  Effective actions, such as nonlinear sigma models, are important tools in
studying low-energy fluctuations of magnetically ordered states. Here we derive
an effective action for the smooth order-parameter distortions of noncoplanar
magnetic textures [with an SO(3) order parameter, as opposed to the
SO(3)/SO(2)=S^2 of collinear magnets] coupled to itinerant electrons.
Noncoplanar magnetic textures commonly arise in the presence of
electron-mediated long-range spin exchange interactions, giving rise to the
emergence of rich electronic phenomena such as anomalous quantum Hall effect.
We parameterize the smooth order-parameter distortions of such a magnetic
texture by certain non-Abelian fields, and derive the action in terms of these
fields by integrating out the itinerant electrons. As a concrete example, we
perform explicit calculations for a triangular-lattice model with tetrahedral
magnetic ordering. The action encodes the charge and spin quantum-Hall
responses as well as the energetics of twisting the magnetic texture.
