Gauged linear sigma models for mirrors of singular double covers and their noncommutative resolutions

I will present a construction of mirrors of singular Calabi–Yau double covers via noncommutative resolutions realized by gauged linear sigma models with hybrid phases. I will focus on the correspondence between $A$ and $B$-periods and present some conjectures about their derived categories and other invariants. This is based on joint work with Bong Lian and Tsung-Ju Lee.