"Y-system and functional dilogarithm identities"의 두 판 사이의 차이
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==introduction== | ==introduction== | ||
| + | * {{수학노트|url=함수_다이로그_항등식(functional_dilogarithm_identity)}} | ||
| + | * <math>\mathbb{Y}(X,X')</math> the order of <math>X</math> and <math>X'</math> matters! | ||
| + | * Caracciolo, R., F. Gliozzi, and R. Tateo in 1999 proves the functional dilogarithm identities associated ADExADE Y-systems assuming the periodicity of restricted T-system | ||
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| − | + | ==main results== | |
| + | ===level restricted Y-system=== | ||
| + | * Suppose that a family of positive real numbers | ||
| + | :<math>\{Y^{(a)}_m(u)\mid a\in I, 1\leq m \leq t_a\ell-1, | ||
| + | u\in \mathbb{Z}\}</math> | ||
| + | satisfy the level <math>\ell</math> restricted Y-system for <math>\mathfrak{g}</math>. | ||
| + | Then, the following identities hold: | ||
| + | \begin{align}\label{t:eq:DI2} | ||
| + | \frac{6}{\pi^2}\sum_{a\in I}\sum_{m=1}^{t_a\ell-1} | ||
| + | \sum_{u=0}^{2(h^{\vee}+\ell)-1} | ||
| + | L\left( | ||
| + | \frac{Y^{(a)}_m(u)}{1+Y^{(a)}_m(u)} | ||
| + | \right) | ||
| + | &= | ||
| + | 2t(\ell h - h^{\vee})\mathrm{rank}\,\mathfrak{g}, | ||
| + | \end{align} | ||
| + | where <math>h</math> is the Coxeter number of <math>{\mathfrak g}</math>. | ||
| + | * in simply-laced case, we get | ||
| + | :<math> | ||
| + | \begin{align} | ||
| + | \frac{6}{\pi^2}\sum_{a\in I}\sum_{m=1}^{\ell-1} | ||
| + | \sum_{u=0}^{2(h+\ell)-1} | ||
| + | L\left( | ||
| + | \frac{Y^{(a)}_m(u)}{1+Y^{(a)}_m(u)} | ||
| + | \right) | ||
| + | &= | ||
| + | 2(\ell h - h)r=2h(\ell-1)r | ||
| + | \end{align} | ||
| + | </math> | ||
| − | + | ===Y-systems for a pair of Dynkin diagrams=== | |
| − | + | * Bloch group element | |
| − | * | + | :<math> |
| − | + | \sum_{(\mathbf{i},u)\in S_{+}} Y_{\mathbf{i}}(u)\wedge (1+Y_{\mathbf{i}}(u))=0\in \Lambda^2 \mathbb{Q}(y)^{\times} | |
| − | + | </math> where <math>S_{+}=\{(\mathbf{i},u) |0\leq u \leq 2(h+h')-1,(\mathbf{i},u)\in P_{+}\}</math>. | |
| − | + | * functional dilogarithm identity | |
| + | :<math> | ||
| + | \sum_{(\mathbf{i},u)\in S_{+}}L\left(\frac{Y_\mathbf{i}(u)}{1+Y_\mathbf{i}(u)}\right)=h r r' L(1) | ||
| + | </math> | ||
| − | |||
| − | + | ==bicoloring== | |
| + | * what's <math>P_{+}</math>? | ||
| + | * We give an alternate bicoloring on the pair of Dynkin diagrams. Let us fix bipartite decompositions of <math>I</math> and <math>I'</math>. | ||
| + | * Let <math> \mathbf{I}= I\times I'</math> and <math>\mathbf{I}=\mathbf{I}_{+}\sqcup \mathbf{I}_{-}</math> where <math>\mathbf{I}_{+}=(I_{+}\times I'_{+}) \sqcup (I_{-}\times I'_{-})</math> and <math>\mathbf{I}_{-}=(I_{+}\times I'_{-}) \sqcup (I_{-}\times I'_{+})</math>. | ||
