  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.
