"Y-system and functional dilogarithm identities"의 두 판 사이의 차이

2020년 11월 13일 (금) 08:38 판

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

related items

computational resource

https://docs.google.com/file/d/0B8XXo8Tve1cxVHpWMS10SmJlSmM/edit

expositions

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.

@@ 135번째 줄: / 135번째 줄: @@
 [[분류:dilogarithm]]
 [[분류:Y-system]]
+[[분류:migrate]]