"Non-unitary c(2,k+2) minimal models"의 두 판 사이의 차이

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

related items

Andrews-Gordon identity

computational resource

https://docs.google.com/file/d/0B8XXo8Tve1cxdXRlbU40OExkeW8/edit

@@ 1번째 줄: / 1번째 줄: @@
-==introduction==
+==non-unitary <math>c(2,k+2)</math> minimal models==
-*  important<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<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
+*  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
-==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>
-*  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>
-*  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>
-#  (*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>
-==history==
-* http://www.google.com/search?hl=en&tbs=tl:1&q=
 ==related items==
@@ 41번째 줄: / 19번째 줄: @@
 * [[Andrews-Gordon identity]]
-==encyclopedia==
-* http://en.wikipedia.org/wiki/
-* http://www.scholarpedia.org/
-* http://www.proofwiki.org/wiki/
-* Princeton companion to mathematics([[2910610/attachments/2250873|Companion_to_Mathematics.pdf]])
-==books==
-* [[2010년 books and articles]]<br>
-* http://gigapedia.info/1/
-* http://gigapedia.info/1/
-* http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=
-==expositions==
-==articles==
-* http://www.ams.org/mathscinet
-* http://www.zentralblatt-math.org/zmath/en/
-* http://arxiv.org/
-* http://www.pdf-search.org/
-* http://pythagoras0.springnote.com/
-* [http://math.berkeley.edu/%7Ereb/papers/index.html http://math.berkeley.edu/~reb/papers/index.html]
-* http://dx.doi.org/
-==question and answers(Math Overflow)==
-* http://mathoverflow.net/search?q=
-* http://mathoverflow.net/search?q=
-==blogs==
-*  구글 블로그 검색<br>
-**  http://blogsearch.google.com/blogsearch?q=<br>
-** http://blogsearch.google.com/blogsearch?q=
-* http://ncatlab.org/nlab/show/HomePage
-==experts on the field==
-* http://arxiv.org/
+==computational resource==
+* https://docs.google.com/file/d/0B8XXo8Tve1cxdXRlbU40OExkeW8/edit
-==links==
-* [http://detexify.kirelabs.org/classify.html Detexify2 - LaTeX symbol classifier]
+[[분류:conformal field theory]]
-* [http://pythagoras0.springnote.com/pages/1947378 수식표현 안내]
+[[분류:math and physics]]
-* [http://www.research.att.com/%7Enjas/sequences/index.html The On-Line Encyclopedia of Integer Sequences]
+[[분류:minimal models]]
-* http://functions.wolfram.com/
+[[분류:migrate]]

"Non-unitary c(2,k+2) minimal models"의 두 판 사이의 차이

2020년 12월 28일 (월) 05:17 기준 최신판

non-unitary \(c(2,k+2)\) minimal models

related items

computational resource

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