| + | * Let <math>\epsilon : \mathbf{I}\to \{1,-1\}</math> be the function defined by <math>\epsilon(\mathbf{i})=\pm 1</math> for <math>\mathbf{i}\in \mathbf{I}_{\pm}</math> and <math>P_{\pm} =\{(\mathbf{i},u)\in \mathbf{I}\times\mathbb{Z}| \epsilon(\mathbf{i})(-1)^u=\pm 1\}</math>. | ||
| + | * Roughly speaking, we want our alternate bicoloring interchanges their colors as <math>u\in \mathbb{Z}</math> changes by 1. | ||
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| − | + | ==an example== | |
| − | + | * compute <math>\mathbb{Y}(A_2,A_1)</math> explicitly | |
| − | + | * <math>y_{m-1}y_{m+1}=y_m+1</math> | |
| − | + | * Start with two variables <math>y_1,y_2</math>. | |
| − | == | + | * <math>y_3y_1=y_2+1</math>. so <math>y_3=\frac{y_2+1}{y_1}</math> |
| − | + | * <math>y_2y_4=y_3+1 </math>implies <math>y_4=\frac{y_3+1}{y_2}=\frac{y_1+y_2+1}{y_1y_2}</math> | |
| − | * <math> | + | * <math>y_3y_5=y_4+1</math> implies <math>y_5=\frac{y_4+1}{y_3}= \frac{y_1+1}{y_2}</math> we are getting Laurent polynomials |
| − | + | * <math>y_4y_6=y_5</math> implies <math>y_6=\frac{y_5+1}{y_4}= \frac{\frac{y_1+1}{y_2}+1}{\frac{y_1+y_2+1}{y_1y_2}}=\frac{y_1(y_1+1)+y_1y_2}{y_1+y_2+1}=y_1</math> | |
| − | + | * [[rank 2 cluster algebra]] | |
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| − | * [[rank 2 cluster algebra | ||
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| − | == | + | ===observations=== |
| + | * we saw that :<math>S=\left\{x,y,\frac{y+1}{x},\frac{x+y+1}{x y},\frac{x+1}{y}\right\}</math> forms a half-period of <math>\mathbb{Y}(A_2,A_1)</math>. | ||
| + | * So we have <math>r=2,h=3</math> and <math>r'=1,h'=2</math>. | ||
| + | * They are all Laurent polynomials in <math>x</math> and <math>y</math>. | ||
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| − | + | ===dilogarithm identities=== | |
| − | + | * From this, one can get functional dilogarithm identities | |
| − | + | \begin{align} | |
| − | + | &\sum_{a\in S}L\left(\frac{a}{1+a}\right) \notag \\ | |
| − | == | + | =& |
| + | L\left(\frac{x}{1+x}\right)+L\left(\frac{y}{1+y}\right)+L\left(\frac{1+y}{x (1+\frac{1+y}{x})}\right)+L\left(\frac{1+x+y}{x y (1+\frac{1+x+y}{x y})}\right)+L\left(\frac{1+x}{(1+\frac{1+x}{y}) y}\right) \notag | ||
| + | \\ | ||
| + | =& L\left(\frac{x}{x+1}\right)+L\left(\frac{y}{y+1}\right)+L\left(\frac{y+1}{x+y+1}\right)+L\left(\frac{x+y+1}{x y+x+y+1}\right)+L\left(\frac{x+1}{x+y+1}\right) \notag \\ | ||
| + | =&3L(1)=\frac{\pi^2}{2} \notag | ||
| + | \end{align} | ||
| + | and | ||
| + | \begin{align} | ||
| + | &\sum_{a\in S}L\left(\frac{1}{1+a}\right) \notag \\ | ||
| + | =& | ||
| + | L\left(\frac{1}{x+1}\right)+L\left(\frac{1}{y+1}\right)+L\left(\frac{1}{\frac{y+1}{x}+1}\right)+L\left(\frac{1}{\frac{x+y+1}{x y}+1}\right)+L\left(\frac{1}{\frac{x+1}{y}+1}\right) \notag \\ | ||
