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

TL;DR

This paper explores generalized mirror symmetries for twisted connected sum $G_2$ manifolds in Type II string theory, providing new examples, testing conjectures, and clarifying the role of the B-field in these dualities.

Contribution

It introduces a framework for understanding two types of mirror symmetries for $G_2$ manifolds, including novel examples and consistency checks with known models.

Findings

Evidence for two kinds of mirror symmetries in $G_2$ compactifications.

Construction of new smooth and singular mirror $G_2$ backgrounds.

Validation of mirror maps using Joyce orbifold examples.

Abstract

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.