Towards Generalized Mirror Symmetry for Twisted Connected Sum $G_2$ Manifolds

摘要

We revisit our construction of mirror symmetries for compactifications ofType II superstrings on twisted connected sum $G_2$ manifolds. For a given$G_2$ manifold, we discuss evidence for the existence of mirror symmetries oftwo kinds: one is an autoequivalence for a given Type II superstring on amirror pair of $G_2$ manifolds, the other is a duality between Type II stringswith different chiralities for another pair of mirror manifolds. We clarify therole of the B-field in the construction, and check that the correspondingmassless spectra are respected by the generalized mirror maps. We discuss hintstowards a homological version based on BPS spectroscopy. We provide severalnovel examples of smooth, as well as singular, mirror $G_2$ backgrounds viapairs of dual projecting tops. We test our conjectures against a Joyce orbifoldexample, where we reproduce, using our geometrical methods, the known mirrormaps that arise from the SCFT worldsheet perspective. Along the way, we discussnon-Abelian gauge symmetries, and argue for the generation of theAffleck-Harvey-Witten superpotential in the pure SYM case.