I will talk about the mysterious correspondence between the topological string and condensed matter physics. Recently, it was pointed out that the quantum eigenvalue problem for a particular Calabi–Yau manifold, known as local $\mathbb{F}_0$, is closely related to the Hofstadter problem for electrons on a two-dimensional square lattice. Then we generalize this result, and find that the local $\cB_3$ geometry, which is a three-point blow-up of local $\mathbb{P}^2$, is associated with electrons on a triangular lattice. This correspondence allows us to use known results in condensed matter physics to investigate the quantum geometry of the toric Calabi–Yau manifold.