"(3,4) Ising minimal model CFT"의 두 판 사이의 차이

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<h5>introduction</h5>
  
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<h5 style="margin: 0px; line-height: 2em;">Ising model as a minimal model</h5>
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* [[Ising models|Ising model]]<br>
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* first review the [[minimal models]] page
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*  Ising model is a unitary minimal model and thus can be understood by the representation of [[Virasoro algebra|Viraroso algebra]]<br>
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*  the representation is given by following data<br><math>m= 3</math><br> central charge <math>c = 1-{6\over m(m+1)} = \frac{1}{2}</math><br><math>h_{p,q}(c) = {(4p-3q)^2-1 \over 48}</math><br><math>p= 1,2</math>, <math>q = 1,\cdots p</math><br><math>(p,q)=(1,1), (2,1), (2,2)</math><br>
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*  possible values of <math>h</math><br><math>0, 1/2, 1/16</math><br>
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<h5 style="margin: 0px; line-height: 2em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">graded dimensions</h5>
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*  associated chiral algebra has three irreducible modules with the following graded dimensions<br><math>\chi_0=q^{-1/48}(1+q^2+q^3+2q^4+2q^5+3q^6+\cdots)</math><br><math>\chi_{\epsilon}=q^{23/48}(1+q+q^2+q^3+2q^4+2q^5+3q^6+\cdots)</math><br><math>\chi_{\sigma}=q^{1/24}(1+q+q^2+2q^3+2q^4+3q^5+4q^6+\cdots)</math><br>
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*  Rocha-Caridi character[RC84]<br>
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<h5 style="margin: 0px; line-height: 2em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">modularity of graded dimensions</h5>
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<math>\chi_M(-1/\tau)=\sum_{N} S_{M,N}\chi_N(\tau)</math>
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<math>\chi_M(\tau+1)=\sum_{N} T_{M,N}\chi_N(\tau)</math>
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<math>T=\left(\begin{array}{ccc}e^{-\pi i/24} & 0 & 0 \\ 0 & e^{23\pi i/24} & 0 \\ 0 & 0 & e^{\pi i/12}\end{array} \right)</math>
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<math>2S=\left(\begin{array}{ccc}1 & 1& \sqrt{2} \\ 1 & 1 & -\sqrt{2} \\ \sqrt{2} & -\sqrt{2} & 0\end{array} \right)</math>
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<h5 style="margin: 0px; line-height: 2em;">matching two sets of funtions</h5>
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* [http://www.research.att.com/%7Enjas/sequences/A000009 http://www.research.att.com/~njas/sequences/A000009]<br>
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<math>f(\tau)=\frac{e^{-\frac{\pi i}{24}}\eta(\frac{\tau+1}{2})}{\eta(\tau)}=q^{-1/48} \prod_{n=1}^{\infty} (1+q^{n-\frac{1}{2}})</math>
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<math>\chi_0+\chi_{\epsilon}</math>
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{1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 2, 1, 2, 2, 3, 2, 3, 3, 4}
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[http://www.research.att.com/%7Enjas/sequences/A027349 http://www.research.att.com/~njas/sequences/A027349]
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<math>f_1(\tau)=\frac{\eta(\frac{\tau}{2})}{\eta(\tau)}=q^{-1/48} \prod_{n=1}^{\infty} (1-q^{n-\frac{1}{2}})</math>
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<math>\chi_0-\chi_{\epsilon}</math>
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{1, -1, 0, -1, 1, -1, 1, -1, 2, -2, 2, -2, 3, -3, 3, -4, 5, -5, 5, -6, 7, -8, 8, -9, 11, -12, 12, -14, 16, -17, 18}
