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| (사용자 2명의 중간 판 17개는 보이지 않습니다) | |||
| 1번째 줄: | 1번째 줄: | ||
==introduction== | ==introduction== | ||
| − | * [[Ising models|Ising model]] | + | * [[Ising models|Ising model]] |
| − | * first | + | * first review the [[minimal models]] page |
| − | * [[Weber functions and conformal field theory]] | + | * [[Weber functions and conformal field theory]] |
| − | * [[rank 1 case]] | + | * [[rank 1 case]] |
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| − | + | ==Ising model as a minimal model== | |
| − | + | * Ising model is a unitary minimal model and thus can be understood by the representation of [[Virasoro algebra|Viraroso algebra]] | |
| + | * the representation is given by following data | ||
| + | ** <math>m= 3</math> | ||
| + | ** <math>c = 1-{6\over m(m+1)} = \frac{1}{2}</math> | ||
| + | ** <math>h_{p,q}(c) = {(4p-3q)^2-1 \over 48}</math> | ||
| + | ** <math>(p,q)=(1,1), (2,1), (2,2)</math> or <math>(p,q)=(1,2), (1,3), (2,3)</math> | ||
| + | * possible values of <math>h</math> | ||
| + | ** <math>0, 1/2, 1/16</math> | ||
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==graded dimensions== | ==graded dimensions== | ||
| − | * associated chiral algebra has three irreducible modules with the following graded dimensions | + | * 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> |
| − | * Rocha-Caridi character[RC84] [[bosonic characters of Virasoro minimal models(Rocha-Caridi formula)]] | + | * Rocha-Caridi character[RC84] [[bosonic characters of Virasoro minimal models(Rocha-Caridi formula)]] |
<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> | <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> | ||
| 30번째 줄: | 32번째 줄: | ||
<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> | <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== | ==modularity of graded dimensions== | ||
| 40번째 줄: | 42번째 줄: | ||
<math>\chi_M(\tau+1)=\sum_{N} T_{M,N}\chi_N(\tau)</math> | <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> | <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> | ||
| 46번째 줄: | 48번째 줄: | ||
<math>2S=\left(\begin{array}{ccc}1 & 1& \sqrt{2} \\ 1 & 1 & -\sqrt{2} \\ \sqrt{2} & -\sqrt{2} & 0\end{array} \right)</math> | <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== | ==matching two sets of funtions== | ||
| − | * [http://www.research.att.com/%7Enjas/sequences/A000009 http://www.research.att.com/~njas/sequences/A000009] | + | * [http://www.research.att.com/%7Enjas/sequences/A000009 http://www.research.att.com/~njas/sequences/A000009] |
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| + | ===1=== | ||
<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> | <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> | ||
<math>\chi_0+\chi_{\epsilon}=\chi_{1,1}+\chi_{1,3}</math> | <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} | {1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 2, 1, 2, 2, 3, 2, 3, 3, 4} | ||
| 66번째 줄: | 68번째 줄: | ||
[http://www.research.att.com/%7Enjas/sequences/A027349 http://www.research.att.com/~njas/sequences/A027349] | [http://www.research.att.com/%7Enjas/sequences/A027349 http://www.research.att.com/~njas/sequences/A027349] | ||
| + | ===2=== | ||
<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> | <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> | ||
| 74번째 줄: | 77번째 줄: | ||
[http://www.research.att.com/%7Enjas/sequences/A081362 http://www.research.att.com/~njas/sequences/A081362] | [http://www.research.att.com/%7Enjas/sequences/A081362 http://www.research.att.com/~njas/sequences/A081362] | ||
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| + | ===3=== | ||
<math>\chi_0=q^{-1/48}(1+q^2+q^3+2q^4+2q^5+3q^6+\cdots)</math> | <math>\chi_0=q^{-1/48}(1+q^2+q^3+2q^4+2q^5+3q^6+\cdots)</math> | ||
<math>\chi_{\epsilon}=q^{23/48}(1+q+q^2+q^3+2q^4+2q^5+3q^6+\cdots)</math> | <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} | {1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 2, 1, 2, 2, 3, 2, 3, 3, 4} | ||
| 88번째 줄: | 91번째 줄: | ||
{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} | {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=%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)] | |
| − | {2,-1,0,0,1,0,1,0,3,...} -> | + | {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> | ||
| − | + | 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,...} -> | + | {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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| + | ===4=== | ||
<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> | <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> | ||
| 110번째 줄: | 115번째 줄: | ||
[http://www.research.att.com/%7Enjas/sequences/A000009 http://www.research.att.com/~njas/sequences/A000009] | [http://www.research.att.com/%7Enjas/sequences/A000009 http://www.research.att.com/~njas/sequences/A000009] | ||
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==related items== | ==related items== | ||
| − | + | * [[Integrable perturbations of Ising model]] | |
* [[determinantal identities and Airy kernel]] | * [[determinantal identities and Airy kernel]] | ||
* [[Weber functions and conformal field theory]] | * [[Weber functions and conformal field theory]] | ||
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| − | + | ==computational resource== | |
| − | + | * https://docs.google.com/file/d/0B8XXo8Tve1cxMU8ycnR2NEczS3c/edit | |
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==articles== | ==articles== | ||
| + | * 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]] | |
| − | + | [[분류:integrable systems]] | |
| + | [[분류:math and physics]] | ||
| + | [[분류:minimal models]] | ||
| + | [[분류:migrate]] | ||
2020년 12월 28일 (월) 04:00 기준 최신판
introduction
- first review the minimal models page
- Weber functions and conformal field theory
- rank 1 case
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}
{2,-1,0,0,1,0,1,0,3,...} -> \[q^{-1/48}(1-1/2q^{1/2}+q^{4/2}+q^{6/2}+3q^{8/2}+\cdots)\]
{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
- Integrable perturbations of Ising model
- determinantal identities and Airy kernel
- Weber functions and conformal field theory
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.