Associative submanifolds in twisted connected sum $G_2$-manifolds

We introduce a method to construct closed rigid associative submanifolds in twisted connected sum $G_2$-manifolds. More precisely, we prove a gluing theorem of asymptotically cylindrical (ACyl) associative submanifolds in ACyl $G_2$-manifolds under a transverse intersection hypothesis. 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 terms of algebraic-geometric conditions and topological conditions, respectively. This yields many rigid associative submanifolds with new topological types $S^3$, $\mathbf{R}{\mathbf{P}}^3$ or $\mathbf{R}{\mathbf{P}}^3\mathrm{\#}\mathbf{R}{\mathbf{P}}^3$.