text: We introduce a global Landau-Ginzburg model which is mirror to several toric Deligne-Mumford stacks and describe the change of the Gromov-Witten theories under discrepant transformations. We prove a formal decomposition of the quantum cohomology D-modules (and of the all-genus Gromov-Witten potentials) under a discrepant toric wall-crossing. In the case of weighted blowups of weak-Fano compact toric stacks along toric centres, we show that an analytic lift of the formal decomposition corresponds, via the Gamma-integral structure, to an Orlov-type semiorthogonal decomposition of topological K-groups. We state a conjectural functoriality of Gromov-Witten theories under discrepant transformations in terms of a Riemann-Hilbert problem.
  @cite_18: We study B-branes in two-dimensional N=(2,2) anomalous models, and their behaviour as we vary bulk parameters in the quantum Kahler moduli space. We focus on the case of (2,2) theories defined by abelian gauged linear sigma models (GLSM). We use the hemisphere partition function as a guide to find how B-branes split in the IR into components supported on Higgs, mixed and Coulomb branches: this generalizes the band restriction rule of Herbst-Hori-Page to anomalous models. As a central example, we work out in detail the case of GLSMs for Hirzebruch-Jung resolutions of cyclic surface singularities. In these non-compact models we explain how to compute and regularize the hemisphere partition function for a brane with compact support, and check that its Higgs branch component explicitly matches with the geometric central charge of an object in the derived category.
  @cite_54: We study B-type D-branes in linear sigma models with Abelian gauge groups. The most important finding is the grade restriction rule. It classifies representations of the gauge group on the Chan-Paton factor, which can be used to define a family of D-branes over a region of the Kahler moduli space that connects special points of different character. As an application, we find a precise, transparent relation between D-branes in various geometric phases as well as free orbifold and Landau-Ginzburg points. The result reproduces and unifies many of the earlier mathematical results on equivalences of D-brane categories, including the McKay correspondence and Orlov's construction.
summary: Recently, Clingempeel-Le Floch-Romo @cite_18 compared the hemisphere partition functions (which in our language correspond to certain solutions of the quantum D-modules) of the cyclic quotient singularities @math and their Hirzebruch-Jung resolutions. They discussed the relation to a semiorthogonal decomposition of the the derived categories, extending the work of Herbst-Hori-Page @cite_54 to the anomalous (discrepant) case. Their examples are complementary to ours: the Hirzebruch-Jung resolutions are type (II-ii) discrepant transformations whereas transformations in Theorem are of type (II-i) or (III) (see Remark for these types).