二维Toda链、交换差分算子与全纯丛

Toda systems for Takiff algebras

We study completely integrable systems attached to Takiff algebras $\mathfrak{g}_N$, extending open Toda systems of split simple Lie algebras $\mathfrak{g}$. With respect to Darboux coordinates on coadjoint orbits $\mathcal{O}$, the potentials of the hamiltonians are products of polynomial and exponential functions. General solutions for equations of motion for $\mathfrak{g}_N$ are obtained using differential operators called jet transformations. These results are applied to a $3$-body problem based on $\mathfrak{sl}(2)$, and to an extension of soliton solutions for $A_\infty$ to associated Ta…

Atiyah-Hitchin括号和开放的Toda格

The Atiyah--Hitchin bracket and the open Toda lattice

有限非周期Toda格点的动力学是有限三对角Jacobi矩阵的等谱变形。这是已知的，因为工作的Stieltjes，这种矩阵是在一对一的对应关系与他们的外尔函数。这些是有理函数，将上半平面映射到自身.我们认为外尔函数的表示是两个多项式的商和指数表示。我们建立了这些表示和最近开发的代数-几何方法的Jacobi矩阵的反问题之间的连接。有理函数空间具有自然的Poisson结构，这是由Atiyah和Hitchin发现的。我们表明，AH结构下的线性分数变换的不变性导致两个系统的正则坐标和两个家庭的交换哈密顿。我们建立了这些系统与雅可比椭圆坐标的关系。

非紧对称空间的几何Toda方程

The geometric Toda equations for noncompact symmetric spaces

本文分类了非复非紧单实李群G对应的齐性空间G/T上τ-本原调和映射所满足的Toda方程版本，解释了其几何意义，并确定了可通过稳定性准则证解存在的完全非紧Toda对，还关联了G-Higgs丛构造。

Constrained Toda hierarchy and turning points of the Ruijsenaars-Schneider model

We introduce a new integrable hierarchy of nonlinear differential-difference equations which we call constrained Toda hierarchy (C-Toda). It can be regarded as a certain subhierarchy of the 2D Toda lattice obtained by imposing the constraint $\bar {\cal L}={\cal L}^†$ on the two Lax operators (in the symmetric gauge). We prove the existence of the tau-function of the C-Toda hierarchy and show that it is the square root of the 2D Toda lattice tau-function. In this and some other respects the C-Toda is a Toda analogue of the CKP hierarchy. It is also shown that zeros of the tau-function of ellip…

超越Pfaff-Toda的矩阵dressing

Matrix Dressing Beyond Pfaff-Toda

本文提出矩阵伪差分算子框架统一Adler-Pfaff与Pfaff-Toda层级，引入矩阵Pfaff-Toda层级，恢复连续与多分量情形，并导出相关tau函数与约化结果。

Semi-discrete Lagrangian 2-forms and the Toda hierarchy

We present a variational theory of integrable differential-difference equations (semi-discrete integrable systems). This is a natural extension of the ideas known by the names "Lagrangian multiforms" and "Pluri-Lagrangian systems", which have previously been established in both the fully discrete and fully continuous cases. The main feature of these ideas is to capture a hierarchy of commuting equations in a single variational principle. Our main example to illustrate the new semi-discrete theory of Lagrangian multiforms is the Toda lattice. This ODE describes the evolution in continuous time …

W_3共形场论、量子Boussinesq理论和边界仿射Toda理论的可积结构

Integrable structure of W_3 Conformal Field Theory, Quantum Boussinesq Theory and Boundary Affine Toda Theory