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[http://www.research.att.com/%7Enjas/sequences/A081362 http://www.research.att.com/~njas/sequences/A081362]
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<math>\chi_0=q^{-1/48}(1+q^2+q^3+2q^4+2q^5+3q^6+\cdots)</math>
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<math>\chi_{\epsilon}=q^{23/48}(1+q+q^2+q^3+2q^4+2q^5+3q^6+\cdots)</math>
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{1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 2, 1, 2, 2, 3, 2, 3, 3, 4}
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{1, -1, 0, -1, 1, -1, 1, -1, 2, -2, 2, -2, 3, -3, 3, -4, 5, -5, 5, -6, 7, -8, 8, -9, 11, -12, 12, -14, 16, -17, 18}
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sum [http://www.wolframalpha.com/input/?i=%281,+0,+0,+1,+0,+1,+0,+1,+1,+1%29%2B%281,+-1,+0,+-1,+1,+-1,+1,+-1,+2,+-2%29 http://www.wolframalpha.com/input/?i=(1,+0,+0,+1,+0,+1,+0,+1,+1,+1)%2B(1,+-1,+0,+-1,+1,+-1,+1,+-1,+2,+-2)]
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{2,-1,0,0,1,0,1,0,3,...} -> <math>q^{-1/48}}(1-1/2q^{1/2}+q^{4/2}+q^{6/2}+3q^{8/2}+\cdots)</math>
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difference [http://www.wolframalpha.com/input/?i=%281,+0,+0,+1,+0,+1,+0,+1,+1,+1%29-%281,+-1,+0,+-1,+1,+-1,+1,+-1,+2,+-2%29 http://www.wolframalpha.com/input/?i=(1,+0,+0,+1,+0,+1,+0,+1,+1,+1)-(1,+-1,+0,+-1,+1,+-1,+1,+-1,+2,+-2)]
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{0,1,0,2,-1,2,-1,2,-1,...} -> <math>\frac{1}{2}q^{-1/48}}(q^{1/2}+2q^{3/2}-q^{4/2}+2q^{5/2}-q^{6/2} +2q^{7/2} -q^{8/2}\cdots)</math>
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<math>f_2(\tau)=\sqrt{2}\frac{\eta(2\tau)}{\eta(\tau)}=\sqrt{2}q^{1/24} \prod_{n=1}^{\infty} (1+q^{n})</math>
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<math>\chi_{\sigma}=q^{1/24}(1+q+q^2+2q^3+2q^4+3q^5+4q^6+\cdots)</math>
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{1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18, 22, 27, 32, 38, 46, 54, 64, 76, 89, 104, 122, 142, 165, 192, 222, 256, 296}
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[http://www.research.att.com/%7Enjas/sequences/A000009 http://www.research.att.com/~njas/sequences/A000009]
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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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* [[determinantal identities and Airy kernel]]
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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://eom.springer.de/ http://eom.springer.de]
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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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* [[2011년 books and articles]]
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* http://library.nu/search?q=
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* http://library.nu/search?q=
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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://math.stackexchange.com/search?q=
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* http://physics.stackexchange.com/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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<h5>experts on the field</h5>
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* http://arxiv.org/
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<h5>links</h5>
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* [http://detexify.kirelabs.org/classify.html Detexify2 - LaTeX symbol classifier]
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* [http://pythagoras0.springnote.com/pages/1947378 수식표현 안내]

