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I will discuss the problem of electrically charged, massless fermions scattering off magnetic monopoles. The interpretation of the outgoing states has long been a puzzle, as they can carry fractional quantum numbers. I will argue that such outgoing particles live in the twisted sector of a topological co-dimension 1 surface, which ends on the monopole. This leads to new predictions for the outgoing state in QED as well as the Standard Model. The surface is often non-invertible, and as such the outgoing radiation not only carries unconventional flavor quantum numbers, but is often trailed by a topological field theory.