"Non-unitary c(2,2k+1) minimal models"의 두 판 사이의 차이

수학노트
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<h5>introduction</h5>
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==introduction==
  
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<h5 style="background-position: 0px 100%; font-size: 1.16em; margin: 0px; color: rgb(34, 61, 103); line-height: 3.42em; font-family: 'malgun gothic',dotum,gulim,sans-serif;">central charge and conformal dimensions</h5>
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==central charge and conformal dimensions==
  
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*  central charge<br><math>c(2,2k+1)=1-\frac{3(2k-1)^2}{2k+1}</math><br>
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*  central charge<math>c(2,2k+1)=1-\frac{3(2k-1)^2}{2k+1}</math>
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*  primary fields have conformal dimensions<br><math>h_j=-\frac{j(2k-1-j)}{2(2k+1)}</math>, <math>j\in \{0,1,\cdots,k-1\}</math> or by setting i=j+1<br><math>h_i=-\frac{(i-1)(2k-i)}{2(2k+1)}</math> <math>i\in \{1,2, \cdots,k\}</math> (this is An's notation in his paper)<br>
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*  primary fields have conformal dimensions<math>h_j=-\frac{j(2k-1-j)}{2(2k+1)}</math>, <math>j\in \{0,1,\cdots,k-1\}</math> or by setting i=j+1<math>h_i=-\frac{(i-1)(2k-i)}{2(2k+1)}</math> <math>i\in \{1,2, \cdots,k\}</math> (this is An's notation in his paper)
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*  effective central charge<br><math>c_{eff}=c-24h_{min}</math><br><math>c_{eff}=\frac{2k-2}{2k+1}</math><br>
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*  effective central charge<math>c_{eff}=c-24h_{min}</math><math>c_{eff}=\frac{2k-2}{2k+1}</math>
  
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+
  
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+
  
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<h5>character formula and Andrew-Gordon identity</h5>
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==character formula and Andrew-Gordon identity==
  
 
* [[Andrews-Gordon identity]]
 
* [[Andrews-Gordon identity]]
 +
*  character functions<math>\chi_i(\tau)=q^{h_i-c/24}\prod_{n\neq 0,\pm i\pmod {2k+1}}(1-q^n)^{-1}</math>
 +
*  to understand the factor <math>q^{h-c/24}</math>, look at the [[finite size effect]] page also
 +
* [[bosonic characters of Virasoro minimal models(Rocha-Caridi formula)]]<math>\chi_{r,s}^{(p,p')}=\frac{q^{\Delta_{r,s}^{(p,p')}}}{(q)_{\infty}}\sum_{n=-\infty}^{\infty}(q^{pp'n^2+(rp'-sp)n}-q^{(pn+r)(p'n+s)})</math><math>\Delta_{r,s}^{(p,p')}=h_{r,s}^{(p,p')}-\frac{c}{24}</math>
  
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*  character functions<br><math>\chi_i(\tau)=q^{h_i-c/24}\prod_{n\neq 0,\pm i\pmod {2k+1}}(1-q^n)^{-1}</math><br>
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Let's specify p=2, p'=2k+1, r=1, s=i
−
*  to understand the factor <math>q^{h-c/24}</math>, look at the [[finite size effect]] page also<br>
 
−
* [[bosonic characters of Virasoro minimal models(Rocha-Caridi formula)]]<br>
 
  
−
 
+
<math>\sum_{n=-\infty}^{\infty}(q^{2(2k+1)n^2+(2k+1-2i)n}-q^{(2n+1)((2k+1)n+i)})=\sum_{n=-\infty}^{\infty}(q^{2(2k+1)n^2+(2k+1-2i)n}-\sum_{n=-\infty}^{\infty}q^{(2n+1)((2k+1)n+i)})</math>
  
−
 
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<math>=\sum_{n=-\infty}^{\infty}(q^{2(2k+1)n^2+(2k+1-2i)n}-\sum_{n=-\infty}^{\infty}q^{(2(-n)+1)(-(2k+1)(-n)+i)})</math>
  
