Non-commutative resolutions and pre-quotients of Calabi-Yau double covers

Following an earlier proposal arXiv:2307.02038 to apply the GLSM formalism to understand the so-called non-commutative resolution, this paper takes one important step further to extend this formalism to a much larger class of non-commutative resolutions. The proposal was initially motivated by the discovery of a new class of mirror pairs singular Calabi-Yau varieties arXiv:2003.07148, given by certain branched double covers over toric varieties of MPCP type. The overarching problem was to understand these mirror pairs from the viewpoint of homological mirror symmetry arXiv:alg-geom/9411018. In the present paper, we propose two main results along this line. First, one new insight is that the `gauge-fixing' condition on the branching locus of the double cover used in arXiv:2003.07148 can be relaxed in an interesting way. This turns out to produce GLSMs that describe a much larger class of non-commutative resolutions, leading to $A$-periods for a larger class of non-commutative resolutions, as well as the GKZ systems for their $A$-periods. Second, we show that the $A$-periods can also be realized as $A$-periods of a certain smooth CICY family in a toric variety of MPCP type, such that a suitable finite quotient of this family recovers the double cover CY we have started with. We call this CICY family the `pre-quotient' of the double cover CY. This realization strongly suggests that pre-quotient may provide an important approach for understanding homological mirror symmetry for singular double cover CY varieties and non-commutative resolutions.