量子Toroidal代数及其推广在可积系统及拓扑顶点中的应用

Quantum Toroidal algebra is a two-parameter analogue of W_{1+∞}. It has abundant structures and has attracted a lot of attention of many mathematicians and physicists in the world. Recently this algebra has been applied to several areas in mathematical physics, such as the refined topological vertex, Nekrasov partition function, Alday-Gaiotto-Tachikawa correspondence and Matrix model. In this project, we will consider the application of fermion representation of quantum toroidal algebra in the integrable system. And we will consider the relation among the quantum toroidal algebra, the gluing of topological vertex and Nekrasov partition function. Also we want to construct the action of elliptic quantum toroidal algebra on 2D and 3D Young diagrams and an elliptic version of refined topological vertex. Moreover, we will explore the application of elliptic quantum toroidal algebra in gauge theory. Furthermore, we will explore the application of quantum toroidal algebra in integrable 3d field theory.

量子Toroidal代数是一个两参数推广的W_{1+∞}代数，它具有非常丰富的数学结构。近些年来它被发现在数学物理很多方面都有重要作用，如精细拓扑顶点及Nekrasov配分函数，Alday-Gaiotto-Tachikawa对应，矩阵模型等。此项目中，我们重点考虑量子Toroidal代数的费米实现在可积系统中的应用，以及依据之前我们构造的顶点算子研究量子Toroidal代数与拓扑顶点粘合及Nekrasov配分函数的关系。我们也要考虑椭圆量子Toroidal代数在2维杨图和3维杨图上的作用，以及借助于椭圆量子Toroidal代数的表示构造椭圆的精细拓扑顶点，并探索它在规范理论中的应用。另外我们也将展开对量子Toroidal代数在可积的3维场论中作用的探索。

量子Toroidal代数也被称为Ding-Iohara代数, 它是一个两参数推广的W_{1+∞}代数, 近些年来它被发现在数学物理很多方面都有重要作用. 我们研究了用特征值表示的beta形变高斯矩阵模型和N×N复矩阵模型; 研究了椭圆Ding-Iohara代数的矢量表示及在2维杨图上的表示；建立了Hom-Yang-Baxter方程的四种等价表述, 并借助于三代数系统研究了Hom-Yang-Baxter方程的有理解.