  Effective dynamics of conduction electrons in antiferromagnetic (AFM)
materials with slowly varying spin texture is developed via non-Abelian gauge
theory. Quite different from the ferromagnetic (FM) case, the spin of a
conduction electron does not follow the background texture even in the
adiabatic limit due to the accumulation of a SU(2) non-Abelian Berry phase.
Correspondingly, it is found that the orbital dynamics becomes spin-dependent
and is affected by two emergent gauge fields. While one of them is the
non-Abelian generalization of what has been discovered in FM systems, the other
leads to an anomalous velocity that has no FM counterpart. Two examples are
provided to illustrate the distinctive spin dynamics of a conduction electron.
