Y-system and functional dilogarithm identities

수학노트
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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


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.
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