| + | =& L\left(\frac{1}{x+1}\right)+L\left(\frac{1}{y+1}\right)+L\left(\frac{x}{x+y+1}\right)+L\left(\frac{x y}{x y+x+y+1}\right)+L\left(\frac{y}{x+y+1}\right) \notag \\ | ||
| + | =&2L(1)=\frac{\pi^2}{3} \notag. | ||
| + | \end{align} | ||
* {{수학노트|url=5항_관계식_(5-term_relation)}} | * {{수학노트|url=5항_관계식_(5-term_relation)}} | ||
| − | * [ | + | * [[Dilogarithm identity for CFT revisited]] |
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==history== | ==history== | ||
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==related items== | ==related items== | ||
| 92번째 줄: | 100번째 줄: | ||
* [[cluster algebra]] | * [[cluster algebra]] | ||
* [[Nahm's equation]] | * [[Nahm's equation]] | ||
| − | * [[Central charge | + | * [[Central charge and dilogarithm]] |
* [[dilogarithm and dilogarithm identities]] | * [[dilogarithm and dilogarithm identities]] | ||
| − | * [[Bloch group | + | * [[Bloch group]] |
| − | + | ==computational resource== | |
| + | * https://docs.google.com/file/d/0B8XXo8Tve1cxVHpWMS10SmJlSmM/edit | ||
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==expositions== | ==expositions== | ||
| 128번째 줄: | 113번째 줄: | ||
* [http://fuji.cec.yamanashi.ac.jp/%7Ering/icra14/lecture/conf/nakanishi.pdf Periodicities in cluster algebras and dilogarithm identities] | * [http://fuji.cec.yamanashi.ac.jp/%7Ering/icra14/lecture/conf/nakanishi.pdf Periodicities in cluster algebras and dilogarithm identities] | ||
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==articles== | ==articles== | ||
| + | * Nakanishi, Tomoki. “Quantum Generalized Cluster Algebras and Quantum Dilogarithms of Higher Degrees.” arXiv:1410.0584 [math], October 2, 2014. http://arxiv.org/abs/1410.0584. | ||
| + | * Nakanishi, Tomoki. “Periodicities in Cluster Algebras and Dilogarithm Identities.” arXiv:1006.0632 [math], June 3, 2010. http://arxiv.org/abs/1006.0632. | ||
| + | * Inoue, Rei, Osamu Iyama, Bernhard Keller, Atsuo Kuniba, and Tomoki Nakanishi. 2013. “Periodicities of T and Y-Systems, Dilogarithm Identities, and Cluster Algebras II: Types C_r, F_4, and G_2.” Publications of the Research Institute for Mathematical Sciences 49 (1): 43–85. doi:10.4171/PRIMS/96. | ||
| + | * Inoue, Rei, Osamu Iyama, Bernhard Keller, Atsuo Kuniba, and Tomoki Nakanishi. 2013. “Periodicities of T and Y-Systems, Dilogarithm Identities, and Cluster Algebras I: Type B_r.” Publications of the Research Institute for Mathematical Sciences 49 (1): 1–42. doi:10.4171/PRIMS/95. | ||
| + | * Nakanishi, Tomoki. “Dilogarithm Identities for Conformal Field Theories and Cluster Algebras: Simply Laced Case.” arXiv:0909.5480 [math], September 30, 2009. http://arxiv.org/abs/0909.5480. | ||
