Hofstadter's butterfly in quantum geometry

Abstract

Wepoint out that the recent conjectural solution to the spectral problem for the Hamiltonian H = ex + e-x + ep + e-p in terms of the refined topological invariants of a localCalabi-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 K'hler modulus of the CY, can be found explicitlywhen the quantumparameter q = eiℏis 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. Themodular double operation, exchanging ℏ and = ℏ 4π2 /ℏ , plays an important role.