Exchange relations and crossing

We discuss the scattering matrix of two-dimensional integrable QFTs whose fields obey non-trivial exchange relations. We show that crossing equations for such models have to be modified, and propose their consistent modification. This modification opens the way to constructing new integrable S matrices. As a check, we consider the crossing equations for the SU(N) chiral Gross-Neveu model, and for the Φ21 deformation of the tricritical Ising model, finding an agreement with the existing proposals. Finally, we reconsider the crossing equations for massless excitations of the mixed-flux AdS3×S3×T4 light-cone gauge superstring sigma model, and conjecture that the massless excitations satisfy non-trivial exchange relations. This changes the crossing equations and leads to a simpler massless dressing factor.