Low-Dimensional Manifolds and Gauge Theory

9704507 Morgan在这个项目中进行的研究是双重的。这两个方面的研究一方面是几何学和拓扑学的交叉，另一方面是理论物理学（量子场论和弦理论）的交叉。一个方面涉及四维流形的规范论不变量，特别是Donaldson多项式不变量和Seiberg-Witten不变量。这里的目的是理解这些不变量与更经典的四流形几何描述的关系，例如，柄体描述。另一方面是研究椭圆纤维流形上的全纯向量束及其与数学物理中f理论中出现的K3曲面族的关系。这里的目的是为假定的f理论对偶性的两边建立数学结果，然后检查这种对偶性的预测是否确实成立。近年来，数学中的几何和拓扑领域与理论物理中的量子场论和弦论领域之间的相互作用重新兴起。这种相互作用对两个学科来说都是卓有成效的——物理学为低维空间的研究带来了新的方法，而数学为理论高能物理的研究提供了框架。本项目将更详细地研究其中两种相互作用。一种是利用理论物理提供的工具研究四维空间。在过去的几年里，这些工具已经产生了许多见解，并且有更多关于四维空间的信息应该可以使用这些工具来访问。该项目的另一方面涉及到一种新的物理理论的数学。这个理论对两种数学对象做出了非常精确的预测，表面上看，这两种对象之间几乎没有什么关系。另一方面，这些数学对象可以用独立于物理学之外的纯数学工具来研究。这些物体的数学分析结果可以用来检验物理理论的预测，就像使用物理实验一样。* * *

9704507 Morgan The study to be carried out in this project is two-fold. Both aspects of the study lie at the intersection of geometry and topology, on the one hand, and theoretical physics (quantum field theory and string theory) on the other. One aspect concerns gauge-theoretic invariants of four-dimensional manifolds, in particular, the Donaldson polynomial invariants and the Seiberg-Witten invariants. Here the aim is to understand the relation of these invariants to more classical geometric descriptions of a four-manifold, for example, a handlebody description. The other aspect is the study of holomorphic vector bundles over elliptically fibered manifolds and the relation of these to families of K3 surfaces that arise in F-theory in mathematical physics. Here the aim is to establish mathematical results for both sides of the putative F-theory duality and then to check that the predictions of this duality do indeed hold. There has been a resurgence in recent years of the interactions between the areas of geometry and topology in mathematics and the areas of quantum field theory and string theory in theoretical physics. This interaction has proved very fruitful for both disciplines -- the physics brings new methods to the study of low dimensional spaces, and the mathematics provides the framework in which the studies in theoretical high-energy physics are done. This project will investigate in more detail two of these interactions. One is the study of four-dimensional spaces, using tools provided by theoretical physics. These tools have yielded much insight in the last few years, and there is much more information about four-dimensional spaces that should be accessible using these tools. The other aspect of the project involves the mathematics of a new physical theory. This theory makes very precise predictions about two kinds of mathematical objects that, on the face of it, bear little relation to each other. On the other hand, these mathemat ical objects can be studied in their own right by purely mathematical tools that have been developed independently of physics. The results of the mathematical analyses of these objects can be used to check the predictions of the physical theory in much the same way as physical experiments are used. ***