Non-unitary c(2,2k+1) minimal models - 편집 역사

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2020-12-28T12:24:58Z

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2020년 11월 14일 (토) 03:00에 imported>Pythagoras0님의 편집

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2020년 11월 14일 (토) 03:00에 imported>Pythagoras0님의 편집

2020-11-14T03:00:59Z

imported>Pythagoras0: /* asymptotic analysis of Andrews-Gordon identity */

2013-07-23T14:14:19Z

asymptotic analysis of Andrews-Gordon identity

imported>Pythagoras0: /* asymptotic analysis of Andrews-Gordon identity */

2013-07-23T14:13:02Z

asymptotic analysis of Andrews-Gordon identity

imported>Pythagoras0: /* asymptotic analysis of Andrews-Gordon identity */

2013-07-23T14:05:07Z

asymptotic analysis of Andrews-Gordon identity

imported>Pythagoras0: /* asymptotic analysis of Andrews-Gordon identity */

2013-07-23T09:09:20Z

asymptotic analysis of Andrews-Gordon identity

2013년 2월 26일 (화) 08:37에 imported>Pythagoras0님의 편집

2013-02-26T08:37:45Z

@@ 1번째 줄: / 1번째 줄: @@
 ==introduction==
-−
-−
-−
 ==central charge and conformal dimensions==
 *  central charge<math>c(2,2k+1)=1-\frac{3(2k-1)^2}{2k+1}</math>
-*  primary fields have conformal dimensions<math>h_j=-\frac{j(2k-1-j)}{2(2k+1)}</math>, <math>j\in \{0,1,\cdots,k-1\}</math> or by setting i=j+1<math>h_i=-\frac{(i-1)(2k-i)}{2(2k+1)}</math> <math>i\in \{1,2, \cdots,k\}</math> (this is An's notation in his paper)
+*  primary fields have conformal dimensions<math>h_j=-\frac{j(2k-1-j)}{2(2k+1)}</math>, <math>j\in \{0,1,\cdots,k-1\}</math> or by setting i=j+1<math>h_i=-\frac{(i-1)(2k-i)}{2(2k+1)}</math> <math>i\in \{1,2, \cdots,k\}</math> (this is An's notation in his paper)
 *  effective central charge<math>c_{eff}=c-24h_{min}</math><math>c_{eff}=\frac{2k-2}{2k+1}</math>
-−
-−
 ==character formula and Andrew-Gordon identity==
@@ 21번째 줄: / 21번째 줄: @@
 * [[Andrews-Gordon identity]]
 *  character functions<math>\chi_i(\tau)=q^{h_i-c/24}\prod_{n\neq 0,\pm i\pmod {2k+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
 * [[bosonic characters of Virasoro minimal models(Rocha-Caridi formula)]]<math>\chi_{r,s}^{(p,p')}=\frac{q^{\Delta_{r,s}^{(p,p')}}}{(q)_{\infty}}\sum_{n=-\infty}^{\infty}(q^{pp'n^2+(rp'-sp)n}-q^{(pn+r)(p'n+s)})</math><math>\Delta_{r,s}^{(p,p')}=h_{r,s}^{(p,p')}-\frac{c}{24}</math>
@@ 42번째 줄: / 42번째 줄: @@
 <math>\chi_{r,s}^{(p,p')}=q^{\Delta_{r,s}^{(p,p')}}\prod_{n\neq 0,\pm i\pmod {2k+1}}(1-q^n)^{-1}</math>
-Using a result from [http://pythagoras0.springnote.com/pages/9409120 베일리 격자(Bailey lattice)]
+Using a result from [http://pythagoras0.springnote.com/pages/9409120 베일리 격자(Bailey lattice)]
 <math>\sum_{n_1\geq\cdots\geq n_{k-1}\geq0}\frac{q^{n_1^2+\cdots+n_{k-1}^2+n_i+\cdots+n_{k-1}}}{(q)_{n_{1}-n_{2}}\cdots (q)_{n_{k-2}-n_{k-1}}(q)_{n_{k-1}}}=\sum_{n=-\infty}^{\infty}(-1)^{n}q^{\frac{(2k+1)n^2}{2}}q^{\frac{n(2k-2i+1)}{2}}=\prod_{n\neq 0,\pm i\pmod {2k+1}}(1-q^n)^{-1}</math>
-−
-−
 ==asymptotic analysis of Andrews-Gordon identity==
-*  non-unitary <math>c(2,k+2)</math> [[minimal models]]
+*  non-unitary <math>c(2,k+2)</math> [[minimal models]]
 :<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>
-*  non-unitary <math>c(2,2k+1)</math> [[minimal models]]
+*  non-unitary <math>c(2,2k+1)</math> [[minimal models]]
 :<math>
 \sum_{i=1}^{2k-2}L\left(\frac{\sin^2\frac{\pi}{2k+1}}{\sin^2\frac{(i+1)\pi}{2k+1}}\right)=\frac{2(2k-2)}{2k+1}\cdot \frac{\pi^2}{6}
@@ 61번째 줄: / 61번째 줄: @@
 </math>
-−
 ==different expressions for central charge==
@@ 68번째 줄: / 68번째 줄: @@
 *  L-values<math>\frac{k}{12}+\frac{2k+1}{12}-\frac{j(N-j)}{2N}</math>
-−
-−
 ==Dirichlet L-function==
@@ 78번째 줄: / 78번째 줄: @@
 <math>L(-1, \chi) = \frac{1}{2f}\sum_{n=1}^{\infty}{\chi(n)}{n}</math>
