Mathematical Sciences: Representations of Infinite-Dimensional Algebras with Applications to Two-Dimensional Quantum Field Theory

本研究涉及的是表征理论， 无限维李代数及其在量子力学中应用 场理论 运动积分研究的新途径 在二维共形场论的变形中， 发展。 仿射Kac-Moody代数的模函子 在合理的水平上，将采用几何 对应的表示的属性。 的 复代数曲线的Langlands-Drinfeld对应 将进行调查。 拓扑场论，与 仿射代数，将研究从上同调点 风景 微分李代数的量子形变 将研究圆上的运算符， W-代数 最后，给出了Verma中奇异向量的计算公式， Virasoro代数上的模将应用于 向量场李代数上同调的计算。 这项研究涉及一个数学对象， 李代数 李代数起源于另一个叫做 李群。 李群的一个例子是一个 一个旋转之后是另一个旋转的球体。 李群和 李代数在涉及以下分析的领域中是重要的： 球面运动

This research is concerned with representation theory of infinite-dimensional Lie algebras with applications to quantum field theory. A new approach to the study of integrals of motion in deformations of two-dimensional conformal field theories will be developed. The modular functor for affine Kac-Moody algebras at the rational level will be pursued, using geometrical properties of the corresponding representations. The Langlands-Drinfeld correspondence for complex algebraic curves will be investigated. Topological field theories, associated to affine algebras, will be studied from the cohomological point of view. The quantum deformation of the Lie algebra of differential operators on the circle will be studied in connection with the W-algebras. Finally, the formulas for singular vectors in Verma modules over the Virasoro algebra will be applied to the computation of the cohomologies of Lie algebras of vector fields. This research is concerned with a mathematical object called a Lie algebra. Lie algebras arise from another object called a Lie group. An example of a Lie group is the rotations of a sphere where one rotation is followed by another. Lie groups and Lie algebras are important in areas involving analysis of spherical motion.