  The B-model approach of topological string theory leads to difference
equations by quantizing algebraic mirror curves. It is known that these quantum
mechanical systems are solved by the refined topological strings. 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. In this
paper, we generalize this idea to a more complicated Calabi--Yau manifold. We
find that the local $\mathcal{B}_3$ 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.
