G2-Instantons on Joyce-Karigiannis Manifolds

g2 -瞬子是一个特定偏微分方程的解，g2 -瞬子方程。该项目旨在回答g2瞬子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。具体来说，需要回答的问题是：这些流形上是否存在g2 -瞬子？如果存在，是有限的还是无限的？一个新颖的贡献是将解析结果从无界空间转移到Joyce-Karigiannis流形，以证明解的存在性。这个过程被称为粘合，以前在其他空间进行过。另一个新颖的贡献是使用六维的厄米杨米尔斯方程的解来获得g2 -瞬子方程的一组解。这部分利用代数几何领域的结果，得到六维的解。它也会反馈到这个领域，通过研究具有特定对称性的六维解。为了获得空间的数值不变量，在纯数学中有一个正在进行的研究工作，即计算g2 -瞬子方程的解。所有这些都是由弦理论推动的，弦理论预测，有一些方法可以从计算g2 -瞬子方程的解中获得一个数值不变量（称为可观测值）。由于与六维的特殊联系，该项目不仅有可能构建示例g2 -瞬子，而且有可能首次在给定空间中找到所有g2 -瞬子。进一步，对g2 -瞬子方程解族的极限行为进行了推测。这个项目可能会产生一些例子，为这个猜想提供进一步的证据。

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.