Geometry and Mathematical Physics of D-Branes

D-膜的几何和数学物理

• 批准号: 0905923
• 负责人: Paul Aspinwall
• 金额: $ 34.76万
• 依托单位: Duke University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-01 至 2014-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0905923&HistoricalAwards=false
• 关键词: Geometry; Mathematical; Physics; Branes

The derived category of coherent sheaves has become a powerful tool in recent years in algebraic geometry. Remarkably this category also appears naturally in the mathematical physics of superstring theory where objects are D-branes, i.e., boundary conditions on open strings, and morphisms are Hilbert spaces associated to open string states. The proposer will investigate several aspects of this connection. The mathematics of tilting collections, quiver representations and matrix factorizations will be applied to notions such as monodromy, stability and moduli spaces to further understand string theory. It is expected that this research will lead to insight into the mathematics of the derived category of Calabi-Yau threefolds which play a central role in this proposal.String theory aims at providing an understanding of the fundamental physics of our universe, particularly the geometry of spacetime at short distances. The mathematics of string theory requires the use of sophisticated techniques from modern geometry. Indeed, ideas from string theory have inspired many recent important developements in the pure mathematics of geometry. This proposal will explore the connection between D-branes, which in a sense are subspaces of spacetime, and algebraic geometry, in which geometry is expressed purely in terms of algebraic equations. It is expected that new results both in pure mathematics and mathematical physics will be obtained.

近年来，凝聚层的导出范畴已成为代数几何中的一个有力工具。值得注意的是，这一范畴也自然地出现在超弦理论的数学物理中，其中的对象是D膜，即，开弦上的边界条件，态射是与开弦状态相关的希尔伯特空间。提议者将调查这种联系的几个方面。倾斜集合，矩阵表示和矩阵分解的数学将应用于单值，稳定性和模空间等概念，以进一步理解弦理论。预计这项研究将导致深入了解卡-丘三重范畴的数学，这在这一提议中起着核心作用。弦理论旨在提供对我们宇宙基本物理的理解，特别是短距离时空的几何。弦理论的数学需要使用现代几何学中的复杂技术。事实上，弦理论的思想激发了几何纯数学的许多重要发展。这个提议将探索D-膜（在某种意义上是时空的子空间）和代数几何（其中几何纯粹用代数方程表示）之间的联系。期望在纯数学和数学物理方面都能得到新的结果。