Aspects of Geometry and Topology Related to High Energy Theoretical Physics

DMS-0103877约翰·W·摩根建议在这项拨款下进行的工作涉及高能物理产生的一些数学对偶的数学表述和证明。第一个要研究的对偶性涉及F-理论/杂交弦对偶。对偶性的数学表述是：对于一定的参数范围，F-理论的各种紧化的经典真空模空间和杂化弦理论的经典真空模空间应该同构。我们猜想，适当的数学表述是杂化真空模空间是F-理论模空间的非正规无限复盖，它伴随着F-理论模空间的基本群的极大抛物子群。这些极大抛物子群是某些因子在无穷远处的邻域的基本群。本文详细分析了椭圆曲线上半稳定主丛的模空间以及微分上同调的概念。第二对偶性是著名的介乎SU(2)Yang-Mills理论和阿贝尔Seiberg-Witten理论之间的量子场论对偶性。这些双重描述仅仅是扭曲版的超对称杨-米尔斯理论的高和低能量极限。对这种对偶性的数学理解的一部分涉及给出扭曲超对称理论的高能量极限和低能量极限的精确数学公式，并展示该扭曲理论中的某些关联函数如何在微扰理论的意义上收敛到通常的唐纳森多项式不变量在高能极限和Seiberg-Witten不变量在低能极限。故事的这一部分肯定要服从严格的数学公式。更雄心勃勃的目标是找到某种数学形式主义，允许在这两个极限之间进行数学上的严格交融--数学上替代数学上未定义的量子场论--这将允许人们比较能量的上限和下限。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，它呈现出了新的面貌。随着量子场论和弦理论对全球问题的研究，几何学和拓扑学以及这些理论物理领域之间的相互作用达到了一个新的水平。对这些(到目前为止还不严格的)物理理论的存在的信念使物理学家做出预测，即数学中的猜想。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，物理学家确认，当这些数学预测可以被严格的数学验证时，他们就走在了正确的轨道上，因为严格的数学，不能利用我们最初用来进行预测的量子场论范式。这种相互作用对数学的兴趣在于，以这种方式预测的各种数学陈述被证明是相当新颖的，产生了前所未有的猜想。过去十年或十五年来，在大片几何学和拓扑学领域的许多进展都可以追溯到这个源头。数学猜想最丰富的来源之一是物理学中的各种二元性。当量子场论或弦理论的经典数学极限不止一个时，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，因为它们是单一量子场论或弦理论的极限。数学问题总是一样的--严格定义什么是关系，然后用数学方法建立它。这里提出的工作正是沿着两个这样的二元性的思路进行的。

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.