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<h5>different expressions for central charge</h5>
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<math>=\sum_{n=-\infty}^{\infty}(q^{2n\left[(2k+1)(2n)+(2k+1-2i)\right]/2}-\sum_{n=-\infty}^{\infty}q^{(2n-1)\left[(2k+1)(2n-1)+2k-2i+1\right]/2}</math>
  
−
*  from above<br><math>h_i-c(2,2k+1)/24</math><br><math>c(2,2k+1)=1-\frac{3(2k-1)^2}{2k+1}</math><br><math>h_i=-\frac{(i-1)(2k-i)}{2(2k+1)}</math>, <math>i\in \{1,2, \cdots,k\}</math><br>
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<math>=\sum_{n=-\infty}^{\infty}(-1)^{n}q^{\frac{(2k+1)n^2}{2}}q^{\frac{n(2k-2i+1)}{2}}</math>
−
*  L-values<br><math>\frac{k}{12}+\frac{2k+1}{12}-\frac{j(N-j)}{2N}</math><br>
 
  
−
 
+
<math>=\sum_{n=-\infty}^{\infty}(-1)^n(q^{\frac{(2k-2i+1)}{2}})^{n}(q^{\frac{(2k+1)}{2}})^{n^2}=\prod_{m=1}^{\infty}(1-q^{(2k+1)m})(1-q^{\frac{2k-2i+1}{2}}q^{\frac{2k+1}{2}(2m-1)})(1-q^{-\frac{2k-2i+1}{2}}q^{\frac{2k+1}{2}(2m-1)})</math>
  
−
 
+
<math>=\prod_{m=1}^{\infty}(1-q^{(2k+1)m})(1-q^{(2k+1)m-i})(1-q^{(2k+1)m-(2k-i+1)})</math>
  
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<h5>Dirichlet L-function</h5>
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Thus,
  
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* [http://pythagoras0.springnote.com/pages/4562847 디리클레 L-함수]
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<math>\chi_{r,s}^{(p,p')}=q^{\Delta_{r,s}^{(p,p')}}\prod_{n\neq 0,\pm i\pmod {2k+1}}(1-q^n)^{-1}</math>
  
−
<math>L(-1, \chi) = \frac{1}{2f}\sum_{n=1}^{\infty}{\chi(n)}{n}</math>
+
Using a result from [http://pythagoras0.springnote.com/pages/9409120 베일리 격자(Bailey lattice)]
  
−
<math>n\geq 1</math> 이라 하자. 일반적으로 <math>\chi\neq 1</math>인 primitive 준동형사상 <math>\chi \colon(\mathbb{Z}/f\mathbb{Z})^\times \to \mathbb C^{*}</math>에 대하여 <math>L(1-n,\chi)</math>의 값은 다음과 같이 주어진다
+
<math>\sum_{n_1\geq\cdots\geq n_{k-1}\geq0}\frac{q^{n_1^2+\cdots+n_{k-1}^2+n_i+\cdots+n_{k-1}}}{(q)_{n_{1}-n_{2}}\cdots (q)_{n_{k-2}-n_{k-1}}(q)_{n_{k-1}}}=\sum_{n=-\infty}^{\infty}(-1)^{n}q^{\frac{(2k+1)n^2}{2}}q^{\frac{n(2k-2i+1)}{2}}=\prod_{n\neq 0,\pm i\pmod {2k+1}}(1-q^n)^{-1}</math>
  
−
<math>L(1-n,\chi)=-\frac{f^{n-1}}{n}\sum_{(a,f)=1}}\chi(a)B_n(\frac{a}{f})</math>
+
  
−
<math>L(-1,\chi)=L(1-2,\chi)=-\frac{f}{2}\sum_{(a,f)=1}}\chi(a)B_2(\frac{a}{f})</math>
+
  
