Exchange relations and crossing

Abstract 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 Φ 21 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 A d S 3 × S 3 × T 4 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.