  We develop an effective gauge theory for three-flavored magnons in frustrated
magnets hosting topological textures with the aid of the quaternion
representation of the SO(3) order parameter. We find that the effect of
topological solitons on magnons is captured generally by the non-Abelian
emergent electromagnetic fields, distinct from the previously established gauge
theory for magnons in collinear magnets where the gauge theory is often
restricted to be Abelian. As concrete examples, $4\pi$-vortices in
two-/three-dimensional magnets and Shankar skyrmions in three-dimensional
magnets are discussed in detail, which are shown to induce, respectively, the
Abelian and the non-Abelian topological magnetic field on magnons, and thereby
engender the topological Hall transport in textured frustrated magnets. Our
work is applicable to a broad class of magnetic materials whose low-energy
manifold is described by the SO(3) order parameter. We envision that the
discovery of the non-Abelian magnonic gauge theory will enrich the field of
magnonics as well as prompt the study of magnon transport in textured
frustrated magnets.