-<math>n\geq 1</math> 이라 하자. 일반적으로 <math>\chi\neq 1</math>인 primitive 준동형사상 <math>\chi \colon(\mathbb{Z}/f\mathbb{Z})^\times \to \mathbb C^{*}</math>에 대하여 <math>L(1-n,\chi)</math>의 값은 다음과 같이 주어진다
+<math>n\geq 1</math> 이라 하자. 일반적으로 <math>\chi\neq 1</math>인 primitive 준동형사상 <math>\chi \colon(\mathbb{Z}/f\mathbb{Z})^\times \to \mathbb C^{*}</math>에 대하여 <math>L(1-n,\chi)</math>의 값은 다음과 같이 주어진다
 :<math>L(1-n,\chi)=-\frac{f^{n-1}}{n}\sum_{(a,f)=1}\chi(a)B_n(\frac{a}{f})</math>
 :<math>L(-1,\chi)=L(1-2,\chi)=-\frac{f}{2}\sum_{(a,f)=1}\chi(a)B_2(\frac{a}{f})</math>
-여기서 <math>B_n(x)</math> 는 [http://pythagoras0.springnote.com/pages/4346717 베르누이 다항식](<math>B_0(x)=1</math>, <math>B_1(x)=x-1/2</math>, <math>B_2(x)=x^2-x+1/6</math>, <math>\cdots</math>)
+여기서 <math>B_n(x)</math> 는 [http://pythagoras0.springnote.com/pages/4346717 베르누이 다항식](<math>B_0(x)=1</math>, <math>B_1(x)=x-1/2</math>, <math>B_2(x)=x^2-x+1/6</math>, <math>\cdots</math>)
-−
-Let N=2k+1
+Let N=2k+1
 <math>\omega=\exp \frac{2\pi i}{2k+1}</math>
@@ 92번째 줄: / 92번째 줄: @@
 G: group of Dirichlet characters of conductor N which maps -1 to 1
-G has order k and cyclic generated by <math>\chi</math>
+G has order k and cyclic generated by <math>\chi</math>
 <math>c_i=\frac{1}{2k}\sum_{s=1}^{k}\omega^{is}L(-1,\chi^s)</math>
 Then,
 <math>c_i=-\frac{2k+1}{12}+\frac{j(N-j)}{2N}</math>
-where j satisfies <math>\chi(j)=\omega^{k-i}</math>
+where j satisfies <math>\chi(j)=\omega^{k-i}</math>
 Vacuum energy is given by
@@ 106번째 줄: / 106번째 줄: @@
 <math>d_i=\frac{1}{2}L(-1,\chi^{k})-c_i</math>
-−
 Since
-<math>L(-1,\chi^{k})=\frac{N-1}{12}=\frac{k}{6}</math>,  the vacuum energy
+<math>L(-1,\chi^{k})=\frac{N-1}{12}=\frac{k}{6}</math>,  the vacuum energy
 <math>d_i=\frac{1}{2}L(-1,\chi^{k})-c_i=\frac{k}{12}+\frac{2k+1}{12}-\frac{j(N-j)}{2N}=\frac{j(2k+1-j)}{2(2k+1)}-\frac{k+1}{12}</math>.
-These are equal to <math>{h_i-c/24}</math>
+These are equal to <math>{h_i-c/24}</math>
-−
-−
-# k := 5 f[k_, j_] := (2 k)/    24 + ((2 k + 1)/12 - (j (2 k + 1 - j))/(2 (2 k + 1))) Table[{j, f[k, j]}, {j, 1, 2 k}] // TableForm Table[{j, -24*f[k, j]}, {j, 1, 2 k}] // TableForm d[k_, j_] := (2 (k - j) + 1)^2/(8 (2 k + 1)) - 1/24 Table[{j, d[k, j]}, {j, 1, 2 k}] // TableForm Table[{j, -24*d[k, j]}, {j, 1, 2 k}] // TableForm cef[k_, j_] := -((j (2 k - 1 - j))/(2 (2 k +          1))) - (1 - (3 (2 k - 1)^2)/(2 k + 1))/24 Table[{j, cef[k, j]}, {j, 0, 2 k - 1}] // TableForm Table[{j, -24*cef[k, j]}, {j, 0, 2 k - 1}] // TableForm
+# k := 5 f[k_, j_] := (2 k)/    24 + ((2 k + 1)/12 - (j (2 k + 1 - j))/(2 (2 k + 1))) Table[{j, f[k, j]}, {j, 1, 2 k}] // TableForm Table[{j, -24*f[k, j]}, {j, 1, 2 k}] // TableForm d[k_, j_] := (2 (k - j) + 1)^2/(8 (2 k + 1)) - 1/24 Table[{j, d[k, j]}, {j, 1, 2 k}] // TableForm Table[{j, -24*d[k, j]}, {j, 1, 2 k}] // TableForm cef[k_, j_] := -((j (2 k - 1 - j))/(2 (2 k +          1))) - (1 - (3 (2 k - 1)^2)/(2 k + 1))/24 Table[{j, cef[k, j]}, {j, 0, 2 k - 1}] // TableForm Table[{j, -24*cef[k, j]}, {j, 0, 2 k - 1}] // TableForm
-−
-−
-# w := Exp[2 Pi*I*1/k] L[j_] := -(2 k + 1)/2*   Sum[DirichletCharacter[2 k + 1, j, a]*     BernoulliB[2, a/(2 k + 1)], {a, 1, 2 k}] c[k_, i_] := 1/(2 k) Sum[w^(i*s)*L[Mod[3*s, 2 k]], {s, 1, k}] Table[DiscretePlot[{Re[DirichletCharacter[2 k + 1, j, a]],    Im[DirichletCharacter[2 k + 1, j, a]]}, {a, 0, 2 k + 1},   PlotLabel -> j], {j, 1, EulerPhi[2 k + 1]}] Table[c[i], {i, 1, 2 k}]
+# w := Exp[2 Pi*I*1/k] L[j_] := -(2 k + 1)/2*   Sum[DirichletCharacter[2 k + 1, j, a]*     BernoulliB[2, a/(2 k + 1)], {a, 1, 2 k}] c[k_, i_] := 1/(2 k) Sum[w^(i*s)*L[Mod[3*s, 2 k]], {s, 1, k}] Table[DiscretePlot[{Re[DirichletCharacter[2 k + 1, j, a]],    Im[DirichletCharacter[2 k + 1, j, a]]}, {a, 0, 2 k + 1},   PlotLabel -> j], {j, 1, EulerPhi[2 k + 1]}] Table[c[i], {i, 1, 2 k}]
 [[분류:conformal field theory]]
 [[분류:math and physics]]
 [[분류:minimal models]]
 [[분류:migrate]]

