QALF hyperkähler metrics

Hyperkähler流形是一种几何空间，具有极其丰富的几何结构。这种丰富的结构使得超凯勒流形特别美丽，并且是更大类几何空间的约束例子：例如，所有的超凯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。从一个完全不同的角度来看，自20世纪80年代以来，人们理解超凯勒流形自然地出现在理论物理学中的许多规范理论的真空（或“平衡态”）空间中，即推广麦克斯韦电磁方程的物理理论。在数学和物理之间富有成效的相互作用中，超凯勒流形的几何性质可以用来导出相应的物理理论的性质，而物理学首先预言了具有杰出性质的超凯勒流形的存在。一个具有挑战性的障碍，我们充分了解超凯勒流形和他们的行为在家庭是事实，超凯勒流形可以“崩溃”，也就是说，他们可以收敛到一个极限空间的较低的维度。从物理学的角度来看，真空的超kähler空间的坍缩产生于相应的物理理论的某些极限，即耦合常数收敛到零或无穷大。具有规定渐近几何的非紧超凯勒流形在这些最近的进展中发挥了关键作用：具有有趣渐近几何的4维超凯勒流形自20世纪80年代以来已经使用各种技术构造，但直到最近才被完全分类。在这个项目中，我们的目标是构造和分类高维hyperkähler流形与一个杰出的渐近几何，我们称之为QALF，完全解决这类空间的存在性和唯一性问题。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。除了纯数学之外，该项目还直接应用于理论物理，其中QALF超凯勒流形作为三维量子规范理论的真空空间而出现。虽然量子理论本身在严格的数学语言中的定义目前还遥不可及，但在这个项目中，我们严格定义了理论真空的超凯勒空间，从中可以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.