Towards generalized mirror symmetry for twisted connected sum G 2 manifolds

We revisit our construction of mirror symmetries for compactifications of Type II superstrings on twisted connected sum G ₂ manifolds. For a given G ₂ manifold, we discuss evidence for the existence of mirror symmetries of two kinds: one is an autoequivalence for a given Type II superstring on a mirror pair of G ₂ manifolds, the other is a duality between Type II strings with different chiralities for another pair of mirror manifolds. We clarify the role of the B-field in the construction, and check that the corresponding massless spectra are respected by the generalized mirror maps. We discuss hints towards a homological version based on BPS spectroscopy. We provide several novel examples of smooth, as well as singular, mirror G ₂ backgrounds via pairs of dual projecting tops. We test our conjectures against a Joyce orbifold example, where we reproduce, using our geometrical methods, the known mirror maps that arise from the SCFT worldsheet perspective. Along the way, we discuss non-Abelian gauge symmetries, and argue for the generation of the Affleck-Harvey-Witten superpotential in the pure SYM case.