Hofstadter's butterfly and Langlands duality

We address the Hofstadter problem on a two-dimensional square lattice system. We propose a novel perspective on its mathematical structure of the corresponding tight-binding Hamiltonian from a viewpoint of the Langlands duality, a mathematical conjecture relevant to a wide range of the modern mathematics including number theory, solvable systems, representations, and geometry. It is known that the Hamiltonian can be algebraically written by means of the quantum group Uq(sl2). We claim that Hofstadter's fractal is deeply related with the Langlands duality of the quantum group. In addition, from this perspective, the existence of the corresponding elliptic curve expression interpreted from the tight-binging Hamiltonian implies a more fascinating connection with the Langlands program and quantum geometry.