| + | * Chapoton, Frédéric. 2005. Functional Identities for the Rogers Dilogarithm Associated to Cluster Y-Systems. Bulletin of the London Mathematical Society 37, no. 5 (October 1): 755 -760. doi:[http://dx.doi.org/10.1112/S0024609305004510 10.1112/S0024609305004510]. | ||
| + | * Caracciolo, R., F. Gliozzi, and R. Tateo. “A Topological Invariant of RG Flows in 2D Integrable Quantum Field Theories.” arXiv:hep-th/9902094, February 12, 1999. http://arxiv.org/abs/hep-th/9902094. | ||
| + | * Frenkel, Edward, and Andras Szenes. “Thermodynamic Bethe Ansatz and Dilogarithm Identities I.” arXiv:hep-th/9506215, July 2, 1995. http://arxiv.org/abs/hep-th/9506215. | ||
| + | * Gliozzi, F., and R. Tateo. “ADE Functional Dilogarithm Identities and Integrable Models.” Physics Letters B 348, no. 1–2 (March 30, 1995): 84–88. doi:[http://dx.doi.org/10.1016/0370-2693(95)00125-5 10.1016/0370-2693(95)00125-5]. | ||
| + | * Kuniba, A., and T. Nakanishi. “Rogers Dilogarithm in Integrable Systems.” arXiv:hep-th/9210025, October 5, 1992. http://arxiv.org/abs/hep-th/9210025. | ||
| + | * Kuniba, Atsuo, and Tomoki Nakanishi. “Spectra in Conformal Field Theories from the Rogers Dilogarithm.” arXiv:hep-th/9206034, June 9, 1992. http://arxiv.org/abs/hep-th/9206034. | ||
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[[분류:개인노트]] | [[분류:개인노트]] | ||
[[분류:cluster algebra]] | [[분류:cluster algebra]] | ||
[[분류:math and physics]] | [[분류:math and physics]] | ||
| − | [[분류: | + | [[분류:dilogarithm]] |
| + | [[분류:Y-system]] | ||
| + | [[분류:migrate]] | ||
2020년 12월 28일 (월) 05:19 기준 최신판
introduction
- 틀:수학노트
- \(\mathbb{Y}(X,X')\) the order of \(X\) and \(X'\) matters!
- Caracciolo, R., F. Gliozzi, and R. Tateo in 1999 proves the functional dilogarithm identities associated ADExADE Y-systems assuming the periodicity of restricted T-system
main results
level restricted Y-system
- Suppose that a family of positive real numbers
\[\{Y^{(a)}_m(u)\mid a\in I, 1\leq m \leq t_a\ell-1, u\in \mathbb{Z}\}\] satisfy the level \(\ell\) restricted Y-system for \(\mathfrak{g}\). Then, the following identities hold: \begin{align}\label{t:eq:DI2} \frac{6}{\pi^2}\sum_{a\in I}\sum_{m=1}^{t_a\ell-1} \sum_{u=0}^{2(h^{\vee}+\ell)-1} L\left( \frac{Y^{(a)}_m(u)}{1+Y^{(a)}_m(u)} \right) &= 2t(\ell h - h^{\vee})\mathrm{rank}\,\mathfrak{g}, \end{align} where \(h\) is the Coxeter number of \({\mathfrak g}\).
- in simply-laced case, we get
\[ \begin{align} \frac{6}{\pi^2}\sum_{a\in I}\sum_{m=1}^{\ell-1} \sum_{u=0}^{2(h+\ell)-1} L\left( \frac{Y^{(a)}_m(u)}{1+Y^{(a)}_m(u)} \right) &= 2(\ell h - h)r=2h(\ell-1)r \end{align} \]
Y-systems for a pair of Dynkin diagrams
- Bloch group element
\[ \sum_{(\mathbf{i},u)\in S_{+}} Y_{\mathbf{i}}(u)\wedge (1+Y_{\mathbf{i}}(u))=0\in \Lambda^2 \mathbb{Q}(y)^{\times} \] where \(S_{+}=\{(\mathbf{i},u) |0\leq u \leq 2(h+h')-1,(\mathbf{i},u)\in P_{+}\}\).
