Singularities and Collapsing in G2 Manifolds

流形是一种数学方法来描述各种空间所产生的应用。这个项目的重点是一类特殊的流形称为G2流形。这些7维空间在理论物理学中起着至关重要的作用，特别是在所谓的M理论中。除了它们的物理兴趣，G2流形提供了丰富的几何结构，一个理想的设置来研究在几何的不同部分，如黎曼几何和自旋几何，子流形几何和规范理论中产生的对象之间的美丽和深刻的相互作用。由于G2流形是奇数维的，复几何的方法不能直接应用，不像卡拉比-丘流形，这是出现在理论物理弦理论中的6维对应物。因此，目前对G2流形的理解是有限的。M理论家感兴趣的G2流形必须包含数学家称之为奇点的特殊点，这一事实带来了额外的困难。受到弦理论和M理论之间的一个预期物理对偶的启发，该项目旨在开发一种新的G2流形结构并探索其后果。其关键思想是通过坍缩现象将G2几何与更好理解的卡-丘几何联系起来：目的是构造7维G2流形，它们看起来非常接近低维的卡-丘流形。IIA型弦理论和圆上紧化的M-理论之间的物理对偶可以被几何地解释为存在坍缩为卡-丘三重的紧G2流形序列。自21世纪初以来，这些坍缩现象的非紧凑例子已经出现在物理和数学文献中。该项目旨在更彻底地研究这些完整的例子，并将它们用作各种胶合结构中紧致G2流形行为的局部模型。该研究项目的目标是：（i）开发一种新的紧G2流形的结构，该结构接近Calabi-Yau 3-fold collapse limit;（ii）扩展该结构，以产生具有孤立圆锥奇点的紧G2流形的第一个已知例子;（iii）研究这些奇异G2流形的光滑化理论，并给出“几何变换”的例子在G2几何中类似于卡-丘几何中的翻转。除了对M-理论中的期望进行严格的数学验证之外，成功实现这些目标还可能产生更多的紧致7-流形，已知它们具有无挠G2结构，推进对高维完全非紧Ricci平坦流形的理解，这些流形具有超越渐近圆锥和渐近圆柱的有趣渐近几何，并有助于理解光滑G2流形模空间的边界。作为这些结构的低维类似物，也将研究K3曲面上Ricci平坦度量序列中发生的冒泡现象，并将其坍缩到低维极限。

Manifolds are a mathematical way to describe various spaces arising from applications. This project focuses on a special class of manifolds called G2 manifolds. These 7-dimensional spaces play a vital role in theoretical physics, particularly in the so-called M-theory. Besides their physical interest, G2 manifolds provide a rich geometric structure, an ideal setting to study beautiful and deep interactions between objects arising in different parts of geometry such as Riemannian and spin geometry, submanifold geometry, and gauge theory. Since G2 manifolds are odd-dimensional, the methods of complex geometry cannot be applied directly, unlike the case of Calabi-Yau manifolds, which are the 6-dimensional counterpart appearing in string theory of theoretical physics. As a result, the current understanding of G2 manifolds is limited. Additional difficulties arise from the fact that the G2 manifolds of interest to M-theorists must contain special points that mathematicians call singularities. Inspired by one of the expected physical dualities between string theory and M-theory, the project aims to develop a new construction of G2 manifolds and explore its consequences. The key idea is to relate G2 geometry to the better understood Calabi-Yau geometry through the phenomenon of collapsing: the aim is to construct 7-dimensional G2 manifolds that look very close to Calabi-Yau manifolds of one lower dimension. The expected physical duality between Type IIA string theory and M-theory compactified on a circle can be interpreted geometrically as the existence of sequences of compact G2 manifolds collapsing to Calabi-Yau three-folds. Non-compact examples of these collapsing phenomena have appeared in the physics and mathematics literature since the early 2000s. The project aims to study these complete examples more thoroughly and use them as local models for the behavior of compact G2 manifolds in various gluing constructions. Goals of the research project are: (i) to develop a new construction of compact G2 manifolds close to a Calabi-Yau 3-fold collapsed limit; (ii) to extend the construction to produce the first known examples of compact G2 manifolds with isolated conical singularities; and (iii) to study the smoothing theory of these singular G2 manifolds and exhibit examples of "geometric transitions" in G2 geometry analogous to flops in Calabi-Yau geometry. Besides the rigorous mathematical verification of expectations in M-theory, success in achieving these goals would potentially generate more diffeomorphism types of compact 7-manifolds known to admit a torsion-free G2 structure, advance understanding of complete non-compact Ricci flat manifolds in higher dimensions with interesting asymptotic geometries beyond the asymptotically conical and asymptotically cylindrical case, and contribute to understanding of the boundary of the moduli space of smooth G2 manifolds. As a lower-dimensional analogue of these constructions, bubbling phenomena occurring in sequences of Ricci-flat metrics on K3 surfaces collapsing to lower-dimensional limits will also be studied.