Aspects of Geometry and Topology Related to High Energy Theoretical Physics

本基金拟开展的工作涉及高能物理中产生的一些数学对偶性的数学公式和证明。第一个要研究的对偶关系涉及到f理论/异质弦对偶。对偶性的数学表述是：在一定的参数范围内，f理论和杂散弦理论的各种紧化的经典真空的模空间应该是同构的。我们推测，适当的数学表述是：异质真空的模空间是与f -理论模空间的基群的极大抛物子群相关的f -理论模空间的非正态无限片覆盖。这些极大抛物子群是无穷远处某些因子的邻域的基本群。本文详细分析了椭圆曲线上半稳定主束的模空间以及微分上同调的概念。第二对偶是著名的量子场论对偶，介于SU(2) Yang-Mills理论和abelian seberg - witten理论之间。这些对偶描述仅仅是超对称杨-米尔斯理论的扭曲版本的高能和低能极限。对这种对偶性的数学理解的一部分包括给出扭曲超对称理论的高能量和低能量极限的精确数学公式，并展示扭曲理论中的某些相关函数如何在摄动理论的意义上收敛于高能量极限中的通常Donaldson多项式不变量和低能量极限中的Seiberg-Witten不变量。故事的这一部分当然可以用严格的数学公式来描述。更雄心勃勃的目标是找到一些数学形式，允许在这两个极限之间进行数学上严格的解释——数学上替代数学上未定义的量子场论——这将允许人们比较高能量和低能极限。数学与高能理论物理之间的联系一直很密切。在过去的几年里，它有了新的面貌。随着对量子场论和弦理论中全局问题的研究，几何和拓扑与这些理论物理领域之间的相互作用达到了一个新的水平。相信这些（迄今为止不严格的）物理理论的存在，导致物理学家做出预测，即数学上的猜想。这些猜想以一种奇怪的方式扮演着物理预测曾经扮演的角色。现在，当这些数学预测可以通过严格的数学来验证时，物理学家们确认他们在正确的轨道上，因为它是严格的，不能利用我们用来做预测的量子场论范式。数学对这种相互作用的兴趣在于，以这种方式预测的各种数学陈述已经被证明是非常新颖的，产生了不同于以往任何东西的猜想。在过去的十年或十五年里，在几何学和拓扑学的广大领域中取得的许多进展都可以追溯到这个源头。数学猜想最富有成果的来源之一是物理学中的各种对偶性。当量子场论或弦论存在不止一个经典数学极限时，就会出现这种情况。在描述经典极限时出现的数学对象是相关的，因为它们是单量子场论或弦理论的极限。数学问题总是一样的——严格定义关系是什么，然后用数学方法建立它。这里提出的工作正是沿着这条线，为两个这样的二元性。

DMS-0103877John W. MorganThe work proposed to be carried out under this grant concerns mathematical formulation and proof of some of the mathematical dualities arising from high energy physics. The first duality to be studied concerns the F-theory/heterotic string duality. The mathematical formulation of the duality is that for a certain range of parameters the moduli spaces of classical vacua for various compactifications of F-theory and of heterotic string theory should be isomorphic. We conjecture that the appropriate mathematical statement is that the moduli spaces of heterotic vacua are non-normal infinite-sheeted coverings of the F-theory moduli space associated with maximal parabolic subgroups of the fundamental group of the F-theory moduli space. These maximal parabolic subgroups are the fundamental groups of neighborhoods of certain divisors at infinity. This study involves a detailed analysis of the moduli space of semi-stable principal bundles over elliptic curves as well as the notions of differential cohomology. The second duality is the famous quantum field theory duality between SU(2) Yang-Mills theory and the abelian Seiberg-Witten theory. These dual descriptions are simply the high and low energy limits of a twisted version of supersymmetric Yang-Mills theory. One part of a mathematical understanding of this duality involves giving precise mathematical formulations of the high and low energy limits of the twisted supersymmetric theory and showing how certain correlation functions in this twisted theory converge in the sense of perturbation theory to the usual Donaldson polynomial invariants in the high energy limit and to the Seiberg-Witten invariants in the low energy limit. This part of the story is surely amenable to rigorous mathematical formulation. The more ambitious goal is to find some mathematical formalism which allows for a mathematically rigorous interpelation between these two limits -- a mathematical substitute for the mathematically undefined quantum field theory -- which would allow one to compare the high and low energy limits. The connection between mathematics and high energy theoretical physics has always been a close one. In the last few years it has taken on new aspects. With the study of global issues in quantum field theory and string theory has come a new level of interaction between geometry and topology and these areas of theoretical physics. Belief in the existence of these (to date non-rigorous) physics theories leads physicists to make predictions, i.e., conjectures in mathematics. In a strange way these conjectures play the role that physical predictions used to play. Now the physicists take confirmation that they are on the right track when these mathematical predictions can be verified by rigorous mathematics, which, since it is rigorous, can not make use of the quantum field theory paradigms that we used to make the prediction in the first place. The interest in mathematics of this interplay is that the sorts of mathematical statements that are predicted in this way have turned out to be quite novel producing conjectures unlike anything that had been seen before. Much of the progress the last ten or fifteen years in large areas of geometry and topology traces back to this source. One of the most fruitful sources of mathematical conjectures has been various dualities in physics. These occur when there is more than one classical mathematical limit of a quantum field theory or string theory. The mathematical objects that arise in the description of the classical limits then are related because they are limits of a single quantum field theory or string theory. The mathematical problem is always the same -- define rigorously what the relationship is and then establish it mathematically. The work proposed here is exactly along these lines for two such dualities.