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non-unitary \(c(2,k+2)\)'minimal models'==
imported>Pythagoras0 잔글 (찾아 바꾸기 – “<h5>” 문자열을 “==” 문자열로) |
imported>Pythagoras0 잔글 (찾아 바꾸기 – “</h5>” 문자열을 “==” 문자열로) |
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| − | ==introduction | + | ==introduction== |
* important<br> | * important<br> | ||
| 9번째 줄: | 9번째 줄: | ||
| − | <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;">non-unitary <math>c(2,k+2)</math>''''''minimal models'''''' | + | <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;">non-unitary <math>c(2,k+2)</math>''''''minimal models''''''== |
* central charge<br><math>c(2,k+2)=1-\frac{3k^2}{k+2}</math><br><math>k \geq 3</math>, odd<br> | * central charge<br><math>c(2,k+2)=1-\frac{3k^2}{k+2}</math><br><math>k \geq 3</math>, odd<br> | ||
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| − | ==history | + | ==history== |
* http://www.google.com/search?hl=en&tbs=tl:1&q= | * http://www.google.com/search?hl=en&tbs=tl:1&q= | ||
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| − | ==related items | + | ==related items== |
* [[Andrews-Gordon identity]] | * [[Andrews-Gordon identity]] | ||
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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 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== |
* http://en.wikipedia.org/wiki/ | * http://en.wikipedia.org/wiki/ | ||
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| − | ==books | + | ==books== |
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| − | ==expositions | + | ==expositions== |
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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 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== |
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| − | ==question and answers(Math Overflow) | + | ==question and answers(Math Overflow)== |
* http://mathoverflow.net/search?q= | * http://mathoverflow.net/search?q= | ||
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| − | ==blogs | + | ==blogs== |
* 구글 블로그 검색<br> | * 구글 블로그 검색<br> | ||
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| − | ==experts on the field | + | ==experts on the field== |
* http://arxiv.org/ | * http://arxiv.org/ | ||
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| − | ==links | + | ==links== |
* [http://detexify.kirelabs.org/classify.html Detexify2 - LaTeX symbol classifier] | * [http://detexify.kirelabs.org/classify.html Detexify2 - LaTeX symbol classifier] | ||
2012년 10월 28일 (일) 15:35 판
introduction
- important
non-unitary \(c(2,k+2)\)'minimal models'==
- central charge
\(c(2,k+2)=1-\frac{3k^2}{k+2}\)
\(k \geq 3\), odd
- primary fields have conformal dimensions
\(h_j=-\frac{j(k-j)}{2(k+2)}\), \(j\in \{0,1,\cdots,[k/2]\}\)
- effective central charge
\(c_{eff}=\frac{k-1}{k+2}\)
- dilogarithm identity
\(\sum_{i=1}^{[k/2]}L(\frac{\sin^2\frac{\pi}{k+2}}{\sin^2\frac{(i+1)\pi}{k+2}})=\frac{k-1}{k+2}\cdot \frac{\pi^2}{6}\)
- character functions
\(\chi_j(\tau)=q^{h_j-c/24}\prod_{n\neq 0,\pm(j+1)}(1-q^n)^{-1}\)
- to understand the factor \(q^{h-c/24}\), look at the finite size effect page also
- quantum dimension and there recurrence relation
\(d_i=\frac{\sin \frac{(i+1)\pi}{k+2}}{\sin \frac{\pi}{k+2}}\) satisfies
\(d_i^2=1+d_{i-1}d_{i+1}\) where \(d_0=1\), \(d_k=1\)
- (*choose k for c (2,k+2) minimal model*)k := 11
(*define Rogers dilogarithm*)
L[x_] := PolyLog[2, x] + 1/2 Log[x] Log[1 - x]
(*quantum dimension for minimal models*)
f[k_, i_] := (Sin[Pi/(k + 2)]/Sin[(i + 1) Pi/(k + 2)])^2
(*effective central charge*)
g[k_] := (k*Pi^2)/(2 (k + 2))
(*compare the results*)
N[Sum[L[f[k, i]], {i, 1, k - 1}] + Pi^2/6, 10]
N[g[k], 10]
d[k_, i_] := Sin[(i + 1) Pi/(k + 2)]/Sin[Pi/(k + 2)]
Table[{i, d[k, i]}, {i, 1, k}] // TableForm
Table[{i, N[(d[k, i])^2 - (1 + d[k, i - 1]*d[k, i + 1]), 10]}, {i, 1,
k}] // TableForm
history
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=
expositions
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)
blogs
- 구글 블로그 검색
- http://ncatlab.org/nlab/show/HomePage
experts on the field
links
\(c(2,k+2)=1-\frac{3k^2}{k+2}\)
\(k \geq 3\), odd
\(h_j=-\frac{j(k-j)}{2(k+2)}\), \(j\in \{0,1,\cdots,[k/2]\}\)
\(c_{eff}=\frac{k-1}{k+2}\)
\(\sum_{i=1}^{[k/2]}L(\frac{\sin^2\frac{\pi}{k+2}}{\sin^2\frac{(i+1)\pi}{k+2}})=\frac{k-1}{k+2}\cdot \frac{\pi^2}{6}\)
\(\chi_j(\tau)=q^{h_j-c/24}\prod_{n\neq 0,\pm(j+1)}(1-q^n)^{-1}\)
\(d_i=\frac{\sin \frac{(i+1)\pi}{k+2}}{\sin \frac{\pi}{k+2}}\) satisfies
\(d_i^2=1+d_{i-1}d_{i+1}\) where \(d_0=1\), \(d_k=1\)
(*define Rogers dilogarithm*)
L[x_] := PolyLog[2, x] + 1/2 Log[x] Log[1 - x]
(*quantum dimension for minimal models*)
f[k_, i_] := (Sin[Pi/(k + 2)]/Sin[(i + 1) Pi/(k + 2)])^2
(*effective central charge*)
g[k_] := (k*Pi^2)/(2 (k + 2))
(*compare the results*)
N[Sum[L[f[k, i]], {i, 1, k - 1}] + Pi^2/6, 10]
N[g[k], 10]
d[k_, i_] := Sin[(i + 1) Pi/(k + 2)]/Sin[Pi/(k + 2)]
Table[{i, d[k, i]}, {i, 1, k}] // TableForm
Table[{i, N[(d[k, i])^2 - (1 + d[k, i - 1]*d[k, i + 1]), 10]}, {i, 1,
k}] // TableForm
- 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=
expositions