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 [2]. 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 also express their virtual dimensions in terms of data on the asymptotic cross-sections.