Deformations of asymptotically cylindrical associative submanifolds

Abstract

This article develops the deformation theory of asymptotically cylindrical (ACyl) associative submanifolds in ACyl $G_2$-manifolds, laying the foundation for the gluing of ACyl associative submanifolds in twisted connected sum $G_2$-manifolds presented by the author in [Ber22]. We study the moduli space of ACyl associative submanifolds with a fixed asymptotic holomorphic curve and a fixed rate, as well as the moduli space where this asymptotic data is allowed to vary, each endowed with a natural topology. We compute their virtual dimensions and show that in the best-case scenario, the latter embeds as a Lagrangian submanifold in the moduli space of holomorphic curves of the limiting Calabi-Yau 3-fold. These results may also be of independent interest for future.