  It has been conjectured that the hemisphere partition function
arXiv:1308.2217, arXiv:1308.2438 in a gauged linear sigma model (GLSM) computes
the central charge arXiv:math/0212237 of an object in the bounded derived
category of coherent sheaves for Calabi--Yau (CY) manifolds. There is also
evidence in arXiv:alg-geom/ 9511001, arXiv:hep-th/0007071. On the other hand,
non-commutative resolutions of singular CY varieties have been studied in the
context of abelian GLSMs arXiv:0709.3855. In this paper, we study an analogous
construction of abelian GLSMs for non-commutative resolutions and propose they
can be used to study a class of recently discovered mirror pairs of singular CY
varieties. Our main result shows that the hemisphere partition functions
(a.k.a.~$A$-periods) in the new GLSM are in fact period integrals
(a.k.a.~$B$-periods) of the singular CY varieties. We conjecture that the two
are completely equivalent: $B$-periods are the same as $A$-periods. We give
some examples to support this conjecture and formulate some expected
homological mirror symmetry (HMS) relation between the GLSM theory and the CY.
As shown in arXiv:2003.07148, the $B$-periods in this case are precisely given
by a certain fractional version of the $B$-series of arXiv:alg-geom/9511001.
Since a hemisphere partition function is defined as a contour integral in a
cone in the complexified secondary fan (or FI-theta parameter space)
arXiv:1308.2438, it can be reduced to a sum of residues (by theorems of
Passare-Tsikh-Zhdanov and Tsikh-Zhdanov). Our conjecture shows that this
residue sum may now be amenable to computations in terms of the $B$-series.