@@ 9번째 줄: / 9번째 줄: @@
 ==central charge and conformal dimensions==
-*  central charge<br><math>c(2,2k+1)=1-\frac{3(2k-1)^2}{2k+1}</math><br>
+*  central charge<math>c(2,2k+1)=1-\frac{3(2k-1)^2}{2k+1}</math>
-*  primary fields have conformal dimensions<br><math>h_j=-\frac{j(2k-1-j)}{2(2k+1)}</math>, <math>j\in \{0,1,\cdots,k-1\}</math> or by setting i=j+1<br><math>h_i=-\frac{(i-1)(2k-i)}{2(2k+1)}</math> <math>i\in \{1,2, \cdots,k\}</math> (this is An's notation in his paper)<br>
+*  primary fields have conformal dimensions<math>h_j=-\frac{j(2k-1-j)}{2(2k+1)}</math>, <math>j\in \{0,1,\cdots,k-1\}</math> or by setting i=j+1<math>h_i=-\frac{(i-1)(2k-i)}{2(2k+1)}</math> <math>i\in \{1,2, \cdots,k\}</math> (this is An's notation in his paper)
-*  effective central charge<br><math>c_{eff}=c-24h_{min}</math><br><math>c_{eff}=\frac{2k-2}{2k+1}</math><br>
+*  effective central charge<math>c_{eff}=c-24h_{min}</math><math>c_{eff}=\frac{2k-2}{2k+1}</math>
@@ 20번째 줄: / 20번째 줄: @@
 * [[Andrews-Gordon identity]]
-*  character functions<br><math>\chi_i(\tau)=q^{h_i-c/24}\prod_{n\neq 0,\pm i\pmod {2k+1}}(1-q^n)^{-1}</math><br>
+*  character functions<math>\chi_i(\tau)=q^{h_i-c/24}\prod_{n\neq 0,\pm i\pmod {2k+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
-* [[bosonic characters of Virasoro minimal models(Rocha-Caridi formula)]]<br><math>\chi_{r,s}^{(p,p')}=\frac{q^{\Delta_{r,s}^{(p,p')}}}{(q)_{\infty}}\sum_{n=-\infty}^{\infty}(q^{pp'n^2+(rp'-sp)n}-q^{(pn+r)(p'n+s)})</math><br><math>\Delta_{r,s}^{(p,p')}=h_{r,s}^{(p,p')}-\frac{c}{24}</math><br>
+* [[bosonic characters of Virasoro minimal models(Rocha-Caridi formula)]]<math>\chi_{r,s}^{(p,p')}=\frac{q^{\Delta_{r,s}^{(p,p')}}}{(q)_{\infty}}\sum_{n=-\infty}^{\infty}(q^{pp'n^2+(rp'-sp)n}-q^{(pn+r)(p'n+s)})</math><math>\Delta_{r,s}^{(p,p')}=h_{r,s}^{(p,p')}-\frac{c}{24}</math>
 Let's specify p=2, p'=2k+1, r=1, s=i
@@ 54번째 줄: / 54번째 줄: @@
 :<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>
 *  non-unitary <math>c(2,2k+1)</math> [[minimal models]]
-$$
+:<math>
 \sum_{i=1}^{2k-2}L\left(\frac{\sin^2\frac{\pi}{2k+1}}{\sin^2\frac{(i+1)\pi}{2k+1}}\right)=\frac{2(2k-2)}{2k+1}\cdot \frac{\pi^2}{6}
-$$
+</math>
-$$
+:<math>
 \sum_{i=1}^{k-1}L\left(\frac{\sin^2\frac{\pi}{2k+1}}{\sin^2\frac{(i+1)\pi}{2k+1}}\right)=\frac{2k-2}{2k+1}\cdot \frac{\pi^2}{6}
-$$
+</math>
@@ 65번째 줄: / 65번째 줄: @@
 ==different expressions for central charge==
-*  from above<br><math>h_i-c(2,2k+1)/24</math><br><math>c(2,2k+1)=1-\frac{3(2k-1)^2}{2k+1}</math><br><math>h_i=-\frac{(i-1)(2k-i)}{2(2k+1)}</math>, <math>i\in \{1,2, \cdots,k\}</math><br>
+*  from above<math>h_i-c(2,2k+1)/24</math><math>c(2,2k+1)=1-\frac{3(2k-1)^2}{2k+1}</math><math>h_i=-\frac{(i-1)(2k-i)}{2(2k+1)}</math>, <math>i\in \{1,2, \cdots,k\}</math>
-*  L-values<br><math>\frac{k}{12}+\frac{2k+1}{12}-\frac{j(N-j)}{2N}</math><br>
+*  L-values<math>\frac{k}{12}+\frac{2k+1}{12}-\frac{j(N-j)}{2N}</math>
@@ 120번째 줄: / 120번째 줄: @@
-# k := 5<br> f[k_, j_] := (2 k)/<br>    24 + ((2 k + 1)/12 - (j (2 k + 1 - j))/(2 (2 k + 1)))<br> Table[{j, f[k, j]}, {j, 1, 2 k}] // TableForm<br> Table[{j, -24*f[k, j]}, {j, 1, 2 k}] // TableForm<br> d[k_, j_] := (2 (k - j) + 1)^2/(8 (2 k + 1)) - 1/24<br> Table[{j, d[k, j]}, {j, 1, 2 k}] // TableForm<br> Table[{j, -24*d[k, j]}, {j, 1, 2 k}] // TableForm<br> cef[k_, j_] := -((j (2 k - 1 - j))/(2 (2 k +<br>          1))) - (1 - (3 (2 k - 1)^2)/(2 k + 1))/24<br> Table[{j, cef[k, j]}, {j, 0, 2 k - 1}] // TableForm<br> Table[{j, -24*cef[k, j]}, {j, 0, 2 k - 1}] // TableForm
+# k := 5 f[k_, j_] := (2 k)/    24 + ((2 k + 1)/12 - (j (2 k + 1 - j))/(2 (2 k + 1))) Table[{j, f[k, j]}, {j, 1, 2 k}] // TableForm Table[{j, -24*f[k, j]}, {j, 1, 2 k}] // TableForm d[k_, j_] := (2 (k - j) + 1)^2/(8 (2 k + 1)) - 1/24 Table[{j, d[k, j]}, {j, 1, 2 k}] // TableForm Table[{j, -24*d[k, j]}, {j, 1, 2 k}] // TableForm cef[k_, j_] := -((j (2 k - 1 - j))/(2 (2 k +          1))) - (1 - (3 (2 k - 1)^2)/(2 k + 1))/24 Table[{j, cef[k, j]}, {j, 0, 2 k - 1}] // TableForm Table[{j, -24*cef[k, j]}, {j, 0, 2 k - 1}] // TableForm
@@ 126번째 줄: / 126번째 줄: @@
-# w := Exp[2 Pi*I*1/k]<br> L[j_] := -(2 k + 1)/2*<br>   Sum[DirichletCharacter[2 k + 1, j, a]*<br>     BernoulliB[2, a/(2 k + 1)], {a, 1, 2 k}]<br> c[k_, i_] := 1/(2 k) Sum[w^(i*s)*L[Mod[3*s, 2 k]], {s, 1, k}]<br> Table[DiscretePlot[{Re[DirichletCharacter[2 k + 1, j, a]],<br>    Im[DirichletCharacter[2 k + 1, j, a]]}, {a, 0, 2 k + 1},<br>   PlotLabel -> j], {j, 1, EulerPhi[2 k + 1]}]<br> Table[c[i], {i, 1, 2 k}]
+# w := Exp[2 Pi*I*1/k] L[j_] := -(2 k + 1)/2*   Sum[DirichletCharacter[2 k + 1, j, a]*     BernoulliB[2, a/(2 k + 1)], {a, 1, 2 k}] c[k_, i_] := 1/(2 k) Sum[w^(i*s)*L[Mod[3*s, 2 k]], {s, 1, k}] Table[DiscretePlot[{Re[DirichletCharacter[2 k + 1, j, a]],    Im[DirichletCharacter[2 k + 1, j, a]]}, {a, 0, 2 k + 1},   PlotLabel -> j], {j, 1, EulerPhi[2 k + 1]}] Table[c[i], {i, 1, 2 k}]
 [[분류:conformal field theory]]
 [[분류:math and physics]]
 [[분류:minimal models]]
 [[분류:migrate]]

