G2-Instantons on Joyce-Karigiannis Manifolds

Joyce-Karigiannis 流形上的 G2-瞬时

• 批准号: 1916384
• 金额: --
• 依托单位: Imperial College London
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2017
• 资助国家: 英国
• 起止时间: 2017 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-1916384
• 关键词: G2; Instantons; Joyce; Karigiannis; Manifolds

G2-instantons are solutions to a particular partial differential equation, the G2-instanton equation. The project aims to answer the question what the set of G2-instantons looks like.The project focuses on one class of seven-dimensional spaces on which to consider the G2-instanton equation, called Joyce-Karigiannis manifolds. In particular, the questions to be answered are: Do there exist G2-instantons on these manifolds? If there exist any, are there finitely many or infinitely many?One novel contribution is to transfer analytic results from unbounded spaces to Joyce-Karigiannis manifolds, in order to show existence of solutions. This process is called gluing, and has previously been carried out on other spaces. Another novel contribution is to use solutions to the well understood Hermitian Yang Mills equation in dimension six, to obtain a family of solutions to the G2-instanton equation. This part makes use of results from the field of algebraic geometry to obtain solutions in dimension six. It will also feed back into this field, by studying solutions in dimension six which have a particular symmetry.There is an ongoing research effort in pure mathematics to count solutions to the G2-instanton equation in order to obtain a numerical invariant of a space. All of this is motivated by string theory, which predicts that there is some way to obtain a numerical invariant (called an observable) from counting solutions to the G2-instanton equation. This project has the potential not only to construct example G2-instantons, but to find all G2-instantons on a given space for the first time, owing to the special connection to six dimensions. Furthermore, there is a conjecture about the limiting behaviour of families of solutions to the G2-instanton equation. This project may produce examples that can give further evidence to this conjecture.

G2-瞬子是一个特殊的偏微分方程，G2-瞬子方程的解。该项目旨在回答G2-瞬子的集合是什么样子的问题。该项目专注于一类七维空间，在其上考虑G2-瞬子方程，称为Joyce-Karigiannis流形。特别是，要回答的问题是：是否存在G2-瞬子在这些流形上？如果有，是无穷多个还是无穷多个？一个新的贡献是转移的分析结果从无界空间的Joyce-Karigiannis流形，以显示存在的解决方案。这个过程被称为胶合，以前已经在其他空间进行过。另一个新的贡献是使用的解决方案，以及理解厄米杨米尔斯方程在6维，以获得一个家庭的解决方案的G2-瞬子方程。这一部分利用代数几何领域的结果来获得六维的解。它也将反馈到这个领域，通过研究在六维中有一个特殊的对称性的解决方案。有一个正在进行的研究工作，在纯数学计数的解决方案，以获得一个空间的数值不变量的G2-瞬子方程。所有这些都是由弦理论激发的，弦理论预言，有某种方法可以通过计算G2-瞬子方程的解来获得一个数值不变量（称为可观测量）。该项目不仅有可能构建示例G2-瞬子，而且由于与六维的特殊联系，首次在给定空间上找到所有G2-瞬子。此外，有一个猜想的限制行为的家庭的解决方案的G2-瞬子方程。这个项目可能会产生一些例子，可以为这个猜想提供进一步的证据。