Singularities and Collapsing in G2 Manifolds

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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.