−
여기서 <math>B_n(x)</math> 는 [http://pythagoras0.springnote.com/pages/4346717 베르누이 다항식](<math>B_0(x)=1</math>, <math>B_1(x)=x-1/2</math>, <math>B_2(x)=x^2-x+1/6</math>, <math>\cdots</math>)
+
==asymptotic analysis of Andrews-Gordon identity==
 +
*  non-unitary <math>c(2,k+2)</math> [[minimal models]]
 +
:<math>\sum_{i=1}^{k-1}L(\frac{\sin^2\frac{\pi}{k+2}}{\sin^2\frac{(i+1)\pi}{k+2}})=\frac{2(k-1)}{k+2}\cdot \frac{\pi^2}{6}</math>
 +
*  non-unitary <math>c(2,2k+1)</math> [[minimal models]]
 +
:<math>
 +
\sum_{i=1}^{2k-2}L\left(\frac{\sin^2\frac{\pi}{2k+1}}{\sin^2\frac{(i+1)\pi}{2k+1}}\right)=\frac{2(2k-2)}{2k+1}\cdot \frac{\pi^2}{6}
 +
</math>
 +
:<math>
 +
\sum_{i=1}^{k-1}L\left(\frac{\sin^2\frac{\pi}{2k+1}}{\sin^2\frac{(i+1)\pi}{2k+1}}\right)=\frac{2k-2}{2k+1}\cdot \frac{\pi^2}{6}
 +
</math>
  
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+
  
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Let N=2k+1
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==different expressions for central charge==
  
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<math>\omega=\exp \frac{2\pi i}{2k+1}</math>
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*  from above<math>h_i-c(2,2k+1)/24</math><math>c(2,2k+1)=1-\frac{3(2k-1)^2}{2k+1}</math><math>h_i=-\frac{(i-1)(2k-i)}{2(2k+1)}</math>, <math>i\in \{1,2, \cdots,k\}</math>
 +
*  L-values<math>\frac{k}{12}+\frac{2k+1}{12}-\frac{j(N-j)}{2N}</math>
  
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G: group of Dirichlet characters of conductor N which maps -1 to 1
+
  
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G has order k and cyclic generated by <math>\chi</math>
+
  
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<math>c_i=\frac{1}{2k}\sum_{s=1}^{k}\omega^{is}L(-1,\chi^s)</math>
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==Dirichlet L-function==
  
−
Then, 
+
* [http://pythagoras0.springnote.com/pages/4562847 디리클레 L-함수]
  
−
<math>c_i=-\frac{2k+1}{12}+\frac{j(N-j)}{2N}</math>
+
<math>L(-1, \chi) = \frac{1}{2f}\sum_{n=1}^{\infty}{\chi(n)}{n}</math>
  
−
where j satisfies <math>\chi(j)=\omega^{k-i}</math>
+
<math>n\geq 1</math> 이라 하자. 일반적으로 <math>\chi\neq 1</math>인 primitive 준동형사상 <math>\chi \colon(\mathbb{Z}/f\mathbb{Z})^\times \to \mathbb C^{*}</math>에 대하여 <math>L(1-n,\chi)</math>의 값은 다음과 같이 주어진다
 +
:<math>L(1-n,\chi)=-\frac{f^{n-1}}{n}\sum_{(a,f)=1}\chi(a)B_n(\frac{a}{f})</math>
 +
:<math>L(-1,\chi)=L(1-2,\chi)=-\frac{f}{2}\sum_{(a,f)=1}\chi(a)B_2(\frac{a}{f})</math>
  
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Vacuum energy is given by
+
여기서 <math>B_n(x)</math> 는 [http://pythagoras0.springnote.com/pages/4346717 베르누이 다항식](<math>B_0(x)=1</math>, <math>B_1(x)=x-1/2</math>, <math>B_2(x)=x^2-x+1/6</math>, <math>\cdots</math>)
  
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<math>d_i=\frac{1}{2}L(-1,\chi^{k})-c_i</math>
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−
 
+
Let N=2k+1
  
−
Since
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<math>\omega=\exp \frac{2\pi i}{2k+1}</math>
  
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<math>L(-1,\chi^{k})=\frac{N-1}{12}=\frac{k}{6}</math>,  the vacuum energy 
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G: group of Dirichlet characters of conductor N which maps -1 to 1
  
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<math>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}</math>.
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G has order k and cyclic generated by <math>\chi</math>
  
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These are equal to <math>{h_i-c/24}</math>
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<math>c_i=\frac{1}{2k}\sum_{s=1}^{k}\omega^{is}L(-1,\chi^s)</math>
  
