$G_2$-瞬子于 $S^7$ 上的Morse指标

中文摘要 AI 辅助

一个经典结果表明，在单位球面 $S^n$（$n\ge5$）上，每个非平坦的Yang--Mills联络的Morse指标至少为 $n+1$。在本文中，我们证明了在四元数Hopf纤维化 $S^7\to S^4$ 下，$S^4$ 上标准反自对偶BPST瞬子的拉回达到了这个下界，其Morse指标为 $8$，零化度为 $20$。这提供了一个不同于 $SO(8)\to S^7$ 上典型联络的例子。Waldron曾将此联络作为标准 $G_2$-瞬子进行研究，并证明了在固定的标准近平行 $G_2$-结构下，其无穷小形变空间的维数为 $15$。我们将此分析扩展到完整的Yang--Mills二阶变分，确定了Coulomb规范下Jacobi算子的负特征空间和整个核。我们的结果确立了该联络具有最小的可能Morse指标，并且其 $20$ 维Yang--Mills核包含Waldron的 $15$ 维形变空间，以及另外五个独立的零模。

英文摘要

A classical result states that every nonflat Yang--Mills connection on the unit round sphere $S^n$, $n\ge5$, has Morse index at least $n+1$. In this paper, we prove that the pullback of the standard anti-self-dual BPST instanton on $S^4$ under the quaternionic Hopf fibration $S^7\to S^4$ attains this lower bound, with Morse index $8$ and nullity $20$. This provides an example distinct from the canonical connection on $SO(8)\to S^7$. Waldron studied this connection as the standard $G_2$-instanton and showed that its infinitesimal deformation space, for the fixed standard nearly parallel $G_2$-structure, has dimension $15$. We extend this analysis to the full Yang--Mills second variation, determining both the negative eigenspace and the entire kernel of the Jacobi operator in Coulomb gauge. Our result establishes that the connection has the smallest possible Morse index and that its $20$-dimensional Yang--Mills kernel contains Waldron's $15$-dimensional deformation space, together with five additional independent zero modes.

发表机构

• Tsinghua University（清华大学）
• Nanjing University of Science and Technology（南京理工大学）

机构由 AI 辅助整理，请以论文原文为准。