We initiate the systematic study of $$G_2$$G2-instantons with SU(2)$$^2$$2-symmetry. As well as developing foundational theory, we give existence, non-existence and classification results for these instantons. We particularly focus on $$\mathbb {R}^4\times S^3$$R4×S3 with its two explicitly known distinct holonomy $$G_2$$G2 metrics, which have different volume growths at infinity, exhibiting the different behaviour of instantons in these settings. We also give an explicit example of sequences of $$G_2$$G2-instantons where “bubbling” and “removable singularity” phenomena occur in the limit.

中文翻译：

---

SU(2)$$^2$$2-不变量$$G_2$$G2-instantons

我们开始对具有 SU(2)$$^2$$2-symmetry 的 $$G_2$$G2-instantons 进行系统研究。除了发展基础理论，我们还给出了这些瞬时子的存在、不存在和分类结果。我们特别关注 $$\mathbb {R}^4\times S^3$$R4×S3 及其两个明确已知的完全不同的 $$G_2$$G2 度量，它们在无穷远时具有不同的体积增长，表现出不同的行为这些设置中的瞬间子。我们还给出了 $$G_2$$G2 瞬子序列的一个明确示例，其中在极限中出现“冒泡”和“可移除奇点”现象。