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+
Then,
  
−
 
+
<math>c_i=-\frac{2k+1}{12}+\frac{j(N-j)}{2N}</math>
  
−
# k := 5<br> f[k_, j_] := (2 k)/<br>    24 + ((2 k + 1)/12 - (j (2 k + 1 - j))/(2 (2 k + 1)))<br> Table[{j, f[k, j]}, {j, 1, 2 k}] // TableForm<br> Table[{j, -24*f[k, j]}, {j, 1, 2 k}] // TableForm<br> d[k_, j_] := (2 (k - j) + 1)^2/(8 (2 k + 1)) - 1/24<br> Table[{j, d[k, j]}, {j, 1, 2 k}] // TableForm<br> Table[{j, -24*d[k, j]}, {j, 1, 2 k}] // TableForm<br> cef[k_, j_] := -((j (2 k - 1 - j))/(2 (2 k +<br>          1))) - (1 - (3 (2 k - 1)^2)/(2 k + 1))/24<br> Table[{j, cef[k, j]}, {j, 0, 2 k - 1}] // TableForm<br> Table[{j, -24*cef[k, j]}, {j, 0, 2 k - 1}] // TableForm
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where j satisfies <math>\chi(j)=\omega^{k-i}</math>
  
−
 
+
Vacuum energy is given by
  
−
 
+
<math>d_i=\frac{1}{2}L(-1,\chi^{k})-c_i</math>
  
−
# w := Exp[2 Pi*I*1/k]<br> L[j_] := -(2 k + 1)/2*<br>   Sum[DirichletCharacter[2 k + 1, j, a]*<br>     BernoulliB[2, a/(2 k + 1)], {a, 1, 2 k}]<br> c[k_, i_] := 1/(2 k) Sum[w^(i*s)*L[Mod[3*s, 2 k]], {s, 1, k}]<br> Table[DiscretePlot[{Re[DirichletCharacter[2 k + 1, j, a]],<br>    Im[DirichletCharacter[2 k + 1, j, a]]}, {a, 0, 2 k + 1},<br>   PlotLabel -> j], {j, 1, EulerPhi[2 k + 1]}]<br> Table[c[i], {i, 1, 2 k}]
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−
 
+
Since
  
−
 
+
<math>L(-1,\chi^{k})=\frac{N-1}{12}=\frac{k}{6}</math>,  the vacuum energy
  
−
 
+
<math>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}</math>.
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<h5>history</h5>
 
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* http://www.google.com/search?hl=en&tbs=tl:1&q=
 
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<h5>related items</h5>
 
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<h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">encyclopedia</h5>
 
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* http://en.wikipedia.org/wiki/
 
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* http://www.scholarpedia.org/
 
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* http://www.proofwiki.org/wiki/
 
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* Princeton companion to mathematics([[2910610/attachments/2250873|Companion_to_Mathematics.pdf]])
 
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<h5>books</h5>
 
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* [[2010년 books and articles]]<br>
 
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* http://gigapedia.info/1/
 
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* http://gigapedia.info/1/
 
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* http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=
 
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<h5>expositions</h5>
 
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<h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">articles</h5>
 
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* http://www.ams.org/mathscinet
 
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* http://www.zentralblatt-math.org/zmath/en/
 
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* http://arxiv.org/
 
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* http://www.pdf-search.org/
 
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* http://pythagoras0.springnote.com/
 
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* [http://math.berkeley.edu/%7Ereb/papers/index.html http://math.berkeley.edu/~reb/papers/index.html]
 
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* http://dx.doi.org/
 
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<h5>question and answers(Math Overflow)</h5>
 
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* http://mathoverflow.net/search?q=
 
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* http://mathoverflow.net/search?q=
 
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<h5>blogs</h5>
 
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*  구글 블로그 검색<br>
 
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**  http://blogsearch.google.com/blogsearch?q=<br>
 
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** http://blogsearch.google.com/blogsearch?q=
 
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* http://ncatlab.org/nlab/show/HomePage
 
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These are equal to <math>{h_i-c/24}</math>
  
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<h5>experts on the field</h5>
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* http://arxiv.org/
+
  
