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: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=655107
• 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.

我的研究在于与理论物理学密切相关的较高维几何形状的领域。大约一百年来，物理学家一直在寻找一种用量子力学统一重力的理论。一个非常有前途的候选人是M理论，它以一种非常特殊的方式弯曲的7维几何形状（称为歧管）来描述宇宙。这些形状称为G2歧管。但是，要使物理理论与现实一致，就需要这些G2歧管具有某些看起来像锥体的角（称为奇异性）。具有锥状点的G2歧管的子集称为G2对照。问题在于，尽管我们知道成千上万的“平滑” G2歧管（那些没有锥形的点）示例，但仍然没有数学证据表明实际上存在适当的G2 Conifolds。从物理论证和我本人和洛伊代的严格数学工作中，它们都可以存在，并且在大量的情况下。这是一个非常重要的问题，因为它将为M-Bealing作为我们物理宇宙的模型的可行性提供结论性的数学合理性。我建议使用的方法是对我和乔伊斯的平滑G2歧管的一种概括，涉及将粘贴到形状上的特定空间家族，这是对爱因斯坦的一般相对性方程的解决方案。研究平滑变形的G2歧管可以形成锥形点的方式涉及考虑到达边界的模量空间上的曲线。我建议通过分析模量空间本身的曲率来研究这个问题。在此曲率上建立上限提供了有关形成类锥形奇异性的定量信息，并对相关物理施加限制。这两种类型的流形都是物理学统一理论的候选人，尤其是超音理论和理论。从数学上讲，G2歧管是非常有趣的对象，因为它们具有许多常见的属性与Calabi-yau歧管，例如特殊类型的子序列（坐在其中的较小形状）和连接（衡量这种形状变化速率的规则）。尽管如此，但仍存在鲜明的对比，因为出于技术原因，G2歧管不能使用与卡拉比YAU歧管成功的工具一起研究，即经典代数几何形状的方法。这是因为，而不是像卡拉比（Calabi-Yau）歧管那样由局部建模，而是由仅在7个实际维度中存在的特殊数字系统进行局部建模。由于无法使用代数几何形状的工具，因此我们需要研究这种歧管，而是使用分析中的技术，即非线性偏微分方程。正是由于这个原因，对G2歧管和G2 Conifolds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较少而不是Calabi-yau歧管的经典领域。