"Non-unitary c(2,k+2) minimal models"의 두 판 사이의 차이
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imported>Pythagoras0 |
imported>Pythagoras0 |
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| − | ==non-unitary <math>c(2,k+2)</math> | + | ==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> | ||
* primary fields have conformal dimensions<br><math>h_j=-\frac{j(k-j)}{2(k+2)}</math>, <math>j\in \{0,1,\cdots,[k/2]\}</math><br> | * primary fields have conformal dimensions<br><math>h_j=-\frac{j(k-j)}{2(k+2)}</math>, <math>j\in \{0,1,\cdots,[k/2]\}</math><br> | ||
* effective central charge<br><math>c_{eff}=\frac{k-1}{k+2}</math><br> | * effective central charge<br><math>c_{eff}=\frac{k-1}{k+2}</math><br> | ||
| − | * dilogarithm identity | + | * dilogarithm identity |
| + | :<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><br> | ||
* character functions<br><math>\chi_j(\tau)=q^{h_j-c/24}\prod_{n\neq 0,\pm(j+1)}(1-q^n)^{-1}</math><br> | * character functions<br><math>\chi_j(\tau)=q^{h_j-c/24}\prod_{n\neq 0,\pm(j+1)}(1-q^n)^{-1}</math><br> | ||
* to understand the factor <math>q^{h-c/24}</math>, look at the [[finite size effect]] page also<br> | * to understand the factor <math>q^{h-c/24}</math>, look at the [[finite size effect]] page also<br> | ||
2013년 7월 23일 (화) 06:04 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
\[\sum_{i=1}^{k-1}L(\frac{\sin^2\frac{\pi}{k+2}}{\sin^2\frac{(i+1)\pi}{k+2}})=\frac{2(k-1)}{k+2}\cdot \frac{\pi^2}{6}\]
- 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