QALF hyperkähler metrics

Hyperkähler流形是具有极其丰富的几何结构的几何空间。这种丰富的结构使得超级卡勒流形特别美丽，并且是较大类几何空间的受限示例：例如，所有超级卡勒流形都是里奇平坦的，因此与爱因斯坦广义相对论方程的解密切相关。从完全不同的角度来看，自 20 世纪 80 年代起人们就认识到，超卡勒流形是理论物理中许多规范理论（即推广麦克斯韦电磁场方程的物理理论）的真空（或“平衡态”）空间自然产生的。在数学和物理学之间富有成效的相互作用中，超卡勒流形的几何性质可以用来导出相应物理理论的性质，而物理学首先预测了具有独特性质的超卡勒流形的存在。我们全面理解超卡勒流形及其在族中的行为的一个具有挑战性的障碍是，超卡勒流形可以“崩溃”，也就是说，它们可以收敛到较低维度的极限空间。从物理学的角度来看，真空超卡勒空间的塌缩出现在相应物理理论的某些极限内，其中耦合常数收敛于零或无穷大。近年来，在尽可能低的维度 4 中超卡勒流形的塌缩简并的研究中已经取得了实质性进展。具有规定渐近几何的非紧超卡勒流形在这些最新进展中发挥了关键作用：具有有趣的渐近几何的 4 维超卡勒流形自 20 世纪 80 年代以来，人们使用一系列不同的技术来构建几何，但直到最近它们才被完全分类。在这个项目中，我们的目标是使用我们称为 QALF 的杰出渐近几何构造和分类高维超卡勒流形，彻底解决此类空间的存在性和唯一性问题。这项研究的应用有很多。在 hyperkähler 几何中，我们的目标是使用新的示例作为构建块来生成更高维 hyperkähler 流形的更复杂的示例，并研究它们在族中的行为。除了纯数学之外，该项目还直接应用于理论物理学，其中 QALF 超卡勒流形作为 3 维量子规范理论的真空空间而出现。虽然目前无法用严格的数学语言来定义量子理论本身，但在这个项目中，我们严格定义了理论的超kähler真空空间，从中可以导出物理理论的属性。

Hyperkähler manifolds are geometric spaces that carry an extremely rich geometric structure. This rich structure makes hyperkähler manifolds particularly beautiful and constrained examples of larger classes of geometric spaces: for example, all hyperkähler manifolds are Ricci-flat and therefore are closely related to solutions to Einstein's equations of General Relativity. From a completely different perspective, it was understood since the 1980's that hyperkähler manifolds arise naturally as the spaces of vacua (or "equilibrium states") of many gauge theories in theoretical physics, that is, physical theories that generalize Maxwell's equations of electro-magnetism. In a fruitful interaction between mathematics and physics, the geometric properties of the hyperkähler manifolds can be used to derived properties of the corresponding physical theory, while physics predicts the existence of hyperkähler manifolds with distinguished properties in the first place. A challenging obstacle to our full understanding of hyperkähler manifolds and their behaviour in families is the fact that hyperkähler manifolds can "collapse", that is, they can converge to a limit space of lower dimension. From the physics perspective, collapse of the hyperkähler spaces of vacua arise in certain limits of the corresponding physical theory where coupling constants converge to zero or infinity.In recent years substantial progress has been made in the study of collapsed degenerations of hyperkähler manifolds in the lowest possible dimension 4. Non-compact hyperkähler manifolds with prescribed asymptotic geometry have played a key role in these recent advances: 4-dimensional hyperkähler manifolds with interesting asymptotic geometry have been constructed since the 1980's using an array of diverse techniques, but only recently they have been completely classified. In this project we aim to construct and classify higher dimensional hyperkähler manifolds with a distinguished asymptotic geometry that we call QALF, solving completely the existence and uniqueness problem for this class of spaces.Applications of this study are numerous. Within hyperkähler geometry, we aim to use the new examples as building blocks to produce more complicated examples of higher dimensional hyperkähler manifolds and to study their behaviour in families. Beyond pure mathematics, the project has direct applications to theoretical physics, where QALF hyperkähler manifolds arise as spaces of vacua of 3-dimensional quantum gauge theories. While the definition of the quantum theory itself in rigorous mathematical language is currently out of reach, in this project we define rigorously the hyperkähler spaces of vacua of the theory, from which properties of the physical theory can then be derived.