Geometry and Mathematical Physics of D-Branes

相干束的派生范畴是近年来代数几何研究的一个有力工具。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，其中对象是d膜，即开放弦上的边界条件，而态射是与开放弦状态相关的希尔伯特空间。提议者将研究这种联系的几个方面。倾斜集合、颤振表示和矩阵分解的数学将应用于单态、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在本提案中发挥核心作用的Calabi-Yau三倍的派生类别的数学。弦理论旨在提供对我们宇宙的基本物理的理解，特别是在短距离的时空几何。弦理论的数学需要使用现代几何中的复杂技术。事实上，弦理论的思想启发了近年来几何学纯数学的许多重要发展。这个提议将探索d膜之间的联系，d膜在某种意义上是时空的子空间，和代数几何之间的联系，在代数几何中，几何纯粹用代数方程来表达。期望在纯数学和数学物理方面都能取得新的成果。

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.