New Directions in Gromov-Witten and Donaldson-Thomas theory

格罗莫夫-维滕和唐纳森-托马斯理论的新方向

基本信息

批准号：
172668-2012
负责人：
Behrend, Kai
金额：
$ 3.28万
依托单位：
University of British Columbia
依托单位国家：
加拿大
项目类别：
Discovery Grants Program - Individual
财政年份：
2015
资助国家：
加拿大
起止时间：
2015-01-01 至 2016-12-31
项目状态：
已结题

来源：
https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=569620
关键词：
New Directions Gromov Witten Donaldson

项目摘要

Gromov-Witten invariants are important in String Theory, the physical theory according to which elementary particles are tiny strings, instead of points, as was believed since Newton. String theory is the best candidate for the so-called Theory of Everything, which unifies all physical theories, in particular Einstein's theory of relativity and quantum field theory. Recently it was discovered that Gromov-Witten invariants are related to the so-called Donaldson-Thomas invariants. This connection is, in fact, still a conjecture, but it is very important, because Donaldson-Thomas invariants can explain many strange phenomena exhibited by Gromov-Witten invariants. In my recent research I discovered something surprising about Donaldson-Thomas invariants. They behave like Euler characteristics. The Euler characteristic of a shape is a number which does not change, if the shape is deformed as if it was made of rubber. The surface of a sphere, for example, has Euler characteristic 2. The surface of a donut shape has Euler characteristic 0. The fact that Donaldson-Thomas invariants are certain kinds of Euler characteristics has important consequences, also for Gromov-Witten invariants and hence for String Theory. The goal for the future is to understand these Euler characteristics more deeply and make them a more flexible tool. They should not just be numbers, but some more complicated structure. Numbers are for counting things, but the things are lost in the process of counting. The goal is to discover the "things" which are "counted" by the Euler characteristics which give rise to Donaldson-Thomas invariants.

Gromov-witten的不变性在弦理论中很重要，这是物理理论，根据牛顿以来，基本粒子是微小的字符串，而不是点，而不是点。弦理论是所有事物所谓的理论的最佳候选者，该理论统一了所有物理理论，特别是爱因斯坦的相对论和量子场理论。最近发现，格罗莫夫（Gromov-Witten）的不变式与所谓的唐纳森 - 托马斯（Donaldson-Thomas）不变性有关。实际上，这种联系仍然是一个猜想，但这非常重要，因为唐纳森 - 托马斯不变式可以解释格罗莫夫·韦滕不变的许多奇怪现象。在最近的研究中，我发现了关于唐纳森 - 托马斯不变的令人惊讶的事情。他们的行为就像欧拉的特征。形状的Euler特性是一个数字，如果形状变形，就好像是由橡胶制成的。例如，球的表面具有欧拉的特征2。甜甜圈形状的表面具有欧拉的特征0。唐纳森 - 托马斯不变的事实是某些类型的欧拉特征，也对gromov-witten不变性和不变性也具有重要的后果。弦理论。未来的目标是更深入地了解这些Euler特征，并使它们成为更灵活的工具。它们不仅应该是数字，而且应该是一些更复杂的结构。数字是用于计数事物的，但是在计数过程中，事物丢失了。目的是发现欧拉（Euler）特征“计算”的“事物”，这些特征引起了唐纳森 - 托马斯的不变式。