← 이전 판	2020년 11월 14일 (토) 03:00 판
1번째 줄:	1번째 줄:
	+	==introduction==

	+
	+
	+
	+
	+
	+
	+	==central charge and conformal dimensions==
	+
	+	* central charge<br><math>c(2,2k+1)=1-\frac{3(2k-1)^2}{2k+1}</math><br>
	+	* primary fields have conformal dimensions<br><math>h_j=-\frac{j(2k-1-j)}{2(2k+1)}</math>, <math>j\in \{0,1,\cdots,k-1\}</math> or by setting i=j+1<br><math>h_i=-\frac{(i-1)(2k-i)}{2(2k+1)}</math> <math>i\in \{1,2, \cdots,k\}</math> (this is An's notation in his paper)<br>
	+	* effective central charge<br><math>c_{eff}=c-24h_{min}</math><br><math>c_{eff}=\frac{2k-2}{2k+1}</math><br>
	+
	+
	+
	+
	+
	+	==character formula and Andrew-Gordon identity==
	+
	+	* [[Andrews-Gordon identity]]
	+	* character functions<br><math>\chi_i(\tau)=q^{h_i-c/24}\prod_{n\neq 0,\pm i\pmod {2k+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>
	+	* [[bosonic characters of Virasoro minimal models(Rocha-Caridi formula)]]<br><math>\chi_{r,s}^{(p,p')}=\frac{q^{\Delta_{r,s}^{(p,p')}}}{(q)_{\infty}}\sum_{n=-\infty}^{\infty}(q^{pp'n^2+(rp'-sp)n}-q^{(pn+r)(p'n+s)})</math><br><math>\Delta_{r,s}^{(p,p')}=h_{r,s}^{(p,p')}-\frac{c}{24}</math><br>
	+
	+	Let's specify p=2, p'=2k+1, r=1, s=i
	+
	+	<math>\sum_{n=-\infty}^{\infty}(q^{2(2k+1)n^2+(2k+1-2i)n}-q^{(2n+1)((2k+1)n+i)})=\sum_{n=-\infty}^{\infty}(q^{2(2k+1)n^2+(2k+1-2i)n}-\sum_{n=-\infty}^{\infty}q^{(2n+1)((2k+1)n+i)})</math>
	+
	+	<math>=\sum_{n=-\infty}^{\infty}(q^{2(2k+1)n^2+(2k+1-2i)n}-\sum_{n=-\infty}^{\infty}q^{(2(-n)+1)(-(2k+1)(-n)+i)})</math>
	+
	+	<math>=\sum_{n=-\infty}^{\infty}(q^{2n\left[(2k+1)(2n)+(2k+1-2i)\right]/2}-\sum_{n=-\infty}^{\infty}q^{(2n-1)\left[(2k+1)(2n-1)+2k-2i+1\right]/2}</math>
	+
	+	<math>=\sum_{n=-\infty}^{\infty}(-1)^{n}q^{\frac{(2k+1)n^2}{2}}q^{\frac{n(2k-2i+1)}{2}}</math>
	+
	+	<math>=\sum_{n=-\infty}^{\infty}(-1)^n(q^{\frac{(2k-2i+1)}{2}})^{n}(q^{\frac{(2k+1)}{2}})^{n^2}=\prod_{m=1}^{\infty}(1-q^{(2k+1)m})(1-q^{\frac{2k-2i+1}{2}}q^{\frac{2k+1}{2}(2m-1)})(1-q^{-\frac{2k-2i+1}{2}}q^{\frac{2k+1}{2}(2m-1)})</math>
	+
	+	<math>=\prod_{m=1}^{\infty}(1-q^{(2k+1)m})(1-q^{(2k+1)m-i})(1-q^{(2k+1)m-(2k-i+1)})</math>
	+
	+	Thus,
	+
	+	<math>\chi_{r,s}^{(p,p')}=q^{\Delta_{r,s}^{(p,p')}}\prod_{n\neq 0,\pm i\pmod {2k+1}}(1-q^n)^{-1}</math>
	+
	+	Using a result from [http://pythagoras0.springnote.com/pages/9409120 베일리 격자(Bailey lattice)]
	+
	+	<math>\sum_{n_1\geq\cdots\geq n_{k-1}\geq0}\frac{q^{n_1^2+\cdots+n_{k-1}^2+n_i+\cdots+n_{k-1}}}{(q)_{n_{1}-n_{2}}\cdots (q)_{n_{k-2}-n_{k-1}}(q)_{n_{k-1}}}=\sum_{n=-\infty}^{\infty}(-1)^{n}q^{\frac{(2k+1)n^2}{2}}q^{\frac{n(2k-2i+1)}{2}}=\prod_{n\neq 0,\pm i\pmod {2k+1}}(1-q^n)^{-1}</math>
	+
	+
	+
	+
	+
	+	==asymptotic analysis of Andrews-Gordon identity==
	+	* non-unitary <math>c(2,k+2)</math> [[minimal models]]
	+	:<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>
	+	* non-unitary <math>c(2,2k+1)</math> [[minimal models]]
	+	$$
	+	\sum_{i=1}^{2k-2}L\left(\frac{\sin^2\frac{\pi}{2k+1}}{\sin^2\frac{(i+1)\pi}{2k+1}}\right)=\frac{2(2k-2)}{2k+1}\cdot \frac{\pi^2}{6}
	+	$$
	+	$$
