Global Mirrors and Discrepant Transformations for Toric Deligne–Mumford Stacks

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 $\hat \Gamma$-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.