  We find a new relation between the spectral problem for Bloch electrons on a
two-dimensional honeycomb lattice in a uniform magnetic field and that for
quantum geometry of a toric Calabi-Yau threefold. We show that a difference
equation for the Bloch electron is identical to a quantum mirror curve of the
Calabi-Yau threefold. As an application, we show that bandwidths of the
electron spectra in the weak magnetic flux regime are systematically calculated
by the topological string free energies at conifold singular points in the
Nekrasov-Shatashvili limit.
