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

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1번째 줄: 1번째 줄:
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==introduction==
  
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* [[Ising models|Ising model]]
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* first review the [[minimal models]] page
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* [[Weber functions and conformal field theory]]
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* [[rank 1 case]]
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==Ising model as a minimal model==
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*  Ising model is a unitary minimal model and thus can be understood by the representation of [[Virasoro algebra|Viraroso algebra]]
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*  the representation is given by following data
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** <math>m= 3</math>
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** <math>c = 1-{6\over m(m+1)} = \frac{1}{2}</math>
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** <math>h_{p,q}(c) = {(4p-3q)^2-1 \over 48}</math>
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** <math>(p,q)=(1,1), (2,1), (2,2)</math> or <math>(p,q)=(1,2), (1,3), (2,3)</math>
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*  possible values of <math>h</math>
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** <math>0, 1/2, 1/16</math>
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==graded dimensions==
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*  associated chiral algebra has three irreducible modules with the following graded dimensions h=0, <math>\chi_0=\chi _{1,1}^{(3,4)}=q^{-1/48}(1+q^2+q^3+2q^4+2q^5+3q^6+\cdots)</math> h=1/2, <math>\chi_{\epsilon}=\chi_{1,3}^{(3,4)}=q^{23/48}(1+q+q^2+q^3+2q^4+2q^5+3q^6+\cdots)</math> h=1/16, <math>\chi_{\sigma}=\chi _{1,2}^{(3,4)}=q^{1/24}(1+q+q^2+2q^3+2q^4+3q^5+4q^6+\cdots)</math>
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*  Rocha-Caridi character[RC84] [[bosonic characters of Virasoro minimal models(Rocha-Caridi formula)]]
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<math>\chi_{\sigma}=\chi _{1,2}^{(3,4)}=\chi _{2,2}^{(3,4)}=\frac{\eta (2\tau )}{\eta (\tau )}=q^{1/24}\sum _{m=-\infty }^{\infty } (-1)^mq^{3 m^2- m }=q^{1/24}\sum_{n\geq 0}\frac{q^{n(n+1)/2}}{(1-q)(1-q^2)\cdots(1-q^n)}</math>
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<math>\mathfrak{f}_2(\tau)=\sqrt{2}\frac{\eta(2\tau)}{\eta(\tau)}=\sqrt{2}q^{1/24} \prod_{n=1}^{\infty} (1+q^{n})=\sqrt{2}q^{1/24}\sum_{n\geq 0}\frac{q^{n(n+1)/2}}{(1-q)(1-q^2)\cdots(1-q^n)}</math>
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==modularity of graded dimensions==
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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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==matching two sets of funtions==
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* [http://www.research.att.com/%7Enjas/sequences/A000009 http://www.research.att.com/~njas/sequences/A000009]
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===1===
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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}=\chi_{1,1}+\chi_{1,3}</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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===2===
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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}=\chi_{1,1}-\chi_{1,3}</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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===3===
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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,...} ->
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:<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,...} ->
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:<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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===4===
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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}=\chi_{1,2}=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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==related items==
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* [[Integrable perturbations of Ising model‎]]
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* [[determinantal identities and Airy kernel]]
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* [[Weber functions and conformal field theory]]
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==computational resource==
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* https://docs.google.com/file/d/0B8XXo8Tve1cxMU8ycnR2NEczS3c/edit
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==articles==
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* McRae, Robert. ‘Integral Forms for Tensor Powers of the Virasoro Vertex Operator Algebra <math>L(\frac{1}{2},0)</math> and Their Modules’. arXiv:1410.5676 [math], 21 October 2014. http://arxiv.org/abs/1410.5676.
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[[분류:conformal field theory]]
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[[분류:integrable systems]]
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[[분류:math and physics]]
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[[분류:minimal models]]
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[[분류:migrate]]

2020년 12월 28일 (월) 04:00 기준 최신판

introduction



Ising model as a minimal model

  • 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
    • \(m= 3\)
    • \(c = 1-{6\over m(m+1)} = \frac{1}{2}\)
    • \(h_{p,q}(c) = {(4p-3q)^2-1 \over 48}\)
    • \((p,q)=(1,1), (2,1), (2,2)\) or \((p,q)=(1,2), (1,3), (2,3)\)
  • possible values of \(h\)
    • \(0, 1/2, 1/16\)


graded dimensions

  • associated chiral algebra has three irreducible modules with the following graded dimensions h=0, \(\chi_0=\chi _{1,1}^{(3,4)}=q^{-1/48}(1+q^2+q^3+2q^4+2q^5+3q^6+\cdots)\) h=1/2, \(\chi_{\epsilon}=\chi_{1,3}^{(3,4)}=q^{23/48}(1+q+q^2+q^3+2q^4+2q^5+3q^6+\cdots)\) h=1/16, \(\chi_{\sigma}=\chi _{1,2}^{(3,4)}=q^{1/24}(1+q+q^2+2q^3+2q^4+3q^5+4q^6+\cdots)\)
  • Rocha-Caridi character[RC84] bosonic characters of Virasoro minimal models(Rocha-Caridi formula)

\(\chi_{\sigma}=\chi _{1,2}^{(3,4)}=\chi _{2,2}^{(3,4)}=\frac{\eta (2\tau )}{\eta (\tau )}=q^{1/24}\sum _{m=-\infty }^{\infty } (-1)^mq^{3 m^2- m }=q^{1/24}\sum_{n\geq 0}\frac{q^{n(n+1)/2}}{(1-q)(1-q^2)\cdots(1-q^n)}\)

\(\mathfrak{f}_2(\tau)=\sqrt{2}\frac{\eta(2\tau)}{\eta(\tau)}=\sqrt{2}q^{1/24} \prod_{n=1}^{\infty} (1+q^{n})=\sqrt{2}q^{1/24}\sum_{n\geq 0}\frac{q^{n(n+1)/2}}{(1-q)(1-q^2)\cdots(1-q^n)}\)



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


1

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

\(\chi_0+\chi_{\epsilon}=\chi_{1,1}+\chi_{1,3}\)


{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

2

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

\(\chi_0-\chi_{\epsilon}=\chi_{1,1}-\chi_{1,3}\)

{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



3

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

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


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



4

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

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

{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



related items


computational resource


articles

  • McRae, Robert. ‘Integral Forms for Tensor Powers of the Virasoro Vertex Operator Algebra \(L(\frac{1}{2},0)\) and Their Modules’. arXiv:1410.5676 [math], 21 October 2014. http://arxiv.org/abs/1410.5676.
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