D-Brane Physics and Calabi-Yau Geometry

弦理论是数学和物理学中最具挑战性和最有趣的领域之一。它被广泛认为是在最基本的层面上建立完整的统一物理模型的最有希望的方法，它也在纯数学中催生了许多想法。弦理论的最初想法是，宇宙的所有基本组成部分都是开放或闭合的弦环。越来越清楚的是，该理论的这种表述在本质上是不完整的。其中一个缺失的部分是被称为“D-膜”的现象，它天真地看起来是各种维度的几何物体：点、弦、膜等，它们与理论的基本弦共存。缺乏一个令人满意的数学框架来研究这些D-膜，以及对它们的物理性质的理解。在这个项目中，将使用代数和几何中的数学技术来揭示精确定义这些D-膜所需的概念。研究人员将根据需要开发新的数学工具，并应用现有的数学工具来解决这个问题。还将使用量子场论的方法和超重力的见解，详细研究非平凡背景下D-膜的物理性质。从D-膜的研究中获得的见解也将被应用到规范理论和弦理论的物理学中。

DMS-0301476David R. Morrison, Paul Aspinwall, and M. Ronen Plesser.String theory is one of the most challenging and interesting fields inboth mathematics and physics. Widely considered the most promisingapproach to formulating a complete unified model of physics at themost fundamental level, it has also spawned many ideas in puremathematics. The original idea of string theory was that all fundamental constituents of the universe were open or closed loops of string. It has become increasingly clear that this formulation of the theory is incomplete in essential ways. One of the missing pieces is the phenomenon known as "D-branes", which naively appear to be geometric objects of variousdimensions: points, strings, membranes, and so on, which coexist with the fundamental strings of the theory.A satisfactory mathematical framework for studying these D-branes, and an understanding of their physical nature, are lacking. In this project,mathematical techniques from algebra and geometry will be used to uncoverthe concepts required for a precise definition of these D-branes. Theresearchers will develop new mathematical tools, as needed, as well asapply existing mathematical tools to attack this problem. The physical nature of D-branes in non-trivial backgrounds will also be investigated in detail, using methods from quantum field theory and insights from supergravity. The insights gained from the study of D-branes will be applied back to the physics of gauge theory and string theory, as well.