Criticality in self-dual sine-Gordon models

We discuss the nature of criticality in the ${\beta }^2=2\pi N$ self-dual extention of the sine-Gordon model. This field theory is related to the two-dimensional classical XY model with a N-fold degenerate symmetry-breaking field. We briefly overview the already studied cases $N=2,4$ and analyze in detail the case N=3 where a single phase transition in the three-state Potts universality class is expected to occur. The Z${}_3$ infrared critical properties of the ${\beta }^2=6\pi$ self-dual sine-Gordon model are derived using two non-perturbative approaches. On one hand, we map the model onto an integrable deformation of the Z${}_4$ parafermion theory. The latter is known to flow to a massless Z${}_3$ infrared fixed point. Another route is based on the connection with a chirally asymmetric, su(2)${}_4$ $\otimes$ su(2)${}_1$ Wess-Zumino-Novikov-Witten model with anisotropic current-current interaction, where we explore the existence of a decoupling (Toulouse) point.