SLE in self-dual critical Z(N) spin systems: CFT predictions

The Schramm-Loewner evolution (SLE) describes the continuum limit of domain walls at phase transitions in two-dimensional statistical systems. We consider here the SLE in Z(N) spin models at their self-dual critical point. For N=2 and N=3 these models correspond to the Ising and three-state Potts model. For N⩾4 the critical self-dual Z(N) spin models are described in the continuum limit by non-minimal conformal field theories with central charge c⩾1. By studying the representations of the corresponding chiral algebra, we show that two particular operators satisfy a two level null vector condition which, for N⩾4, presents an additional term coming from the extra symmetry currents action. For N=2,3 these operators correspond to the boundary conditions changing operators associated to the SLE16/3 (Ising model) and to the SLE24/5 and SLE10/3 (three-state Potts model). We suggest a definition of the interfaces within the Z(N) lattice models. The scaling limit of these interfaces is expected to be described at the self-dual critical point and for N⩾4 by the SLE4(N+1)/(N+2) and SLE4(N+2)/(N+1) processes.