Supersymmetric Gauge Theory, Donaldson-Thomas Invariants and Hyperkahler Geometry

摘要奖：DMS-1006046首席研究员：安德鲁·内茨克这项研究是基于物理学和几何学之间的接口的最新发展。在最近与Davide Gaiotto和Greg Moore的工作中，主要研究人员使用超对称规范理论的技术来解决Calabi-Yau三重中稳定几何物体的计数问题。作为应用，他们证明了期望的计数，即所谓的“广义Donaldson-Thomas不变量”，实际上印记在某些辅助模空间上的Hyperkahler度量中。这种联系阐明了以前计数问题的神秘方面；特别是它给出了一种新的几何理解，即控制这些不变量如何跳跃，即相关几何对象如何分裂和连接的“穿越墙公式”。与此同时，它提供了一种全新的方式来看待所讨论的Hyperkahler指标。拟议的研究建立在最近在不同方向上的这项工作的基础上。该项目的大部分内容是PI、Davide Gaiotto和Greg Moore之间持续合作的一部分。首先，他们将应用他们的新构造来获得比以前关于完全超kahler度量更明确的信息，最终目标是对K3曲面上的Ricci-Flat度量进行新的描述。其次，他们将探索其构造的扩展，以涵盖与SU(2)以外的群相关的Higgs丛的模空间上的度量。第三，他们将使用规范理论的视角来研究推广了Donaldson-Thomas的猜想新不变量的跨墙性质。在与Sergio Cecotti和Cumrun Vafa的合作中，PI还将从规范理论的解释中寻找对Donaldson-Thomas不变量的新限制。上个世纪基础物理学的最高成就是发展了“量子场论”，物理学家用来描述亚原子粒子的行为的工具包。量子场论的许多方法看起来与数学家通常的方法截然不同。然而，人们逐渐意识到，这些想法中的许多确实适用于“纯”数学问题：例如，关于几何的问题有时可以被重新表述为关于亚原子物理的问题！特别是，最近(由PI与合作者Greg Moore和Davide Gaiotto一起)发现，通过研究某些四维和三维量子系统在极低能量下的行为，人们可以获得关于某些空间(超卡勒空间)的几何的详细信息，这些空间近年来得到了数学家们的深入研究。这似乎是一个更加丰富的故事的开始：通过利用量子系统的更深层次的性质，PI旨在获得关于相应几何图形的更深层次的信息。在数学的相关领域有许多应用，包括旨在建立几何和数论之间的桥梁的“几何朗兰兹计划”。

AbstractAward: DMS-1006046Principal Investigator: Andrew Neitzke This research is based on current developments at the interface between physics and geometry. In recent work with Davide Gaiotto and Greg Moore, the principal investigator has used techniques of supersymmetric gauge theory to attack the problem of counting stable geometric objects in Calabi-Yau threefolds. As an application they showed that the desired counts, known as "generalized Donaldson-Thomas invariants," are actually imprinted into hyperkahler metrics on certain auxiliary moduli spaces. This connection illuminates previously mysterious aspects of the counting problem; in particular it gives a new geometric understanding of the "wall-crossing formula" which governs how these invariants jump, i.e. how the relevant geometric objects can split and join. At the same time, it gives a totally new way of looking at the hyperkahler metrics in question. The proposed research builds on this recent work in various directions. Much of the program is part of a continuing collaboration between the PI, Davide Gaiotto and Greg Moore. First, they will apply their new construction to get more explicit information than was previously available about complete hyperkahler metrics, with the ultimate goal being a new description of the Ricci-flat metric on a K3 surface. Second, they will explore extensions of their construction to encompass metrics on moduli spaces of Higgs bundles associated to groups other than SU(2). Third, they will use the gauge theory perspective to study wall-crossing properties of conjectural new invariants which extend Donaldson-Thomas. In collaboration with Sergio Cecotti and Cumrun Vafa, the PI will also look for new restrictions on the Donaldson-Thomas invariants coming from their gauge-theoretic interpretation.The crowning achievement of fundamental physics over the last century was the development of "quantum field theory", the toolkit which physicists use to describe the behavior of subatomic particles. Many of the methods of quantum field theory look radically different from the usual methods of mathematicians. Nevertheless it has been gradually appreciated that many of these ideas do have applications to problems of "pure" mathematics: for example, questions about geometry can sometimes be rephrased as questions about subatomic physics! In particular, recently it was discovered (by the PI together with collaborators Greg Moore and Davide Gaiotto) that by studying the behavior of certain four-dimensional and three-dimensional quantum systems at very low energies, one can get detailed information about the geometry of certain spaces ("hyperkahler spaces") which have been intensely studied by mathematicians in recent years. This appears to be the beginning of a much richer story: by using deeper properties of the quantum systems, the PI aims to get deeper information about the corresponding geometry. There are numerous applications to related areas of mathematics, including the "geometric Langlands program" which aims to create a bridge between geometry and number theory.