Non-commutative resolutions and periods of singular Calabi-Yau

I will introduce non-commutative resolutions of singular Calabi–Yau double covers, to investigate the notion of B-brane on these spaces. The non-commutative geometry is studied via a GLSM description, and B-branes in these GLSMs provide the non-commutative analogues of sheaves on smooth Calabi–Yau manifolds.Computing the central charges of the B-branes, I will show that they are annihilated by the GKZ system of the mirror singular Calabi–Yau, when the latter is known. We show that(i) There always exists a smooth Calabi–Yau complete intersection which satisfies the same GKZ system; (ii) The B-branes on the non-commutative resolution form the invariant sub-category of a certain equivariant category of coherent sheaves on the smooth complete intersection. This is a universal phenomenon, which allows us to find the GKZ system for the mirror Calabi–Yau, even when the mirror geometry is not known.Based on joint work with Tsung–Ju Lee, Bong Lian and Mauricio Romo.