Using the mirror theorem [CCIT15], 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 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.
The theory of virtual fundamental class defines important
invariants such as the Gromov–Witten and the Donaldson–
Thomas invariants. It has been generalized to the cosection
localized virtual cycle which has applications in Seiberg–
Witten, Fan–Jarvis–Ruan–Witten and other invariants. In
this paper, we prove the formulas of virtual pullback, torus
localization and wall crossing for cosection localized virtual