	+	\sum_{i=1}^{k-1}L\left(\frac{\sin^2\frac{\pi}{2k+1}}{\sin^2\frac{(i+1)\pi}{2k+1}}\right)=\frac{2k-2}{2k+1}\cdot \frac{\pi^2}{6}
	+	$$
	+
	+
	+
	+	==different expressions for central charge==
	+
	+	* from above<br><math>h_i-c(2,2k+1)/24</math><br><math>c(2,2k+1)=1-\frac{3(2k-1)^2}{2k+1}</math><br><math>h_i=-\frac{(i-1)(2k-i)}{2(2k+1)}</math>, <math>i\in \{1,2, \cdots,k\}</math><br>
	+	* L-values<br><math>\frac{k}{12}+\frac{2k+1}{12}-\frac{j(N-j)}{2N}</math><br>
	+
	+
	+
	+
	+
	+	==Dirichlet L-function==
	+
	+	* [http://pythagoras0.springnote.com/pages/4562847 디리클레 L-함수]
	+
	+	<math>L(-1, \chi) = \frac{1}{2f}\sum_{n=1}^{\infty}{\chi(n)}{n}</math>
	+
	+	<math>n\geq 1</math> 이라 하자. 일반적으로 <math>\chi\neq 1</math>인 primitive 준동형사상 <math>\chi \colon(\mathbb{Z}/f\mathbb{Z})^\times \to \mathbb C^{*}</math>에 대하여 <math>L(1-n,\chi)</math>의 값은 다음과 같이 주어진다
	+	:<math>L(1-n,\chi)=-\frac{f^{n-1}}{n}\sum_{(a,f)=1}\chi(a)B_n(\frac{a}{f})</math>
	+	:<math>L(-1,\chi)=L(1-2,\chi)=-\frac{f}{2}\sum_{(a,f)=1}\chi(a)B_2(\frac{a}{f})</math>
	+
	+	여기서 <math>B_n(x)</math> 는 [http://pythagoras0.springnote.com/pages/4346717 베르누이 다항식](<math>B_0(x)=1</math>, <math>B_1(x)=x-1/2</math>, <math>B_2(x)=x^2-x+1/6</math>, <math>\cdots</math>)
	+
	+
	+
	+	Let N=2k+1
	+
	+	<math>\omega=\exp \frac{2\pi i}{2k+1}</math>
	+
	+	G: group of Dirichlet characters of conductor N which maps -1 to 1
	+
	+	G has order k and cyclic generated by <math>\chi</math>
	+
	+	<math>c_i=\frac{1}{2k}\sum_{s=1}^{k}\omega^{is}L(-1,\chi^s)</math>
	+
	+	Then,
	+
	+	<math>c_i=-\frac{2k+1}{12}+\frac{j(N-j)}{2N}</math>
	+
	+	where j satisfies <math>\chi(j)=\omega^{k-i}</math>
	+
	+	Vacuum energy is given by
	+
	+	<math>d_i=\frac{1}{2}L(-1,\chi^{k})-c_i</math>
	+
	+
	+
	+	Since
	+
	+	<math>L(-1,\chi^{k})=\frac{N-1}{12}=\frac{k}{6}</math>, the vacuum energy
	+
	+	<math>d_i=\frac{1}{2}L(-1,\chi^{k})-c_i=\frac{k}{12}+\frac{2k+1}{12}-\frac{j(N-j)}{2N}=\frac{j(2k+1-j)}{2(2k+1)}-\frac{k+1}{12}</math>.
	+
	+	These are equal to <math>{h_i-c/24}</math>
	+
	+
	+
	+
	+
	+	# k := 5<br> f[k_, j_] := (2 k)/<br> 24 + ((2 k + 1)/12 - (j (2 k + 1 - j))/(2 (2 k + 1)))<br> Table[{j, f[k, j]}, {j, 1, 2 k}] // TableForm<br> Table[{j, -24f[k, j]}, {j, 1, 2 k}] // TableForm<br> d[k_, j_] := (2 (k - j) + 1)^2/(8 (2 k + 1)) - 1/24<br> Table[{j, d[k, j]}, {j, 1, 2 k}] // TableForm<br> Table[{j, -24d[k, j]}, {j, 1, 2 k}] // TableForm<br> cef[k_, j_] := -((j (2 k - 1 - j))/(2 (2 k +<br> 1))) - (1 - (3 (2 k - 1)^2)/(2 k + 1))/24<br> Table[{j, cef[k, j]}, {j, 0, 2 k - 1}] // TableForm<br> Table[{j, -24*cef[k, j]}, {j, 0, 2 k - 1}] // TableForm
	+
	+
	+
	+
	+
	+	# w := Exp[2 PiI1/k]<br> L[j_] := -(2 k + 1)/2<br> Sum[DirichletCharacter[2 k + 1, j, a]<br> BernoulliB[2, a/(2 k + 1)], {a, 1, 2 k}]<br> c[k_, i_] := 1/(2 k) Sum[w^(is)L[Mod[3*s, 2 k]], {s, 1, k}]<br> Table[DiscretePlot[{Re[DirichletCharacter[2 k + 1, j, a]],<br> Im[DirichletCharacter[2 k + 1, j, a]]}, {a, 0, 2 k + 1},<br> PlotLabel -> j], {j, 1, EulerPhi[2 k + 1]}]<br> Table[c[i], {i, 1, 2 k}]
	+	[[분류:conformal field theory]]
	+	[[분류:math and physics]]
	+	[[분류:math and physics]]
	+	[[분류:minimal models]]
	+	[[분류:migrate]]

