Morse Index of a $G_2$-Instanton over $S^7$

RESUMO

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.