"Non-unitary c(2,k+2) minimal models"의 두 판 사이의 차이
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Pythagoras0 (토론 | 기여) |
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| − | ==non- | + | ==non-unitary <math>c(2,k+2)</math> minimal models== |
* central charge<math>c(2,k+2)=1-\frac{3k^2}{k+2}</math><math>k \geq 3</math>, odd | * central charge<math>c(2,k+2)=1-\frac{3k^2}{k+2}</math><math>k \geq 3</math>, odd | ||
| − | * primary fields | + | * primary fields have conformal dimensions<math>h_j=-\frac{j(k-j)}{2(k+2)}</math>, <math>j\in \{0,1,\cdots,[k/2]\}</math> |
* effective central charge<math>c_{eff}=\frac{k-1}{k+2}</math> | * effective central charge<math>c_{eff}=\frac{k-1}{k+2}</math> | ||
* 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> | :<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> | ||
* character functions<math>\chi_j(\tau)=q^{h_j-c/24}\prod_{n\neq 0,\pm(j+1)}(1-q^n)^{-1}</math> | * character functions<math>\chi_j(\tau)=q^{h_j-c/24}\prod_{n\neq 0,\pm(j+1)}(1-q^n)^{-1}</math> | ||
| − | * to understand the | + | * to understand the factor <math>q^{h-c/24}</math>, look at the [[finite size effect]] page also |
* quantum dimension and there recurrence relation<math>d_i=\frac{\sin \frac{(i+1)\pi}{k+2}}{\sin \frac{\pi}{k+2}}</math> satisfies<math>d_i^2=1+d_{i-1}d_{i+1}</math> where <math>d_0=1</math>, <math>d_k=1</math> | * quantum dimension and there recurrence relation<math>d_i=\frac{\sin \frac{(i+1)\pi}{k+2}}{\sin \frac{\pi}{k+2}}</math> satisfies<math>d_i^2=1+d_{i-1}d_{i+1}</math> where <math>d_0=1</math>, <math>d_k=1</math> | ||
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| − | # (*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, | + | # (*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 |
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==related items== | ==related items== | ||
| 19번째 줄: | 19번째 줄: | ||
* [[Andrews-Gordon identity]] | * [[Andrews-Gordon identity]] | ||
| − | + | ||
==computational resource== | ==computational resource== | ||
| − | * https://docs.google.com/file/d/0B8XXo8Tve1cxdXRlbU40OExkeW8/ | + | * https://docs.google.com/file/d/0B8XXo8Tve1cxdXRlbU40OExkeW8/edit |
2020년 12월 28일 (월) 05:17 기준 최신판
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