The Mathematics of String Theory and Gauge Theory

这笔赠款将用于组织2012年5月3日至5日在伦敦城市大学和伦敦国王学院举办的讲习班的费用。会议的主要主题将是与弦理论和量子场论密切相关的数学方面的最新进展。我们的目标是将该领域的领先国际研究人员聚集在一起，展示他们的最新成果，分析该领域的最新技术，并鼓励合作，以确定和解决该研究领域的突出问题。一方面，数学与弦理论和量子场论之间的接口，经常在数学中产生新的进展，如镜像对称性、Seiberg-Witten不变量，或可积系统的研究。量子场论可以具有局域或规范对称性，这为它们提供了特别丰富的数学结构。弦理论提供了解决这一领域长期存在的问题的新方法，例如通过研究规范理论或规范/弦对应之间的对偶性。然而，关于量规理论的许多问题仍然没有得到回答。许多数学家和物理学家都认识到，对这些问题的新见解可能会在未来导致这两个学科的深刻新发展。理解规范理论的非微扰性质已经被认为是现代数学的一个关键问题--事实上，四维杨-米尔斯理论的解是克莱研究所千年数学奖之一。最近，重大的新进展使数学家和物理学家能够解决我们对量子规范理论理解中的一些长期空白。仅举三个例子：首先，定域化技术已经被用来计算S^4或S^3上的超对称规范理论的配分函数，不仅允许更好地处理这些规范理论，而且允许计算相关的数学不变量，如Khovanov-Rozansky同调。其次，ADS/CFT的对应最近导致了使用诸如辛流形上的等变局部化等先进技术来研究复杂的几何结构，以及对新的可积系统及其杨氏表示的研究。第三，在过去的一两年里，规范理论的超对称真空和算符的研究得到了新的推动，一方面是因为我们对相干层派生范畴中稳定物体的理解有所改善，另一方面是因为我们找到了一种在超对称规范理论的真空下实现可积结构的形变量子化的方法。拟议中的会议将集中在最近发生这种新发展的四个领域：-定域化、几何不变量和超对称规范理论。-ADS/CFT通信。-规范理论的超对称真空和算子。-新的可积系统和强耦合规范理论。我们选择这四个主题不仅是因为最近一两年有了相当大的进展，而且因为我们认为一个领域的想法有相当大的空间可以从另一个领域的相互作用中受益，并可能刺激进一步的重大突破。

This grant will cover the costs of organizing a workshop to be held between the 3rd and the 5th of May 2012 at City University London and King's College London. The overarching theme of the meeting will be the recent advances in mathematics that have a close connection to string theory and quantum field theories. We aim to bring together the leading international researchers in the field to present their latest results, analyze the state-of-the-art of the field, and inspire collaborations to identify and tackle the outstanding problems in this research area. The interface between mathematics on the one hand and string theory and quantum field theories on the other, has often given rise to novel advances in mathematics, such as mirror symmetry, Seiberg-Witten invariants, or the study of integrable systems. Quantum field theories can have local, or gauge, symmetries, which provide them with a particularly rich mathematical structure. String theory has provided novel ways of tackling long-standing problems in this area, for example through the study of dualities between gauge theories or the gauge/string correspondence. However, many questions about gauge theories remain unanswered. There is a shared recognition amongst many mathematicians and physicists that new insights into these questions are likely to lead to profound new developments in both disciplines in the future. Understanding the non-perturbative properties of gauge theories has been identified as a key problem of modern mathematics - indeed the solution of the four-dimensional Yang-Mills theory is one of the Clay Institute's Millennium Mathematics prizes.Recently, significant new advances have occurred which have allowed mathematicians and physicists to address some of the long-standing gaps in our understanding of quantum gauge theories. To name just three: Firstly, localisation techniques have been used to compute the partition functions of supersymmetric gauge theories on S^4 or S^3, allowing for not only a much better handle on these gauge theories, but also allowing for the calculation of related mathematical invairants such as the Khovanov-Rozansky homology. Secondly, the AdS/CFT correspondence has recently led to both a study of sophisticated geometrical structures using advanced techniques such as equivariant localisation on symplectic manifolds and an investigation of new integrable systems and their Yangian representations. Thirdly, the investigation of supersymmetric vacua and operators of gauge theories has, in the last year or two, been given a new impetus as a result of our improved understanding of stable objects in the derived category of coherent sheafs on the one hand and a way to implement a type of deformation quantisation of the integrable structure underlyig the vacua of supersymmetric gauge theories. The proposed meeting will focus on four areas in which such novel developments have recently taken place:- Localisation, geometrical invariants and supersymmetric gauge theories.- The AdS/CFT correspondence.- Supersymmetric vacua and operators of gauge theories.- New integrable systems and strongly coupled gauge theories.We have chosen these four topics not only because there have been considerable advances in the recent year or two but also because we think there is considerable scope for the ideas from one area to benefit from interactions with another, and may stimulate further significant breakthroughs.