Non-Abelian geometric phases and conductance of spin-$\frac{3}{2}$ holes

Angular momentum $J=$$\frac{3}{2}$ holes in semiconductor heterostructures are shown to accumulate non-Abelian geometric phases as a consequence of their motion. We provide a general framework for analyzing such a system and compute conductance oscillations for a simple ring geometry. We also analyze a figure-eight geometry that captures intrinsically non-Abelian interference effects.