Hofstadter's butterfly in quantum geometry

We point out that the recent conjectural solution to the spectral problem for the Hamiltonian H = e(x) + e (x) + e(p) + e (p) in terms of the refined topological invariants of a local Calabi-Yau (CY) geometry has an intimate relation with two-dimensional non-interacting electrons moving in a periodic potential under a uniform magnetic field. In particular, we find that the quantum A-period, determining the relation between the energy eigenvalue and the Kehler modulus of the CY, can be found explicitly when the quantum parameter q = e(i (h) over bar) is a root of unity, that its branch cuts are given by Hofstadter's butterfly, and that its imaginary part counts the number of states of the Hofstadter Hamiltonian. The modular double operation, exchanging (h) over bar and = <(<(h)over tilde>)over bar> = 4 pi(2)/(h) over bar, plays an important role.