Local solution to the $G_{2}-$monopole equation with prescribed tangent cone and $G_{2}-$structure

TL;DR

This paper proves that for any local $G_{2}$-structure and Hermitian-Yang-Mills connection on $S^{6}$, there exists a local solution to the $G_{2}$-monopole equation that approaches the HYM-connection asymptotically.

Contribution

It establishes the existence of local solutions to the $G_{2}$-monopole equation with prescribed tangent cone and $G_{2}$-structure, expanding understanding of monopole solutions in $G_{2}$-geometry.

Findings

Existence of local solutions to the $G_{2}$-monopole equation.

Solutions are asymptotic to given Hermitian-Yang-Mills connections.

Applicable to any locally defined $G_{2}$-structure on $S^{6}$.

Abstract

For any locally defined $G_{2}-$structure and Hermitian-Yang-Mills connection on $S^{6}$, the $G_{2}-$monopole equation always admits a local solution that is asymptotic to the HYM-connection.