Thin instantons in $G_2$-manifolds and Seiberg-Witten invariants

Abstract

For two nearby disjoint coassociative submanifolds $C$ and $C'$ in a $G_2$-manifold, we construct thin instantons with boundaries lying on $C$ and $C'$ from regular J-holomorphic curves in $C$. We explain their relationship with the Seiberg-Witten invariants for $C$.