We demonstrate that crossing symmetry of S-matrices can be violated in theories with non-invertible symmetries. Focusing on integrable flows to gapped phases in two dimensions, we show that S-matrices derived previously from the bootstrap approach are incompatible with non-invertible symmetries along the flow. We present consistent alternatives, which however violate crossing symmetry and obey modified rules dictated by fusion categories. We also show how these modified crossing rules can be used to constrain the space of amplitudes with a given categorical symmetry.

In this talk I will explore the space of two-to-two S-matrices in two-dimensional theories with a global O(N) symmetry, as restricted by the general principles of unitarity, crossing and analyticity. I will describe various features of the allowed space and identify some special points on its boundary with known integrable theories. Finally, I will present a useful dual formulation of the S-matrix bootstrap problem. Based on arXiv:1909.06495.