带次可加标号的箭图：分类和可积性
Quivers with subadditive labelings: classification and integrability
摘要：
第二作者引入了箭图的严格次可加、次可加和弱次可加标号，推广了Vinberg对无向图的定义。在我们以前的工作中，我们已经表明，箭图与严格次加标签正是表现Zamolodchikov周期性的箭图。在本文中，我们分类所有的箭图与次加标签。我们猜想他们表现出某种形式的可积性，即，作为$T$ -系统动力学进行，在每个顶点的值满足线性递归。相反，我们表明，在这个意义上的每一个可积的必然是我们的分类中的$19$项目之一。对于$\hat A \otimes A$类型的箭图，我们用圆柱多米诺平铺的分拆函数表示递归的系数，称为\emph{Goncharov-Kenyon Hamiltonians}。我们还考虑热带$T$ -系统的类型$\hat A \otimes A$和解释如何仿射切片表现出孤子的行为，即孤子的分辨率和速度守恒。在整个过程中，我们推测如何在纸上的结果预计推广从$\hat A \otimes A$到所有其他颤抖在我们的分类。
英文摘要：
Strictly subadditive, subadditive and weakly subadditive labelings of quivers were introduced by the second author, generalizing Vinberg's definition for undirected graphs. In our previous work we have shown that quivers with strictly subadditive labelings are exactly the quivers exhibiting Zamolodchikov periodicity. In this paper, we classify all quivers with subadditive labelings. We conjecture them to exhibit a certain form of integrability, namely, as the $T$-system dynamics proceeds, the values at each vertex satisfy a linear recurrence. Conversely, we show that every quiver integrable in this sense is necessarily one of the $19$ items in our classification. For the quivers of type $\hat A \otimes A$ we express the coefficients of the recurrences in terms of the partition functions for domino tilings of a cylinder, called \emph{Goncharov-Kenyon Hamiltonians}. We also consider tropical $T$-systems of type $\hat A \otimes A$ and explain how affine slices exhibit solitonic behavior, i.e. soliton resolution and speed conservation. Throughout, we conjecture how the results in the paper are expected to generalize from $\hat A \otimes A$ to all other quivers in our classification.
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