Non-unitary c(2,k+2) minimal models - 편집 역사

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Pythagoras0: /* introduction */

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introduction

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@@ 1번째 줄: / 1번째 줄: @@
-==non-unitary <math>c(2,k+2)</math> minimal models==
+==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
-*  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>
+*  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>
 *  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>
 *  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 factor <math>q^{h-c/24}</math>, look at the [[finite size effect]] page also
+*  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>
-−
-#  (*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
+#  (*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
-−
 ==related items==
@@ 19번째 줄: / 19번째 줄: @@
 * [[Andrews-Gordon identity]]
-−
 ==computational resource==
 * https://docs.google.com/file/d/0B8XXo8Tve1cxdXRlbU40OExkeW8/edit

← 이전 판		2020년 11월 14일 (토) 08:52 판
1번째 줄:		1번째 줄:
−	~~==introduction==~~
−
−	* important
−
−
	==non-unitary <math>c(2,k+2)</math> minimal models==		==non-unitary <math>c(2,k+2)</math> minimal models==

@@ 6번째 줄: / 6번째 줄: @@
 ==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<math>c(2,k+2)=1-\frac{3k^2}{k+2}</math><math>k \geq 3</math>, odd
-*  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<math>h_j=-\frac{j(k-j)}{2(k+2)}</math>, <math>j\in \{0,1,\cdots,[k/2]\}</math>
-*  effective central charge<br><math>c_{eff}=\frac{k-1}{k+2}</math><br>
+*  effective central charge<math>c_{eff}=\frac{k-1}{k+2}</math>
 *  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>
+:<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<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<math>\chi_j(\tau)=q^{h_j-c/24}\prod_{n\neq 0,\pm(j+1)}(1-q^n)^{-1}</math>
-*  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
-*  quantum dimension and there recurrence relation<br><math>d_i=\frac{\sin \frac{(i+1)\pi}{k+2}}{\sin \frac{\pi}{k+2}}</math> satisfies<br><math>d_i^2=1+d_{i-1}d_{i+1}</math> where <math>d_0=1</math>, <math>d_k=1</math><br>
+*  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>
-#  (*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 (*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

← 이전 판		2020년 11월 13일 (금) 15:03 판
35번째 줄:		35번째 줄:
	[[분류:math and physics]]		[[분류:math and physics]]
	[[분류:minimal models]]		[[분류:minimal models]]
		+	[[분류:migrate]]

@@ 4번째 줄: / 4번째 줄: @@
-==non-unitary <math>c(2,k+2)</math>''''''minimal models''''''==
+==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>
 *  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>
-*  dilogarithm identity<br><math>\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}</math><br>
+*  dilogarithm identity
 *  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>

← 이전 판		2013년 2월 26일 (화) 08:37 판
33번째 줄:		33번째 줄:
	[[분류:conformal field theory]]		[[분류:conformal field theory]]
	[[분류:math and physics]]		[[분류:math and physics]]
		+	[[분류:minimal models]]

@@ 1번째 줄: / 1번째 줄: @@
 ==introduction==
-*  important<br>
+* important
@@ 22번째 줄: / 17번째 줄: @@
 #  (*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>
@@ 43번째 줄: / 25번째 줄: @@
+==computational resource==
+* https://docs.google.com/file/d/0B8XXo8Tve1cxdXRlbU40OExkeW8/edit
 [[분류:conformal field theory]]
 [[분류:math and physics]]

← 이전 판		2012년 10월 29일 (월) 17:53 판
128번째 줄:		128번째 줄:
	* http://functions.wolfram.com/		* http://functions.wolfram.com/
	[[분류:conformal field theory]]		[[분류:conformal field theory]]
		+	[[분류:math and physics]]

← 이전 판		2012년 10월 29일 (월) 15:31 판
127번째 줄:		127번째 줄:
	* [http://www.research.att.com/%7Enjas/sequences/index.html The On-Line Encyclopedia of Integer Sequences]		* [http://www.research.att.com/%7Enjas/sequences/index.html The On-Line Encyclopedia of Integer Sequences]
	* http://functions.wolfram.com/		* http://functions.wolfram.com/
		+	[[분류:conformal field theory]]