Year: 2026
Type: preprint
Abstract: In this paper, we determine the nonpositive spectrum of the Yang-Mills Jacobi operator in Coulomb gauge at the standard $G_2$-instanton on $S^7$. We prove that the only negative eigenvalue is $-3$, with multiplicity $8$, and the kernel has dimension $20$. Thus the connection attains the sharp lower bound for the Morse index of nonflat Yang-Mills connections on $S^7$. We also give an explicit description of the kernal of the Jacobi operator, which contains Waldron's $15$-dimensional deformation space for the fixed nearly parallel $G_2$-structure. The 5 additional zero modes are isometry-induced variations that do not satisfy the corresponding linearized instanton equation.
Source: arXiv (Cornell University)
Authors: Han, Xiaoli, Wen, Yang
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7213278338