Integrable difference equations and characters of affine Lie algebras

该建议使用从数学物理模型中获得的见解，以理解纯数学中发现的代数结构。这些模型来自一类特殊的模型，称为可积模型。这意味着，由于它们的高度对称性，它们有足够数量的守恒定律——推广能量守恒定律——可以精确地求解。这样的模型可以在统计力学（离散的，可能有限的系统）和量子场论（连续的，无限维的系统）中找到。在数学中有许多结构可以通过使用这些系统的解的技术和结果来研究。它们出现在组合学、数论、非交换代数的表示和几何中，仅举几例。该项目的成果将促进对所有这些领域的理解。通常，可积系统，如统计力学中的量子自旋链，共形场论及其大量变形，可以用表征理论术语来描述。例如，海森堡自旋链的传递矩阵可以被定义为量子仿射代数的有限维模块的q字符。可积量子场论的Hilbert空间可以表示为（变形的）Virasoro代数扩展的无限维模块。传递矩阵和Virasoro模块的性质满足差分方程或可以被证明为离散可积方程的方程。转移矩阵满足离散的hirota型方程，该方程可以解释为簇代数突变。本文提出了以下课题：(1)研究WZW理论（Demazure模的推广）中渐变共形块的非交换生成函数所满足的差分方程。对于Kirillov-Reshetikhin模的特征，用q系统对应的量子聚类代数中的聚类变量来表示这些生成函数；(2)这些方程的显式解作为费米子特征公式；(3)差分方程的可积结构；(4)给出仿射代数模或Virasoro模特征的函数的稳定极限；(5)离散可积方程的解，称为t系统和量子t系统（在量子簇代数意义上）及其与Nakajima的t，q字符的关系；(6)高维可积差分方程，将t系统推广为Plucker关系，用代数形式表达了五角形映射的高维类似物的离散结构；(7)给出有限、仿射和量子代数的惠特克向量、函数和量子Toda哈密顿量显式解的统计路径模型。

The proposal uses insights gained from models used in mathematical physics in order to understand algebraic structures found in pure mathematics. The models are from a special class of models, called integrable models. This means that due to their high degree of symmetry, they have a sufficient number of conservation laws -- generalizing conservation of energy -- that they can be solved exactly. Such models can be found in statistical mechanics (discrete, possibly finite systems) and in quantum field theory (continuous, infinite-dimensional systems). There are many structures in mathematics that can be studied by using techniques and results from the solutions of such systems. They appear in combinatorics, number theory, representations of non-commutative algebras, and geometry, to name a few. The results of this project will advance understanding in all these areas. Frequently, integrable systems such as quantum spin chains in statistical mechanics, and conformal field theories and their massive deformations, can be described in representation-theoretical terms. For example, the transfer matrix of the Heisenberg spin chains can be given a meaning as a q-character of a finite-dimensional module of quantum affine algebras. The Hilbert space of integrable quantum field theories can be expressed as an infinite-dimensional modules of extensions of the (deformed) Virasoro algebra. Transfer matrices, and the characters of the Virasoro modules, satisfy difference equations or equations which can be shown to be discrete integrable equations. The transfer matrices satisfy a discrete Hirota-type equation which have interpretations as cluster algebra mutations. The following projects are proposed here: (1) The study of the difference equations satisfied by the non-commutative generating functions of graded conformal blocks of WZW theories (generalizations of Demazure modules). These generating functions are expressed in terms of the cluster variables in the quantum cluster algebra corresponding to Q-systems for characters of Kirillov-Reshetikhin modules; (2) The explicit solutions of these equations as fermionic character formulas; (3) The integrable structure of the difference equations; (4) The stabilized limits of these functions which give characters of affine algebra modules or Virasoro modules; (5) Solutions of discrete integrable equations known as T-systems and quantum T-systems (in the sense of quantum cluster algebras) and their relation to Nakajima's t,q-characters; (6) Higher-dimensional integrable difference equations, generalizing the T-systems viewed as Plucker relations, expressing the discrete structure of the higher-dimensional analogs of the pentagram maps in algebraic terms; and (7) Statistical path models which give explicit solutions for Whittaker vectors, functions and quantum Toda Hamiltonians for finite, affine and quantum algebras.