Mathematical Sciences: Geometric Structure of Conformal Field Theories

本研究涉及共形场理论的几何结构。主要研究内容如下：(1)二叉树和李括号；(2)行列式线丛和顶点算符代数；(3)两个以上扭曲元素的对偶的几何证明；(4)有限维李代数、顶点算符代数和高维量子场论的几何定义的抽象形式；(5)指数化的顶点算子代数；(6)指数化的顶点算子代数与椭圆上同调。所支持的研究涉及当前数学活动的一个重要焦点，其中群表示理论、量子场论和代数几何都走到了一起。这项工作对理论物理以及数学的几个领域都有重要的影响。

This research is concerned with the geometric structure of conformal field theory. The principal investigator will study the following topics: (1) binary trees and the Lie bracket; (2) determinant line bundles and vertex operator algebras; (3) a geometric proof of duality for more than two twisted elements; (4) an abstract form of the geometric definitions of finite dimensional Lie algebras, vertex operator algebras and higher dimensional quantum field theory; (5) exponentiating vertex operator algebras; (6) exponentiations of vertex operator algebras and elliptic cohomology. The research supported concerns an important focus of current mathematical activity, where group representation theory, quantum field theory, and algebraic geometry all come together. This work has important implications for theoretical physics, as well as several areas of mathematics.