← 이전 판		2020년 11월 14일 (토) 03:00 판
1번째 줄:		1번째 줄:
−	~~==introduction==~~

−
−
−
−
−
−
−	~~==central charge and conformal dimensions==~~
−
−	* central charge<br><math>c(2,2k+1)=1-\frac{3(2k-1)^2}{2k+1}</math><br>
−	* primary fields have conformal dimensions<br><math>h_j=-\frac{j(2k-1-j)}{2(2k+1)}</math>, <math>j\in \{0,1,\cdots,k-1\}</math> or by setting i=j+1<br><math>h_i=-\frac{(i-1)(2k-i)}{2(2k+1)}</math> <math>i\in \{1,2, \cdots,k\}</math> (this is An's notation in his paper)<br>
−	* effective central charge<br><math>c_{eff}=c-24h_{min}</math><br><math>c_{eff}=\frac{2k-2}{2k+1}</math><br>
−
−
−
−
−
−	~~==character formula and Andrew-Gordon identity==~~
−
−	* [[Andrews-Gordon identity]]
−	* character functions<br><math>\chi_i(\tau)=q^{h_i-c/24}\prod_{n\neq 0,\pm i\pmod {2k+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>
−	* [[bosonic characters of Virasoro minimal models(Rocha-Caridi formula)]]<br><math>\chi_{r,s}^{(p,p')}=\frac{q^{\Delta_{r,s}^{(p,p')}}}{(q)_{\infty}}\sum_{n=-\infty}^{\infty}(q^{pp'n^2+(rp'-sp)n}-q^{(pn+r)(p'n+s)})</math><br><math>\Delta_{r,s}^{(p,p')}=h_{r,s}^{(p,p')}-\frac{c}{24}</math><br>
−
−	~~Let's specify p=2, p'=2k+1, r=1, s=i~~
−
−	~~<math>\sum_{n=-\infty}^{\infty}(q^{2(2k+1)n^2+(2k+1-2i)n}-q^{(2n+1)((2k+1)n+i)})=\sum_{n=-\infty}^{\infty}(q^{2(2k+1)n^2+(2k+1-2i)n}-\sum_{n=-\infty}^{\infty}q^{(2n+1)((2k+1)n+i)})</math>~~
−
−	~~<math>=\sum_{n=-\infty}^{\infty}(q^{2(2k+1)n^2+(2k+1-2i)n}-\sum_{n=-\infty}^{\infty}q^{(2(-n)+1)(-(2k+1)(-n)+i)})</math>~~
−
−	~~<math>=\sum_{n=-\infty}^{\infty}(q^{2n\left[(2k+1)(2n)+(2k+1-2i)\right]/2}-\sum_{n=-\infty}^{\infty}q^{(2n-1)\left[(2k+1)(2n-1)+2k-2i+1\right]/2}</math>~~
−
−	~~<math>=\sum_{n=-\infty}^{\infty}(-1)^{n}q^{\frac{(2k+1)n^2}{2}}q^{\frac{n(2k-2i+1)}{2}}</math>~~
−
−	<math>=\sum_{n=-\infty}^{\infty}(-1)^n(q^{\frac{(2k-2i+1)}{2}})^{n}(q^{\frac{(2k+1)}{2}})^{n^2}=\prod_{m=1}^{\infty}(1-q^{(2k+1)m})(1-q^{\frac{2k-2i+1}{2}}q^{\frac{2k+1}{2}(2m-1)})(1-q^{-\frac{2k-2i+1}{2}}q^{\frac{2k+1}{2}(2m-1)})</math>
−
−	~~<math>=\prod_{m=1}^{\infty}(1-q^{(2k+1)m})(1-q^{(2k+1)m-i})(1-q^{(2k+1)m-(2k-i+1)})</math>~~
−
−	~~Thus,~~
−
−	~~<math>\chi_{r,s}^{(p,p')}=q^{\Delta_{r,s}^{(p,p')}}\prod_{n\neq 0,\pm i\pmod {2k+1}}(1-q^n)^{-1}</math>~~
−
−	~~Using a result from [http://pythagoras0.springnote.com/pages/9409120 베일리 격자(Bailey lattice)]~~
−
−	<math>\sum_{n_1\geq\cdots\geq n_{k-1}\geq0}\frac{q^{n_1^2+\cdots+n_{k-1}^2+n_i+\cdots+n_{k-1}}}{(q)_{n_{1}-n_{2}}\cdots (q)_{n_{k-2}-n_{k-1}}(q)_{n_{k-1}}}=\sum_{n=-\infty}^{\infty}(-1)^{n}q^{\frac{(2k+1)n^2}{2}}q^{\frac{n(2k-2i+1)}{2}}=\prod_{n\neq 0,\pm i\pmod {2k+1}}(1-q^n)^{-1}</math>
−
−
−
−
−
−	~~==asymptotic analysis of Andrews-Gordon identity==~~
−	* non-unitary <math>c(2,k+2)</math> [[minimal models]]
−	~~:<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>~~
−	* non-unitary <math>c(2,2k+1)</math> [[minimal models]]
−	$$
−	~~\sum_{i=1}^{2k-2}L\left(\frac{\sin^2\frac{\pi}{2k+1}}{\sin^2\frac{(i+1)\pi}{2k+1}}\right)=\frac{2(2k-2)}{2k+1}\cdot \frac{\pi^2}{6}~~
−	$$
−	$$