- functional dilogarithm identity
\[ \sum_{(\mathbf{i},u)\in S_{+}}L\left(\frac{Y_\mathbf{i}(u)}{1+Y_\mathbf{i}(u)}\right)=h r r' L(1) \]
bicoloring
- what's \(P_{+}\)?
- We give an alternate bicoloring on the pair of Dynkin diagrams. Let us fix bipartite decompositions of \(I\) and \(I'\).
- Let \( \mathbf{I}= I\times I'\) and \(\mathbf{I}=\mathbf{I}_{+}\sqcup \mathbf{I}_{-}\) where \(\mathbf{I}_{+}=(I_{+}\times I'_{+}) \sqcup (I_{-}\times I'_{-})\) and \(\mathbf{I}_{-}=(I_{+}\times I'_{-}) \sqcup (I_{-}\times I'_{+})\).
- Let \(\epsilon : \mathbf{I}\to \{1,-1\}\) be the function defined by \(\epsilon(\mathbf{i})=\pm 1\) for \(\mathbf{i}\in \mathbf{I}_{\pm}\) and \(P_{\pm} =\{(\mathbf{i},u)\in \mathbf{I}\times\mathbb{Z}| \epsilon(\mathbf{i})(-1)^u=\pm 1\}\).
- Roughly speaking, we want our alternate bicoloring interchanges their colors as \(u\in \mathbb{Z}\) changes by 1.
an example
- compute \(\mathbb{Y}(A_2,A_1)\) explicitly
- \(y_{m-1}y_{m+1}=y_m+1\)
- Start with two variables \(y_1,y_2\).
- \(y_3y_1=y_2+1\). so \(y_3=\frac{y_2+1}{y_1}\)
- \(y_2y_4=y_3+1 \)implies \(y_4=\frac{y_3+1}{y_2}=\frac{y_1+y_2+1}{y_1y_2}\)
- \(y_3y_5=y_4+1\) implies \(y_5=\frac{y_4+1}{y_3}= \frac{y_1+1}{y_2}\) we are getting Laurent polynomials
- \(y_4y_6=y_5\) implies \(y_6=\frac{y_5+1}{y_4}= \frac{\frac{y_1+1}{y_2}+1}{\frac{y_1+y_2+1}{y_1y_2}}=\frac{y_1(y_1+1)+y_1y_2}{y_1+y_2+1}=y_1\)
- rank 2 cluster algebra
observations
- we saw that \[S=\left\{x,y,\frac{y+1}{x},\frac{x+y+1}{x y},\frac{x+1}{y}\right\}\] forms a half-period of \(\mathbb{Y}(A_2,A_1)\).
- So we have \(r=2,h=3\) and \(r'=1,h'=2\).
- They are all Laurent polynomials in \(x\) and \(y\).
