Toric Deligne-Mumford stacks and the better behaved version of the GKZ hypergeometric system

TL;DR

This paper extends the combinatorial understanding of orbifold cohomology and K-theory for Deligne-Mumford toric stacks with deformation parameters, and constructs a mirror symmetry map providing solutions to a refined GKZ hypergeometric system.

Contribution

It generalizes the combinatorial description of orbifold cohomology and K-theory for toric stacks with deformation parameters, and introduces a mirror symmetry map for the better behaved GKZ system.

Findings

Provides a combinatorial description of orbifold cohomology and K-theory with deformation parameters.

Constructs a topological mirror symmetry map for the improved GKZ hypergeometric system.

Produces a complete set of solutions to the better behaved GKZ system.

Abstract

We generalize the combinatorial description of the orbifold (Chen--Ruan) cohomology and of the Grothendieck ring of a Deligne--Mumford toric stack and its associated stacky fan in a lattice $N$ in the presence of a deformation parameter $\beta \in N \otimes {\mathbb C}.$ As an application, we construct a topological mirror symmetry map that produces a complete system of $\Gamma$--series solutions to the better behaved version of the GKZ hypergeometric system for $\beta \in N \otimes {\mathbb C}.$