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It has been conjectured that the hemisphere partition functionarXiv:1308.2217, arXiv:1308.2438 in a gauged linear sigma model (GLSM) computesthe central charge arXiv:math/0212237 of an object in the bounded derivedcategory of coherent sheaves for Calabi--Yau (CY) manifolds. There is alsoevidence in arXiv:alg-geom/ 9511001, arXiv:hep-th/0007071. On the other hand,non-commutative resolutions of singular CY varieties have been studied in thecontext of abelian GLSMs arXiv:0709.3855. In this paper, we study an analogousconstruction of abelian GLSMs for non-commutative resolutions and propose theycan be used to study a class of recently discovered mirror pairs of singular CYvarieties. 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 twoare completely equivalent: $B$-periods are the same as $A$-periods. We givesome examples to support this conjecture and formulate some expectedhomological 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 givenby 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 acone in the complexified secondary fan (or FI-theta parameter space)arXiv:1308.2438, it can be reduced to a sum of residues (by theorems ofPassare-Tsikh-Zhdanov and Tsikh-Zhdanov). Our conjecture shows that thisresidue sum may now be amenable to computations in terms of the $B$-series.

Non-commutative resolutions as mirrors of singular Calabi--Yau varieties