dilogarithm identities
- From this, one can get functional dilogarithm identities
\begin{align} &\sum_{a\in S}L\left(\frac{a}{1+a}\right) \notag \\ =& L\left(\frac{x}{1+x}\right)+L\left(\frac{y}{1+y}\right)+L\left(\frac{1+y}{x (1+\frac{1+y}{x})}\right)+L\left(\frac{1+x+y}{x y (1+\frac{1+x+y}{x y})}\right)+L\left(\frac{1+x}{(1+\frac{1+x}{y}) y}\right) \notag \\ =& L\left(\frac{x}{x+1}\right)+L\left(\frac{y}{y+1}\right)+L\left(\frac{y+1}{x+y+1}\right)+L\left(\frac{x+y+1}{x y+x+y+1}\right)+L\left(\frac{x+1}{x+y+1}\right) \notag \\ =&3L(1)=\frac{\pi^2}{2} \notag \end{align} and \begin{align} &\sum_{a\in S}L\left(\frac{1}{1+a}\right) \notag \\ =& L\left(\frac{1}{x+1}\right)+L\left(\frac{1}{y+1}\right)+L\left(\frac{1}{\frac{y+1}{x}+1}\right)+L\left(\frac{1}{\frac{x+y+1}{x y}+1}\right)+L\left(\frac{1}{\frac{x+1}{y}+1}\right) \notag \\ =& L\left(\frac{1}{x+1}\right)+L\left(\frac{1}{y+1}\right)+L\left(\frac{x}{x+y+1}\right)+L\left(\frac{x y}{x y+x+y+1}\right)+L\left(\frac{y}{x+y+1}\right) \notag \\ =&2L(1)=\frac{\pi^2}{3} \notag. \end{align}
history
- 9-term relation and its accessibility
- cluster algebra
- Nahm's equation
- Central charge and dilogarithm
- dilogarithm and dilogarithm identities
- Bloch group
computational resource
expositions
- Dilogarithm identities in conformal field theory and cluster algebras
- Periodicities in cluster algebras and dilogarithm identities
articles
- Nakanishi, Tomoki. “Quantum Generalized Cluster Algebras and Quantum Dilogarithms of Higher Degrees.” arXiv:1410.0584 [math], October 2, 2014. http://arxiv.org/abs/1410.0584.
- Nakanishi, Tomoki. “Periodicities in Cluster Algebras and Dilogarithm Identities.” arXiv:1006.0632 [math], June 3, 2010. http://arxiv.org/abs/1006.0632.
- Inoue, Rei, Osamu Iyama, Bernhard Keller, Atsuo Kuniba, and Tomoki Nakanishi. 2013. “Periodicities of T and Y-Systems, Dilogarithm Identities, and Cluster Algebras II: Types C_r, F_4, and G_2.” Publications of the Research Institute for Mathematical Sciences 49 (1): 43–85. doi:10.4171/PRIMS/96.
- Inoue, Rei, Osamu Iyama, Bernhard Keller, Atsuo Kuniba, and Tomoki Nakanishi. 2013. “Periodicities of T and Y-Systems, Dilogarithm Identities, and Cluster Algebras I: Type B_r.” Publications of the Research Institute for Mathematical Sciences 49 (1): 1–42. doi:10.4171/PRIMS/95.
- Nakanishi, Tomoki. “Dilogarithm Identities for Conformal Field Theories and Cluster Algebras: Simply Laced Case.” arXiv:0909.5480 [math], September 30, 2009. http://arxiv.org/abs/0909.5480.
- Chapoton, Frédéric. 2005. Functional Identities for the Rogers Dilogarithm Associated to Cluster Y-Systems. Bulletin of the London Mathematical Society 37, no. 5 (October 1): 755 -760. doi:10.1112/S0024609305004510.
- Caracciolo, R., F. Gliozzi, and R. Tateo. “A Topological Invariant of RG Flows in 2D Integrable Quantum Field Theories.” arXiv:hep-th/9902094, February 12, 1999. http://arxiv.org/abs/hep-th/9902094.
- Frenkel, Edward, and Andras Szenes. “Thermodynamic Bethe Ansatz and Dilogarithm Identities I.” arXiv:hep-th/9506215, July 2, 1995. http://arxiv.org/abs/hep-th/9506215.
- Gliozzi, F., and R. Tateo. “ADE Functional Dilogarithm Identities and Integrable Models.” Physics Letters B 348, no. 1–2 (March 30, 1995): 84–88. doi:10.1016/0370-2693(95)00125-5.
- Kuniba, A., and T. Nakanishi. “Rogers Dilogarithm in Integrable Systems.” arXiv:hep-th/9210025, October 5, 1992. http://arxiv.org/abs/hep-th/9210025.
- Kuniba, Atsuo, and Tomoki Nakanishi. “Spectra in Conformal Field Theories from the Rogers Dilogarithm.” arXiv:hep-th/9206034, June 9, 1992. http://arxiv.org/abs/hep-th/9206034.