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

本文研究无限维李代数的表示理论及其在量子场论中的应用。提出了一种研究二维共形场理论形变中的运动积分的新方法。利用相应表示的几何性质，我们将寻求仿射Kac-Moody代数在有理水平上的模函子。我们将研究复代数曲线的朗兰兹-德因费尔德对应。与仿射代数相关的拓扑场论将从上同调的观点来研究。我们将结合W-代数来研究圆上微分算子李代数的量子形变。最后，将Virasoro代数上Verma模的奇异向量公式用于计算向量场的李代数的上同调。这项研究涉及一种名为李代数的数学对象。李代数起源于另一个叫做李群的物体。李群的一个例子是球体的旋转，其中一个旋转之后是另一个旋转。李群和李代数在涉及球面运动分析的领域中很重要。

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.