Quantum K-Theory

本研究项目处于拓扑学、几何学和数学物理的交叉领域。研究者最近给出了一类代数变量的镜像对称猜想的证明，其中包括四维复射影空间中五次方程的经典情况。目前的项目是该工作的延续，并在量子上同调理论中追求几个新的方向。镜像对称是物理学家首先发现的一种现象，与流行的10维宇宙弦模型有关。在这个模型中，六个不可见的维度以所谓的Calabi-Yau流形出现；他们观察到一个被称为Calabi-Yau流形的共形场论的a模型与另一个共形场论的“镜像”流形的b模型是一致的。在数学术语中，这意味着在Calabi-Yau流形的辛拓扑与其镜像流形的复几何之间存在定义良好的对应关系。

9704774 Givental This research project lies at the interface of topology, geometry, and mathematical physics. The investigator has recently given a proof of the mirror symmetry conjecture for a class of algebraic varieties, including the classical case of quintics in four dimensional complex projective space. The current project is a continuation of that work and pursues several new directions in quantum cohomology theory. Mirror symmetry is a phenomenon first uncovered by physicists in connection with the popular 10-dimensional string model of the universe. In this model, the six invisible dimensions arise as so called Calabi-Yau manifolds; they observed that a model of conformal field theory known as the A-model of a Calabi-Yau manifold coincides with another conformal field theory model, the B-model, of its 'mirror' manifold. In mathematical terms this means that there is a well-defined correspondence between the symplectic topology of a Calabi-Yau manifold and the complex geometry of its mirror manifold.