"Non-unitary c(2,k+2) minimal models"의 두 판 사이의 차이
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imported>Pythagoras0 |
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==introduction== | ==introduction== | ||
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# (*choose k for c (2,k+2) minimal model*)k := 11<br> (*define Rogers dilogarithm*)<br> L[x_] := PolyLog[2, x] + 1/2 Log[x] Log[1 - x]<br> (*quantum dimension for minimal models*)<br> f[k_, i_] := (Sin[Pi/(k + 2)]/Sin[(i + 1) Pi/(k + 2)])^2<br> (*effective central charge*)<br> g[k_] := (k*Pi^2)/(2 (k + 2))<br> (*compare the results*)<br> N[Sum[L[f[k, i]], {i, 1, k - 1}] + Pi^2/6, 10]<br> N[g[k], 10]<br> d[k_, i_] := Sin[(i + 1) Pi/(k + 2)]/Sin[Pi/(k + 2)]<br> Table[{i, d[k, i]}, {i, 1, k}] // TableForm<br> Table[{i, N[(d[k, i])^2 - (1 + d[k, i - 1]*d[k, i + 1]), 10]}, {i, 1,<br> k}] // TableForm<br> | # (*choose k for c (2,k+2) minimal model*)k := 11<br> (*define Rogers dilogarithm*)<br> L[x_] := PolyLog[2, x] + 1/2 Log[x] Log[1 - x]<br> (*quantum dimension for minimal models*)<br> f[k_, i_] := (Sin[Pi/(k + 2)]/Sin[(i + 1) Pi/(k + 2)])^2<br> (*effective central charge*)<br> g[k_] := (k*Pi^2)/(2 (k + 2))<br> (*compare the results*)<br> N[Sum[L[f[k, i]], {i, 1, k - 1}] + Pi^2/6, 10]<br> N[g[k], 10]<br> d[k_, i_] := Sin[(i + 1) Pi/(k + 2)]/Sin[Pi/(k + 2)]<br> Table[{i, d[k, i]}, {i, 1, k}] // TableForm<br> Table[{i, N[(d[k, i])^2 - (1 + d[k, i - 1]*d[k, i + 1]), 10]}, {i, 1,<br> k}] // TableForm<br> | ||
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| − | + | ==computational resource== | |
| − | + | * https://docs.google.com/file/d/0B8XXo8Tve1cxdXRlbU40OExkeW8/edit | |
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[[분류:conformal field theory]] | [[분류:conformal field theory]] | ||
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[[분류:math and physics]] | [[분류:math and physics]] | ||
2013년 1월 26일 (토) 11:52 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