Associative submanifolds in twisted connected sum $G_2$-manifolds

Abstract: We introduce a new method to construct closed rigid (unobstructed) associative submanifolds in the twisted connected sum $G_2$-manifolds [Kov03, KL11, CHNP15]. More precisely, we prove a gluing theorem of asymptotically cylindrical (ACyl) associative submanifolds in ACyl $G_2$-manifolds under a hypothesis that can be interpreted as a transverse Lagrangian intersection condition. This is analogous to the gluing theorem for $G_2$-instantons introduced in [SW15]. We rephrase the hypothesis in the special cases where the ACyl associative submanifolds are obtained from holomorphic curves or special Lagrangians in ACyl Calabi-Yau $3$-folds. In this way we find many new associative submanifolds which are diffeomorphic to $S^3$, $\mathbf R\mathbf P^3$ or $\mathbf R\mathbf P^3#\mathbf R\mathbf P^3$. To lay the foundation for the gluing theorem we also study the moduli space of ACyl associative submanifolds with a natural topology. We prove that the moduli space is locally homeomorphic to the zero set of a smooth map between two finite dimensional spaces whose index depends only on the asymptotic cross section.