On Hypergeometric Duality Conjecture

TL;DR

This paper proves a duality conjecture for GKZ hypergeometric systems by providing an explicit formula and connecting it to cohomology pairings on toric stacks, advancing understanding of hypergeometric PDEs and their geometric interpretations.

Contribution

It offers an explicit formula for the duality of GKZ hypergeometric systems and links it to cohomological pairings on toric Deligne-Mumford stacks, confirming a conjecture.

Findings

Explicit formula for GKZ duality

Identification with cohomology pairing in the limit

Connection to $ ext{Gamma}$-series solutions

Abstract

We give an explicit formula for the duality, previously conjectured by Horja and Borisov, of two systems of GKZ hypergeometric PDEs. We prove that in the appropriate limit this duality can be identified with the inverse of the Euler characteristics pairing on cohomology of certain toric Deligne-Mumford stacks, by way of $\Gamma$-series cohomology valued solutions to the equations.