Exceptional geometric structures required for string theory and M-theory: moduli spaces and formation of singularities

弦理论和 M 理论所需的特殊几何结构：模空间和奇点的形成

• 批准号: RGPIN-2014-05050
• 负责人: Karigiannis, Spiro
• 金额: $ 1.02万
• 依托单位: University of Waterloo
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=611406
• 关键词: Exceptional; geometric; structures; required; string

My research lies in an area of higher dimensional geometry that is closely linked to theoretical physics. For about one hundred years, physicists have been searching for a theory that mathematically unifies gravity with quantum mechanics. An extremely promising candidate is M-theory, which describes the universe in terms of a 7-dimensional geometric shape (called a manifold) that is curved in a very special way. These shapes are called G2 manifolds. However, for the physical theory to be consistent with reality one requires these G2 manifolds to have certain corners (called singularities) which look like cones. The subset of G2 manifolds that have cone-like points are called G2 conifolds. The problem is that although we know thousands of examples of "smooth" G2 manifolds (those without cone-like points), there is still no mathematical proof that proper G2 conifolds actually exist. They definitely are expected to exist, and in abundance, both from physical arguments and from rigorous mathematical work of myself and Lotay. The principal short-term goal of my proposed research project is to construct the first ever examples of G2 conifolds, thereby providing a rigorous mathematical proof of their existence. This is an extremely important problem to solve, because it would give conclusive mathematical justification for the feasibility of M-theory as a model of our physical universe. The method I propose to use is a generalization of a method of constructing smooth G2 manifolds of myself and Joyce, which involves glueing onto the shape a particular family of spaces that are solutions to Einstein's equations of general relativity. Another important short-term goal of my proposed research project is to understand the set of all possible G2 manifolds (called the "moduli space"), which is itself a geometric shape of very high dimension. Studying the way in which a smoothly deforming G2 manifold can develop cone-like points involves considering curves on the moduli space that reach the boundary. I propose to investigate this question by analyzing the curvature of the moduli space itself. Establishing upper bounds on this curvature gives quantitative information about the formation of cone-like singularities and imposes restrictions on the associated physics. The long-term mathematical goal is to understand the structure of G2 manifolds as well as we understand Calabi-Yau manifolds, which are 6-dimensional shapes with similar properties that are much better understood. Both of these types of manifolds are candidates for grand unified theories in physics, particularly superstring theory and M-theory. Mathematically, G2 manifolds are very interesting objects because they share many common properties with Calabi-Yau manifolds, such as special types of submanifolds (smaller shapes sitting inside them) and connections (rules for measuring the rates of change on such shapes). In spite of this, there is sharp contrast, however, because for technical reasons G2 manifolds cannot be studied using the same tools that have been successful for Calabi-Yau manifolds, namely methods of classical algebraic geometry. This is because, rather than being locally modelled by the complex numbers like the Calabi-Yau manifolds are, they are locally modelled by an exceptional number system that exists only in 7 real dimensions. Since tools of algebraic geometry are not available, we need to study such manifolds instead using techniques from analysis, namely nonlinear partial differential equations. It is precisely for this reason that the mathematical analysis of G2 manifolds and G2 conifolds is so technically difficult, and why there are so many fewer mathematicians working in this modern area as opposed to the classical area of Calabi-Yau manifolds.

我的研究属于与理论物理学密切相关的高维几何领域。大约一百年来，物理学家一直在寻找一种在数学上将引力与量子力学统一起来的理论。一个非常有前途的候选者是 M 理论，它用以一种非常特殊的方式弯曲的 7 维几何形状（称为流形）来描述宇宙。这些形状称为 G2 流形。然而，为了使物理理论与现实保持一致，需要这些 G2 流形具有某些看起来像圆锥体的角（称为奇点）。具有类锥点的 G2 流形子集称为 G2 锥折。问题是，尽管我们知道数以千计的“平滑”G2 流形（没有类锥点的流形）的例子，但仍然没有数学证明真正存在正确的 G2 锥折。无论是从物理论证还是从我和洛泰严格的数学工作来看，它们肯定是存在的，而且是大量存在的。 我提出的研究项目的主要短期目标是构造第一个 G2 锥折的例子，从而为它们的存在提供严格的数学证明。这是一个需要解决的极其重要的问题，因为它将为 M 理论作为我们物理宇宙模型的可行性提供决定性的数学论证。我建议使用的方法是构造我自己和乔伊斯的平滑 G2 流形的方法的概括，其中涉及将作为爱因斯坦广义相对论方程解的特定空间族粘合到形状上。 我提出的研究项目的另一个重要的短期目标是了解所有可能的 G2 流形的集合（称为“模空间”），它本身就是一个非常高维度的几何形状。研究平滑变形 G2 流形形成锥状点的方式涉及考虑模空间上到达边界的曲线。我建议通过分析模空间本身的曲率来研究这个问题。建立该曲率的上限给出了有关锥状奇点形成的定量信息，并对相关物理施加了限制。 长期的数学目标是理解 G2 流形的结构，就像我们理解 Calabi-Yau 流形一样，Calabi-Yau 流形是具有相似属性的 6 维形状，可以更好地理解。这两种类型的流形都是物理学大统一理论的候选者，特别是超弦理论和 M 理论。从数学上讲，G2 流形是非常有趣的对象，因为它们与 Calabi-Yau 流形具有许多共同的属性，例如特殊类型的子流形（位于其中的较小形状）和连接（测量此类形状变化率的规则）。尽管如此，还是存在鲜明的对比，因为由于技术原因，G2 流形无法使用与 Calabi-Yau 流形相同的成功工具（即经典代数几何方法）来研究。这是因为，它们不是像 Calabi-Yau 流形那样由复数进行局部建模，而是由仅存在于 7 个实维中的特殊数字系统进行局部建模。由于代数几何工具不可用，我们需要研究此类流形，而不是使用分析技术，即非线性偏微分方程。正是由于这个原因，G2 流形和 G2 锥折的数学分析在技术上如此困难，也是为什么与卡拉比-丘流形的经典领域相比，在这个现代领域工作的数学家要少得多的原因。