Algebraic Geometry and Quantum Field Theory of D-Branes

摘要奖：DMS-0606578首席研究员：大卫·R·莫里森、保罗·S·阿斯平沃尔、M·罗宁·普莱瑟尔大体上来说，D-膜是一根开放的弦可以在其上结束的东西。对这个概念的任何详细探索很快就会发现，这个概念太粗糙了，弦论几何的所有微妙之处在描述D-膜时都发挥了最大作用。作者将同时研究D-膜的数学和物理性质，在数学方面借鉴代数几何(特别是复三重)、同调代数和衍生范畴的技术，从物理方面借鉴量子场论和超引力的技术。这项研究是对当前理论物理中一个重要主题的数学方面的更广泛探索的一部分：使用一维物体(“弦”)而不是传统的零维物体(“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，但也是引力的模型，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)。过去十年的重要发现之一是弦可能有被限制在特定方向上的端点。弦的端点被称为D-膜，这些将是本项目详细数学研究的主题。

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.