Reconstructing GKZ via Topological Recursion
Hiroyuki Fuji, Kohei Iwaki, Masahide Manabe, Ikuo Satake · Communications in Mathematical Physics · 2019
In this article, a novel description of the hypergeometric differential equation found from Gel’fand–Kapranov–Zelevinsky’s system (referred to as 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 a quantum curve from the classical limit $$\hbar \rightarrow 0$$ via the topological recursion. In this analysis, the spectral curve (referred to as GKZ curve) plays a central role, and it can be described by 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 phenomenon for the equivariant $${\mathbb {C}}\mathbf{P }^{1}$$ 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.