−
 
+
# 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
  
−
 
+
  
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<h5>links</h5>
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−
* [http://detexify.kirelabs.org/classify.html Detexify2 - LaTeX symbol classifier]
+
# 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}]
−
* [http://pythagoras0.springnote.com/pages/1947378 수식표현 안내]
+
[[분류:conformal field theory]]
−
* [http://www.research.att.com/%7Enjas/sequences/index.html The On-Line Encyclopedia of Integer Sequences]
+
[[분류:math and physics]]
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* http://functions.wolfram.com/
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[[분류:minimal models]]
 +
[[분류:migrate]]

2020년 12월 28일 (월) 05:24 기준 최신판

introduction

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

Let's specify p=2, p'=2k+1, r=1, s=i

\(\sum_{n=-\infty}^{\infty}(q^{2(2k+1)n^2+(2k+1-2i)n}-q^{(2n+1)((2k+1)n+i)})=\sum_{n=-\infty}^{\infty}(q^{2(2k+1)n^2+(2k+1-2i)n}-\sum_{n=-\infty}^{\infty}q^{(2n+1)((2k+1)n+i)})\)

\(=\sum_{n=-\infty}^{\infty}(q^{2(2k+1)n^2+(2k+1-2i)n}-\sum_{n=-\infty}^{\infty}q^{(2(-n)+1)(-(2k+1)(-n)+i)})\)

\(=\sum_{n=-\infty}^{\infty}(q^{2n\left[(2k+1)(2n)+(2k+1-2i)\right]/2}-\sum_{n=-\infty}^{\infty}q^{(2n-1)\left[(2k+1)(2n-1)+2k-2i+1\right]/2}\)

\(=\sum_{n=-\infty}^{\infty}(-1)^{n}q^{\frac{(2k+1)n^2}{2}}q^{\frac{n(2k-2i+1)}{2}}\)

\(=\sum_{n=-\infty}^{\infty}(-1)^n(q^{\frac{(2k-2i+1)}{2}})^{n}(q^{\frac{(2k+1)}{2}})^{n^2}=\prod_{m=1}^{\infty}(1-q^{(2k+1)m})(1-q^{\frac{2k-2i+1}{2}}q^{\frac{2k+1}{2}(2m-1)})(1-q^{-\frac{2k-2i+1}{2}}q^{\frac{2k+1}{2}(2m-1)})\)

\(=\prod_{m=1}^{\infty}(1-q^{(2k+1)m})(1-q^{(2k+1)m-i})(1-q^{(2k+1)m-(2k-i+1)})\)

Thus,

\(\chi_{r,s}^{(p,p')}=q^{\Delta_{r,s}^{(p,p')}}\prod_{n\neq 0,\pm i\pmod {2k+1}}(1-q^n)^{-1}\)

Using a result from 베일리 격자(Bailey lattice)

\(\sum_{n_1\geq\cdots\geq n_{k-1}\geq0}\frac{q^{n_1^2+\cdots+n_{k-1}^2+n_i+\cdots+n_{k-1}}}{(q)_{n_{1}-n_{2}}\cdots (q)_{n_{k-2}-n_{k-1}}(q)_{n_{k-1}}}=\sum_{n=-\infty}^{\infty}(-1)^{n}q^{\frac{(2k+1)n^2}{2}}q^{\frac{n(2k-2i+1)}{2}}=\prod_{n\neq 0,\pm i\pmod {2k+1}}(1-q^n)^{-1}\)



asymptotic analysis of Andrews-Gordon identity

\[\sum_{i=1}^{k-1}L(\frac{\sin^2\frac{\pi}{k+2}}{\sin^2\frac{(i+1)\pi}{k+2}})=\frac{2(k-1)}{k+2}\cdot \frac{\pi^2}{6}\]

\[ \sum_{i=1}^{2k-2}L\left(\frac{\sin^2\frac{\pi}{2k+1}}{\sin^2\frac{(i+1)\pi}{2k+1}}\right)=\frac{2(2k-2)}{2k+1}\cdot \frac{\pi^2}{6} \] \[ \sum_{i=1}^{k-1}L\left(\frac{\sin^2\frac{\pi}{2k+1}}{\sin^2\frac{(i+1)\pi}{2k+1}}\right)=\frac{2k-2}{2k+1}\cdot \frac{\pi^2}{6} \]


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

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

\(n\geq 1\) 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}]
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