Algebraic Geometry and Quantum Field Theory of D-Branes

摘要奖：DMS-0606578主要研究者：大卫R.作者：莫里森阿斯平沃尔，M. D-膜，粗略地说，是开弦可以终止于其上的东西。对这个概念的任何详细探索很快就会发现，粗糙的概念太粗糙了，弦论几何的所有微妙之处在描述D-膜时都发挥了全部作用。 提出者将研究D-膜的数学和物理性质，从代数几何（特别是复三重），同调代数和数学方面的衍生类别，以及量子场论和物理方面的超引力中提取技术。 提出者期望，对这个故事中互补的数学和物理方面的研究将相互照亮，并随之在纯数学和理论物理方面获得新的结果。这项研究是对理论物理学当前一个重要主题的物理方面的更广泛探索的一部分：使用一维对象（“弦”）而不是传统的零维对象（“点”）来模拟自然界的粒子和力。 这类物理模型本质上是量子力学的，但也模拟了引力，从而解决了理论物理学中的一个重大难题（即量子力学和爱因斯坦引力理论的明显不相容）。 过去十年的一个重要发现是弦可能有端点，这些端点被限制在特定的方向上。弦端点“连接”的地方被称为D-膜，这些将是当前项目中详细的物理研究的主题。

AbstractAward: DMS-0606578Principal Investigator: David R. Morrison, Paul S. Aspinwall, M. Ronen PlesserA D-brane, roughly, is something on which an open string may end.Any detailed exploration of this notion quickly reveals that therough notion is too crude, and that all of the subtleties of thegeometry of string theory come into play in full force indescribing D-branes. The proposers will investigate both themathematical and physical properties of D-branes, drawing upontechniques from algebraic geometry (particularly complexthreefolds), homological algebra, and derived categories on themathematics side, and on quantum field theory and supergravityfrom the physics side. The proposers anticipate that the studyof the complementary mathematical and physical aspects of thisstory will shed light on each other, and that new results in bothpure mathematics and theoretical physics will be obtained as aconsequence.This study is part of a broader exploration of the mathematicalaspects of an important current theme in theoretical physics:modeling the particles and forces of nature using one-dimensionalobjects ("strings") instead of the traditional zero-dimensionalobjects ("points"). Physical models of this kind arequantum-mechanical in nature but also model the gravitationalforce, thus reconciling one of the great puzzles in theoreticalphysics (namely, the apparent incompatibility of quantummechanics and Einstein's theory of gravity). One of theimportant discoveries of the past decade is that strings may haveendpoints which are constrained to lie in particular directions.The places where the endpoints of strings "attach" are known asD-branes, and these will be the subject of detailed mathematicalstudy in the current project.