2011년 5월 6일 (금) 11:55 판

introduction

 

 

Ising model as a minimal model
  • first review the minimal models page
  • Ising model is a unitary minimal model and thus can be understood by the representation of Viraroso algebra
  • the representation is given by following data
    \(m= 3\)
    central charge \(c = 1-{6\over m(m+1)} = \frac{1}{2}\)
    \(h_{p,q}(c) = {(4p-3q)^2-1 \over 48}\)
    \(p= 1,2\), \(q = 1,\cdots p\)
    \((p,q)=(1,1), (2,1), (2,2)\)
  • possible values of \(h\)
    \(0, 1/2, 1/16\)

 

 

 

graded dimensions
  • associated chiral algebra has three irreducible modules with the following graded dimensions
    \(\chi_0=q^{-1/48}(1+q^2+q^3+2q^4+2q^5+3q^6+\cdots)\)
    \(\chi_{\epsilon}=q^{23/48}(1+q+q^2+q^3+2q^4+2q^5+3q^6+\cdots)\)
    \(\chi_{\sigma}=q^{1/24}(1+q+q^2+2q^3+2q^4+3q^5+4q^6+\cdots)\)
  • Rocha-Caridi character[RC84]

 

 

modularity of graded dimensions

\(\chi_M(-1/\tau)=\sum_{N} S_{M,N}\chi_N(\tau)\)

\(\chi_M(\tau+1)=\sum_{N} T_{M,N}\chi_N(\tau)\)

 

\(T=\left(\begin{array}{ccc}e^{-\pi i/24} & 0 & 0 \\ 0 & e^{23\pi i/24} & 0 \\ 0 & 0 & e^{\pi i/12}\end{array} \right)\)

\(2S=\left(\begin{array}{ccc}1 & 1& \sqrt{2} \\ 1 & 1 & -\sqrt{2} \\ \sqrt{2} & -\sqrt{2} & 0\end{array} \right)\)

 

 

matching two sets of funtions

 

\(f(\tau)=\frac{e^{-\frac{\pi i}{24}}\eta(\frac{\tau+1}{2})}{\eta(\tau)}=q^{-1/48} \prod_{n=1}^{\infty} (1+q^{n-\frac{1}{2}})\)

\(\chi_0+\chi_{\epsilon}\)

 

{1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 2, 1, 2, 2, 3, 2, 3, 3, 4}

http://www.research.att.com/~njas/sequences/A027349

\(f_1(\tau)=\frac{\eta(\frac{\tau}{2})}{\eta(\tau)}=q^{-1/48} \prod_{n=1}^{\infty} (1-q^{n-\frac{1}{2}})\)

\(\chi_0-\chi_{\epsilon}\)

{1, -1, 0, -1, 1, -1, 1, -1, 2, -2, 2, -2, 3, -3, 3, -4, 5, -5, 5, -6, 7, -8, 8, -9, 11, -12, 12, -14, 16, -17, 18}

http://www.research.att.com/~njas/sequences/A081362

 

 

\(\chi_0=q^{-1/48}(1+q^2+q^3+2q^4+2q^5+3q^6+\cdots)\)

\(\chi_{\epsilon}=q^{23/48}(1+q+q^2+q^3+2q^4+2q^5+3q^6+\cdots)\)

 

{1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 2, 1, 2, 2, 3, 2, 3, 3, 4}

{1, -1, 0, -1, 1, -1, 1, -1, 2, -2, 2, -2, 3, -3, 3, -4, 5, -5, 5, -6, 7, -8, 8, -9, 11, -12, 12, -14, 16, -17, 18}

sum http://www.wolframalpha.com/input/?i=(1,+0,+0,+1,+0,+1,+0,+1,+1,+1)%2B(1,+-1,+0,+-1,+1,+-1,+1,+-1,+2,+-2)

{2,-1,0,0,1,0,1,0,3,...} -> \(q^{-1/48}}(1-1/2q^{1/2}+q^{4/2}+q^{6/2}+3q^{8/2}+\cdots)\)

difference http://www.wolframalpha.com/input/?i=(1,+0,+0,+1,+0,+1,+0,+1,+1,+1)-(1,+-1,+0,+-1,+1,+-1,+1,+-1,+2,+-2)

 

{0,1,0,2,-1,2,-1,2,-1,...} -> \(\frac{1}{2}q^{-1/48}}(q^{1/2}+2q^{3/2}-q^{4/2}+2q^{5/2}-q^{6/2} +2q^{7/2} -q^{8/2}\cdots)\)

 

 

\(f_2(\tau)=\sqrt{2}\frac{\eta(2\tau)}{\eta(\tau)}=\sqrt{2}q^{1/24} \prod_{n=1}^{\infty} (1+q^{n})\)

\(\chi_{\sigma}=q^{1/24}(1+q+q^2+2q^3+2q^4+3q^5+4q^6+\cdots)\)

{1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18, 22, 27, 32, 38, 46, 54, 64, 76, 89, 104, 122, 142, 165, 192, 222, 256, 296}

http://www.research.att.com/~njas/sequences/A000009

 

 

history

 

 

related items

 

 

encyclopedia

 

 

books

 

 

 

expositions

 

 

articles

 

 

 

question and answers(Math Overflow)

 

 

 

blogs

 

 

experts on the field

 

 

links
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