G2-Instantons on Joyce-Karigiannis Manifolds

G2-瞬子是一种特殊偏微分方程的解，即G2-瞬子方程。该项目旨在回答G2-瞬子集是什么样子的问题。该项目专注于一类可用于考虑G2-瞬子方程的七维空间，称为Joyce-Karigiannis流形。特别是，需要回答的问题是：这些流形上是否存在G2瞬子？如果存在解，它是有限多还是无穷多？一个新颖的贡献是将分析结果从无界空间转移到Joyce-Karigiannis流形上，以证明解的存在性。这一过程被称为粘合，以前曾在其他空间进行过。另一个新的贡献是利用广为人知的六维厄米-杨-米尔斯方程的解，得到G2-瞬子方程的一族解。这一部分利用代数几何领域的结果来获得六维的解。它还将通过研究具有特定对称性的6维的解来反馈到这个领域。在纯数学中，有一项正在进行的研究工作，即对G2瞬子方程的解进行计数，以获得空间的数值不变量。所有这一切都是由弦理论推动的，该理论预测，有某种方法可以从G2-瞬子方程的计数解中获得数值不变量(称为观测值)。由于与六维空间的特殊联系，该项目不仅有可能构造示例G2瞬子，而且首次发现给定空间上的所有G2瞬子。此外，还有一个关于G2-瞬子方程解族的极限行为的猜想。这个项目可能会给出进一步证明这一猜想的例子。

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.