−	~~\sum_{i=1}^{k-1}L\left(\frac{\sin^2\frac{\pi}{2k+1}}{\sin^2\frac{(i+1)\pi}{2k+1}}\right)=\frac{2k-2}{2k+1}\cdot \frac{\pi^2}{6}~~
−	$$
−
−
−
−	~~==different expressions for central charge==~~
−
−	* from above<br><math>h_i-c(2,2k+1)/24</math><br><math>c(2,2k+1)=1-\frac{3(2k-1)^2}{2k+1}</math><br><math>h_i=-\frac{(i-1)(2k-i)}{2(2k+1)}</math>, <math>i\in \{1,2, \cdots,k\}</math><br>
−	* L-values<br><math>\frac{k}{12}+\frac{2k+1}{12}-\frac{j(N-j)}{2N}</math><br>
−
−
−
−
−
−	~~==Dirichlet L-function==~~
−
−	* [http://pythagoras0.springnote.com/pages/4562847 디리클레 L-함수]
−
−	~~<math>L(-1, \chi) = \frac{1}{2f}\sum_{n=1}^{\infty}{\chi(n)}{n}</math>~~
−
−	<math>n\geq 1</math> 이라 하자. 일반적으로 <math>\chi\neq 1</math>인 primitive 준동형사상 <math>\chi \colon(\mathbb{Z}/f\mathbb{Z})^\times \to \mathbb C^{*}</math>에 대하여 <math>L(1-n,\chi)</math>의 값은 다음과 같이 주어진다
−	~~:<math>L(1-n,\chi)=-\frac{f^{n-1}}{n}\sum_{(a,f)=1}\chi(a)B_n(\frac{a}{f})</math>~~
−	~~:<math>L(-1,\chi)=L(1-2,\chi)=-\frac{f}{2}\sum_{(a,f)=1}\chi(a)B_2(\frac{a}{f})</math>~~
−
−	여기서 <math>B_n(x)</math> 는 [http://pythagoras0.springnote.com/pages/4346717 베르누이 다항식](<math>B_0(x)=1</math>, <math>B_1(x)=x-1/2</math>, <math>B_2(x)=x^2-x+1/6</math>, <math>\cdots</math>)
−
−
−
−	~~Let N=2k+1~~
−
−	~~<math>\omega=\exp \frac{2\pi i}{2k+1}</math>~~
−
−	~~G: group of Dirichlet characters of conductor N which maps -1 to 1~~
−
−	~~G has order k and cyclic generated by <math>\chi</math>~~
−
−	~~<math>c_i=\frac{1}{2k}\sum_{s=1}^{k}\omega^{is}L(-1,\chi^s)</math>~~
−
−	~~Then,~~
−
−	~~<math>c_i=-\frac{2k+1}{12}+\frac{j(N-j)}{2N}</math>~~
−
−	~~where j satisfies <math>\chi(j)=\omega^{k-i}</math>~~
−
−	~~Vacuum energy is given by~~
−
−	~~<math>d_i=\frac{1}{2}L(-1,\chi^{k})-c_i</math>~~
−
−
−
−	~~Since~~
−
−	~~<math>L(-1,\chi^{k})=\frac{N-1}{12}=\frac{k}{6}</math>, the vacuum energy~~
−
−	~~<math>d_i=\frac{1}{2}L(-1,\chi^{k})-c_i=\frac{k}{12}+\frac{2k+1}{12}-\frac{j(N-j)}{2N}=\frac{j(2k+1-j)}{2(2k+1)}-\frac{k+1}{12}</math>.~~
−
−	~~These are equal to <math>{h_i-c/24}</math>~~
−
−
−
−
−
−	# k := 5<br> f[k_, j_] := (2 k)/<br> 24 + ((2 k + 1)/12 - (j (2 k + 1 - j))/(2 (2 k + 1)))<br> Table[{j, f[k, j]}, {j, 1, 2 k}] // TableForm<br> Table[{j, -24f[k, j]}, {j, 1, 2 k}] // TableForm<br> d[k_, j_] := (2 (k - j) + 1)^2/(8 (2 k + 1)) - 1/24<br> Table[{j, d[k, j]}, {j, 1, 2 k}] // TableForm<br> Table[{j, -24d[k, j]}, {j, 1, 2 k}] // TableForm<br> cef[k_, j_] := -((j (2 k - 1 - j))/(2 (2 k +<br> 1))) - (1 - (3 (2 k - 1)^2)/(2 k + 1))/24<br> Table[{j, cef[k, j]}, {j, 0, 2 k - 1}] // TableForm<br> Table[{j, -24*cef[k, j]}, {j, 0, 2 k - 1}] // TableForm
−
−
−
−
−
−	# w := Exp[2 PiI1/k]<br> L[j_] := -(2 k + 1)/2<br> Sum[DirichletCharacter[2 k + 1, j, a]<br> BernoulliB[2, a/(2 k + 1)], {a, 1, 2 k}]<br> c[k_, i_] := 1/(2 k) Sum[w^(is)L[Mod[3*s, 2 k]], {s, 1, k}]<br> Table[DiscretePlot[{Re[DirichletCharacter[2 k + 1, j, a]],<br> Im[DirichletCharacter[2 k + 1, j, a]]}, {a, 0, 2 k + 1},<br> PlotLabel -> j], {j, 1, EulerPhi[2 k + 1]}]<br> Table[c[i], {i, 1, 2 k}]
−	~~[[분류:conformal field theory]]~~
−	~~[[분류:math and physics]]~~
−	~~[[분류:math and physics]]~~
−	~~[[분류:minimal models]]~~

