The extended Picard–Fuchs Yukawa coupling conjecture for local mirror symmetry

Consider the A-model on a Calabi–Yau threefold $X$. Let $u_i$ be the logarithm of the B-model complex deformation parameter $z_i$ obtained from the toric construction of the mirror Calabi–Yau manifold $\hat{X}$. Extended Picard–Fuchs conjecture. The B-model Yukawa coupling $C_{u_i u_j u_k}$ of $\hat{X}$ obtained from the A-model Yukawa coupling of $X$ is a rational function in $z_i=\exp(u_i)$. Moreover, its denominator includes the divisor of the defining equation of the discriminant locus of $\hat{X}$. This conjecture proposes the natural intersection-theoretic behavior needed for the extended Picard–Fuchs system and is intended to characterize the singularity structure of local mirror symmetry.