Noninvertible Symmetries, Anomalies, and Scattering Amplitudes

We show that crossing symmetry of S matrices is modified in certain theories with noninvertible symmetries or anomalies. Focusing on integrable flows to gapped phases in two dimensions, we find that S matrices derived previously from the bootstrap approach are incompatible with noninvertible symmetries along the flow. We present consistent alternatives, which, however, violate standard crossing symmetry and obey modified rules dictated by fusion categories. We extend these rules to theories with discrete anomalies.