@@ 54번째 줄: / 54번째 줄: @@
 :<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>
 *  non-unitary <math>c(2,2k+1)</math> [[minimal models]]
 $$
 \sum_{i=1}^{k-1}L\left(\frac{\sin^2\frac{\pi}{2k+1}}{\sin^2\frac{(i+1)\pi}{2k+1}}\right)=\frac{2k-2}{2k+1}\cdot \frac{\pi^2}{6}

← 이전 판	2020년 11월 16일 (월) 17:56 판
(차이 없음)

@@ 51번째 줄: / 51번째 줄: @@
 ==asymptotic analysis of Andrews-Gordon identity==
 *  non-unitary <math>c(2,k+2)</math> [[minimal models]]
-:<math>\sum_{i=1}^{[k/2]}L\left(\frac{\sin^2\frac{\pi}{k+2}}{\sin^2\frac{(i+1)\pi}{k+2}}\right)=\frac{k-1}{k+2}\cdot \frac{\pi^2}{6}</math><br>
+:<math>\sum_{i=1}^{[k/2]}L\left(\frac{\sin^2\frac{\pi}{k+2}}{\sin^2\frac{(i+1)\pi}{k+2}}\right)=\frac{k-1}{k+2}\cdot \frac{\pi^2}{6}</math>
 *  non-unitary <math>c(2,2k+1)</math> [[minimal models]]
-:<math>\sum_{i=1}^{2k-1}L\left(\frac{\sin^2\frac{\pi}{2k+1}}{\sin^2\frac{(i+1)\pi}{2k+1}}\right)=\frac{2k-2}{2k+1}\cdot \frac{\pi^2}{6}</math><br>
+:<math>\sum_{i=1}^{2k-1}L\left(\frac{\sin^2\frac{\pi}{2k+1}}{\sin^2\frac{(i+1)\pi}{2k+1}}\right)=\frac{2k-2}{2k+1}\cdot \frac{\pi^2}{6}</math>

← 이전 판		2013년 2월 26일 (화) 08:37 판
128번째 줄:		128번째 줄:
	[[분류:math and physics]]		[[분류:math and physics]]
	[[분류:math and physics]]		[[분류:math and physics]]
		+	[[분류:minimal models]]