Supersymmetric Gauge Theory, Donaldson-Thomas Invariants and Hyperkahler Geometry

AbstractAward：DMS-1006046首席研究员：Andrew Neitzke本研究基于物理学和几何学之间界面的当前发展。 在最近与大卫·盖奥托（Davide Gaiotto）和格雷格·摩尔（Greg Moore）的合作中，主要研究者使用超对称规范理论的技术来解决卡-丘三重计数稳定几何物体的问题。 作为一个应用，他们表明，所需的计数，被称为“广义唐纳森-托马斯不变量”，实际上是印到超卡勒度量的某些辅助模空间。 这种联系照亮了计数问题以前神秘的方面;特别是它给出了一个新的几何理解的“墙穿越公式”，其中管理如何这些不变量跳，即如何相关的几何对象可以分裂和加入。 与此同时，它提供了一个全新的方式来看待hyperkahler指标的问题。 拟议的研究建立在这一最近的工作在各个方向。 该计划的大部分是PI，Davide Gaiotto和Greg摩尔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更明确的关于完整超卡勒度量的信息，最终目标是对K3曲面上的Ricci平坦度量进行新的描述。 其次，他们将探索他们的建设扩展，以涵盖与SU（2）以外的群相关联的希格斯束的模空间上的度量。 第三，他们将使用规范理论的角度来研究的跨壁性质的新的不变量的推广唐纳森-托马斯。 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唐纳森-托马斯不变量的新限制。上个世纪基础物理学的最高成就是“量子场论”的发展，这是物理学家用来描述亚原子粒子行为的工具包。 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，人们逐渐认识到，这些思想中的许多确实可以应用于“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，最近发现（由PI与合作者Greg摩尔和Davide Gaiotto一起），通过研究某些四维和三维量子系统在非常低的能量下的行为，可以获得有关某些空间（“超卡勒空间”）几何的详细信息，这些空间近年来一直被数学家深入研究。 这似乎是一个更丰富的故事的开始：通过使用量子系统的更深层次的属性，PI旨在获得有关相应几何的更深层次的信息。 它在数学的相关领域有许多应用，包括“几何朗兰兹纲领”，旨在建立几何和数论之间的桥梁。

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.