2023/07/05 by Lee, Tsung-Ju, Lian, Bong H., Romo, Mauricio
#Algebraic Geometry (math.AG) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th)
paper · doi:10.48550/arxiv.2307.02038
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.