Affine algebra representations and discrete integrability

本课题主要研究了统计力学中的离散可积模型、仿射代数和量子仿射代数的表示理论和聚类代数等动力系统的组合学之间的关系。从物理角度看，该项目对弦理论中的壁交公式和凝聚态物理中量子霍尔效应的准粒子波函数具有应用价值。研究的问题包括仿射李代数模的费米子构造及其聚变产物；应用统计物理方法给出离散可积系统（如t系统或q系统）的簇代数中簇变量的显式解，从而给出相关正猜想的证明；以及这些系统的非交换推广，与Kontsevich非交换过壁公式有关，或者描述量子离散Liouville或Hirota方程的量子簇代数。这项研究处于数学和物理的边缘。它寻求应用统计力学的技术来解决仿射李代数的组合学和表示理论及其量化问题。这些问题的主要特征是可积性，也就是说，它们产生于具有高度对称性的系统。通常这种对称性允许以物理配分函数的形式找到显式解，这显然是系统构型上的正和。这种正性性质通常是基础数学对象的一种推测性质。

The research project exploits relations among (1) discrete integrable models in statistical mechanics, (2) representation theory of affine and quantum affine algebras, and (3) the combinatorics of dynamical systems such as cluster algebras. From the physical point of view, the project has applications to wall crossing formulas in string theory and the wave functions of quasi-particles responsible for the quantum Hall effect in condensed matter physics. The problems investigated include fermionic constructions of affine Lie algebra modules and their fusion products; Applications of methods from statistical physics to give explicit solutions for the cluster variables in cluster algebras related to discrete integrable systems such as T-systems or Q-systems, thereby giving proofs of the relevant positivity conjectures; and non-commutative generalizations of these systems, related to the Kontsevich non-commutative wall-crossing formula, or the quantum cluster algebras which describe quantum discrete Liouville or Hirota equations.This research is at the boundary between mathematics and physics. It seeks to apply techniques from statistical mechanics to solving problems in combinatorics and representation theory of affine Lie algebras and their quantization. The prime characteristic of these problems is integrability, that is, they arise from systems with a high degree of symmetry. Often this symmetry allows for finding explicit solutions in the form of physical partition functions, which are manifestly positive sums over configurations of a system. This positivity property is frequently a conjectured property of the underlying mathematical objects.