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.

我的研究领域是与理论物理学密切相关的高维几何学。大约100年来，物理学家一直在寻找一种理论，在数学上将引力与量子力学统一起来。一个非常有希望的候选者是M理论，它用一个以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。这些形状被称为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的角（称为奇点）。G2流形中具有锥状点的子集称为G2锥。问题是，尽管我们知道成千上万的“光滑”G2流形（没有锥状点的流形）的例子，但仍然没有数学证据证明真正的G2锥确实存在。它们肯定是存在的，而且是丰富的，无论是从物理论证还是从我和洛泰的严格数学工作中。 我所提出的研究项目的主要短期目标是构建有史以来第一个G2 conifolds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确凿的数学证明。我建议使用的方法是我和乔伊斯构造光滑G2流形的方法的推广，其中包括将一个特定的空间族粘在形状上，这些空间族是爱因斯坦广义相对论方程的解。 我提出的研究项目的另一个重要的短期目标是理解所有可能的G2流形的集合（称为“模空间”），它本身就是一个非常高维的几何形状。研究光滑变形的G2流形可以发展锥状点的方式涉及到考虑到达边界的模空间上的曲线。我建议通过分析模空间本身的曲率来研究这个问题。建立这个曲率的上界给出了关于锥状奇点形成的定量信息，并对相关的物理学施加了限制。 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，这是一种具有类似性质的6维形状，更容易理解。这两种类型的流形都是物理学中大统一理论的候选者，特别是超弦理论和M理论。在数学上，G2流形是非常有趣的对象，因为它们与Calabi-Yau流形有许多共同的性质，例如特殊类型的子流形（位于其中的较小形状）和连接（测量这些形状变化率的规则）。尽管如此，有鲜明的对比，但是，因为技术原因G2流形不能使用相同的工具，已成功的卡-丘流形，即方法的经典代数几何研究。这是因为，它们不是像卡-丘流形那样由复数局部建模，而是由一个只存在于7真实的维中的特殊数字系统局部建模。由于代数几何的工具是不可用的，我们需要研究这样的流形，而不是使用技术分析，即非线性偏微分方程。正是由于这个原因，G2流形和G2锥的数学分析在技术上是如此困难，以及为什么在这个现代领域工作的数学家比在经典的卡-丘流形领域工作的数学家少得多。