Global Mirrors and Discrepant Transformations for Toric Deligne-Mumford Stacks

Abstract

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.

111 pages, 16 figures, published version

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• Phases Of N=2 Theories In 1+1 Dimensions With Boundary
• Hochschild homology and semiorthogonal decompositions
• Log Crepant Birational Maps and Derived Categories
• Asymptotic Expansion and the LG/(Fano, General Type) Correspondence
• Gamma II for toric varieties from integrals on T-dual branes and homological mirror symmetry

Cited by in corpus (1)

• Quantum flips I: local model