Journal of Differential Geometry

Hodge-theoretic mirror symmetry for toric stacks

Tom Coates, Alessio Corti, Hiroshi Iritani, and Hsian-Hua Tseng

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Using the mirror theorem [15], we give a Landau–Ginzburg mirror description for the big equivariant quantum cohomology of toric Deligne–Mumford stacks. More precisely, we prove that the big equivariant quantum $D$-module of a toric Deligne–Mumford stack is isomorphic to the Saito structure associated to the mirror Landau–Ginzburg potential. We give a Gelfand–Kapranov–Zelevinsky (GKZ) style presentation of the quantum $D$-module, and a combinatorial description of quantum cohomology as a quantum Stanley–Reisner ring. We establish the convergence of the mirror isomorphism and of quantum cohomology in the big and equivariant setting.

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J. Differential Geom., Volume 114, Number 1 (2020), 41-115.

Received: 16 October 2016
First available in Project Euclid: 28 December 2019

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Coates, Tom; Corti, Alessio; Iritani, Hiroshi; Tseng, Hsian-Hua. Hodge-theoretic mirror symmetry for toric stacks. J. Differential Geom. 114 (2020), no. 1, 41--115. doi:10.4310/jdg/1577502022.

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