本文研究了由W_3代数生成的具有扩展共形对称性的共形场论的Yang-Baxter可积结构。我们明确地构造了作用在W_3代数的不可约最高权模上的各种T-和Q-算子。这些算子可以看作是与量子代数U_q（\hat{sl}（3））相关的可积格系统的交换转移矩阵和Q-矩阵的连续场论模拟。我们制定了几个详细的Q-运营商的某些分析特性的示意图，并提出了确切的渐近展开的T-和Q-运营商在大值的谱参数。特别地，我们证明了T-算子的渐近展开产生了W_3 CFT的运动局部积分的无穷集，它在经典极限下再现了与经典Boussinesq方程相关的守恒Hamilton算子的无穷集.我们进一步研究了Q算子的真空本征值（对应于W_3模的最高权向量），并证明了它们与具有零体积质量的非平衡边界仿射Toda场论中边界指数场的期望值简单相关。

一个相对论性锥函数及其Whittaker极限

A Relativistic Conical Function and its Whittaker Limits

In previous work we introduced and studied a function $R(a_{+},a_{-},{\bf c};v,\hat{v})$ that is a generalization of the hypergeometric function ${}_2F_1$ and the Askey-Wilson polynomials. When the coupling vector ${\bf c}\in{\mathbb C}^4$ is specialized to $(b,0,0,0)$ , $b\in{\mathbb C}$ , we obtain a function ${\mathcal R}(a_{+},a_{-},b;v,2\hat{v})$ that generalizes the conical function specialization of ${}_2F_1$ and the $q$ -Gegenbauer polynomials. The function ${\mathcal R}$ is the joint eigenfunction of four analytic difference operators associated with the relativistic Calogero-Moser sy…

全纯丛与标量差分算子：单点构造

Holomorphic bundles and Scalar Difference Operators: One-Point Constructions

有效地构造了秩l>1的一维差分算子的交换环及其变形。我们的分析结构是基于所谓的'' Tyurin参数''的稳定框架全纯向量丛代数曲线的亏格等于g和陈数等于lg。这些参数大量使用本作者已经在1978年至1980年的微分算子。在离散情况下，它们的变形由2D Toda Lattice hierarhy而不是KP控制。在l=2，g=1的情况下，这里出现了新的可积系统。高秩差分算子的理论比以前文献中仅使用谱曲线上的2点构造（即“无限点”的数目等于2）的秩1情形丰富得多。对于每个偶数秩l=2k，一点构造出现在这个问题中。只有在这种情况下，交换环依赖于功能参数。两点构造将在接下来的工作中研究：即使对于更高的秩l>1，这种情况也可以在Theta函数中解决。对于秩l>1的单点结构则不是这样。

可积族、矩阵模型和τ-函数的卷积对称性

Convolution symmetries of integrable hierarchies, matrix models and τ-functions

KP型和2KP-Toda型可积族的广义卷积对称将希尔伯特空间$\HH= L^2(S^1)$元素的傅立叶系数乘以一个指定的常数序列。这导致了希尔伯特空间格拉斯曼$\Gr_{\HH_+}(\HH)$上的相应变换，从而导致了决定KP和2-Toda族解的Sato-Segal-Wilson τ-函数上的相应变换。相应的作用在相关的费米子Fock空间也是对角的标准正交基确定占领网站和分区标记。定义这些交换流的初始点的元素元素$W \in \Gr_{\HH_+}(\HH)$的普吕克坐标是相关τ函数的单Schur函数和双Schur函数中的系数，因此在此作用下乘以相应的对角因子。将这种变换应用于矩阵积分，我们得到了新的外耦合型矩阵模型，它们也是KP或2KP-Toda τ-函数。类似地得到了τ函数的更一般的多重积分表示及其有限行列式表达式。

Recursion and Hamiltonian operators for integrable nonabelian difference equations

In this paper, we carry out the algebraic study of integrable differential-difference equations whose field variables take values in an associative (but not commutative) algebra. We adapt the Hamiltonian formalism to nonabelian difference Laurent polynomials and describe how to obtain a recursion operator from the Lax representation of an integrable nonabelian differential-difference system. As an application, we propose a novel family of integrable equations: the nonabelian Narita-Itoh-Bogoyavlensky lattice, for which we construct their recursion operators and Hamiltonian operators and prove …