The berry phase of D0-branes

We study SU(2) Yang-Mills quantum mechanics with N = 2, 4, 8 and 16 supercharges. This describes the non-relativistic dynamics of a pair of D0-branes moving in d = 3, 4, 6 and 10 spacetime dimensions respectively. We show that as the D0-branes orbit, states undergo a Berry holonomy described by the four Hopf maps. For the N = 2 theory, the associated Hopf map is the Z(2) Mobius bundle and its effect is to turn the D0-branes into anyons with exchange statistics e(i pi/2). For the N = 4, 8 and 16 theories, the Hopf maps give rise to Berry connections that are familiar to physicists: the U(1) Dirac monopole; the SU(2) Yang monopole; and the SO(8) octonionic monopole.