

\(n\geq 1\) 이라 하자. 일반적으로 \(\chi\neq 1\)인 primitive 준동형사상 \(\chi \colon(\mathbb{Z}/f\mathbb{Z})^\times \to \mathbb C^{*}\)에 대하여 \(L(1-n,\chi)\)의 값은 다음과 같이 주어진다

\(L(1-n,\chi)=-\frac{f^{n-1}}{n}\sum_{(a,f)=1}}\chi(a)B_n(\frac{a}{f})\)

\(L(-1,\chi)=L(1-2,\chi)=-\frac{f}{2}\sum_{(a,f)=1}}\chi(a)B_2(\frac{a}{f})\)

여기서 \(B_n(x)\) 는 베르누이 다항식(\(B_0(x)=1\), \(B_1(x)=x-1/2\), \(B_2(x)=x^2-x+1/6\), \(\cdots\))

 

Let N=2k+1

\(\omega=\exp \frac{2\pi i}{2k+1}\)

G: group of Dirichlet characters of conductor N which maps -1 to 1

G has order k and cyclic generated by \(\chi\)

\(c_i=\frac{1}{2k}\sum_{s=1}^{k}\omega^{is}L(-1,\chi^s)\)

Then, 

\(c_i=-\frac{2k+1}{12}+\frac{j(N-j)}{2N}\)

where j satisfies \(\chi(j)=\omega^{k-i}\)

Vacuum energy is given by

\(d_i=\frac{1}{2}L(-1,\chi^{k})-c_i\)

 

Since

\(L(-1,\chi^{k})=\frac{N-1}{12}=\frac{k}{6}\),  the vacuum energy 

\(d_i=\frac{1}{2}L(-1,\chi^{k})-c_i=\frac{k}{12}+\frac{2k+1}{12}-\frac{j(N-j)}{2N}=\frac{j(2k+1-j)}{2(2k+1)}-\frac{k+1}{12}\).

These are equal to \({h_i-c/24}\)

 

 

1. k := 5
   f[k_, j_] := (2 k)/
      24 + ((2 k + 1)/12 - (j (2 k + 1 - j))/(2 (2 k + 1)))
   Table[{j, f[k, j]}, {j, 1, 2 k}] // TableForm
   Table[{j, -24*f[k, j]}, {j, 1, 2 k}] // TableForm
   d[k_, j_] := (2 (k - j) + 1)^2/(8 (2 k + 1)) - 1/24
   Table[{j, d[k, j]}, {j, 1, 2 k}] // TableForm
   Table[{j, -24*d[k, j]}, {j, 1, 2 k}] // TableForm
   cef[k_, j_] := -((j (2 k - 1 - j))/(2 (2 k +
            1))) - (1 - (3 (2 k - 1)^2)/(2 k + 1))/24
   Table[{j, cef[k, j]}, {j, 0, 2 k - 1}] // TableForm
   Table[{j, -24*cef[k, j]}, {j, 0, 2 k - 1}] // TableForm

 

 

1. w := Exp[2 Pi*I*1/k]
   L[j_] := -(2 k + 1)/2*
     Sum[DirichletCharacter[2 k + 1, j, a]*
       BernoulliB[2, a/(2 k + 1)], {a, 1, 2 k}]
   c[k_, i_] := 1/(2 k) Sum[w^(i*s)*L[Mod[3*s, 2 k]], {s, 1, k}]
   Table[DiscretePlot[{Re[DirichletCharacter[2 k + 1, j, a]],
      Im[DirichletCharacter[2 k + 1, j, a]]}, {a, 0, 2 k + 1},
     PlotLabel -> j], {j, 1, EulerPhi[2 k + 1]}]
   Table[c[i], {i, 1, 2 k}]

 

 

 

history

• http://www.google.com/search?hl=en&tbs=tl:1&q=

 

 

 

 

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

• 구글 블로그 검색
  
  • http://blogsearch.google.com/blogsearch?q=
    
  • http://blogsearch.google.com/blogsearch?q=
• 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/