Reconstructing GKZ via topological recursion

TL;DR

This paper presents a new topological recursion approach to reconstruct the GKZ hypergeometric differential equation as a quantum curve, linking it to Gromov-Witten theory, mirror symmetry, and physical dualities.

Contribution

It introduces a novel description of the GKZ equation via topological recursion and spectral curves, connecting it to quantum curves and physical interpretations in string theory.

Findings

Reconstruction of GKZ equation as a WKB expansion

Identification of the GKZ equation with the quantum curve for brane partition functions

Analysis of Stokes matrices and wall-crossing phenomena in equivariant models

Abstract

In this article, a novel description of the hypergeometric differential equation found from Gel'fand-Kapranov-Zelevinsky's system (referred to GKZ equation) for Givental's $J$-function in the Gromov-Witten theory will be proposed. The GKZ equation involves a parameter $\hbar$, and we will reconstruct it as the WKB expansion from the classical limit $\hbar\to 0$ via the topological recursion. In this analysis, the spectral curve (referred to GKZ curve) plays a central role, and it can be defined as the critical point set of the mirror Landau-Ginzburg potential. Our novel description is derived via the duality relations of the string theories, and various physical interpretations suggest that the GKZ equation is identified with the quantum curve for the brane partition function in the cohomological limit. As an application of our novel picture for the GKZ equation, we will discuss the Stokes matrix for the equivariant $\mathbb{C}\textbf{P}^{1}$ model and the wall-crossing formula for the total Stokes matrix will be examined. And as a byproduct of this analysis we will study Dubrovin's conjecture for this equivariant model.