Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 032, 111 pages      arXiv:1906.00801      https://doi.org/10.3842/SIGMA.2020.032
Contribution to the Special Issue on Integrability, Geometry, Moduli in honor of Motohico Mulase for his 65th birthday

Global Mirrors and Discrepant Transformations for Toric Deligne-Mumford Stacks

Hiroshi Iritani
Department of Mathematics, Graduate School of Science, Kyoto University, Kitashirakawa-Oiwake-cho, Sakyo-ku, Kyoto, 606-8502, Japan

Received June 13, 2019, in final form March 29, 2020; Published online April 22, 2020

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 $\widehat{\Gamma}$-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.

Key words: quantum cohomology; mirror symmetry; toric variety; Landau-Ginzburg model; Gamma-integral structure.

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