## Hodge-theoretic mirror symmetry for toric stacks.(English)Zbl 1464.14044

Summary: Using the mirror theorem [T. Coates et al., Compos. Math. 151, No. 10, 1878–1912 (2015; Zbl 1330.14093)], 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.

### MSC:

 14J33 Mirror symmetry (algebro-geometric aspects) 14D23 Stacks and moduli problems 14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials 14M25 Toric varieties, Newton polyhedra, Okounkov bodies 53D37 Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category

Zbl 1330.14093
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