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Hyperkähler流形成为更大类几何空间的特别美丽和受限的例子：例如，所有的Hyperkähler流形都是Ricci平坦的，因此与爱因斯坦广义相对论方程的解密切相关。从一个完全不同的角度来看，自1980年代的S以来，人们理解超卡勒流形是作为理论物理中许多规范理论的真空(或“平衡态”)空间自然产生的，即推广麦克斯韦电磁方程的物理理论。在数学和物理之间卓有成效的互动中，超kähler流形的几何性质可以用来推导相应的物理理论的性质，而物理学首先预测了具有特殊性质的超kähler流形的存在。我们完全理解Hyperkähler流形及其在家族中的行为的一个具有挑战性的障碍是，Hyperkähler流形可以“坍塌”，也就是说，它们可以收敛到更低维的极限空间。从物理的角度来看，真空的Hyperkähler空间在相应的物理理论的某些极限下会出现坍塌，当耦合常数收敛到零或无穷大时。近年来，关于最低维Hyperkähler流形的坍塌退化的研究取得了实质性的进展。具有指定的渐近几何的非紧致的Hyperkähler流形在最近的进展中起到了关键作用：自1980年代S以来，人们利用各种不同的技术构造了具有有趣的渐近几何的四维Hyperkähler流形，但直到最近才完全分类。在这个项目中，我们的目的是构造和分类具有一种特殊的渐近几何的高维Hyperkähler流形，我们称之为QALF，完全解决了这类空间的存在唯一性问题。在Hyperkähler几何中，我们的目标是使用新的例子作为构建块来产生更复杂的高维Hyperkähler流形的例子，并研究它们在族中的行为。除了纯粹的数学，该项目还直接应用于理论物理，在理论物理中，QALF Hyperkähler流形作为三维量子规范理论的真空空间而出现。虽然目前还不能用严格的数学语言来定义量子理论本身，但在这个项目中，我们严格地定义了该理论的真空的超kähler空间，然后可以从它推导出物理理论的性质。

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.