Large-N limit of the XY-Ashkin-Teller model: solution in two dimensions

We introduce a vector model coupled to an N-colour Ashkin-Teller model and solve its phase diagram in the large-N limit in two dimensions. It is shown that the transition line starts from the axis of Ising couplings with a be haviour which can be critical or first order depending on the strength of t he four-spin coupling and of the coupling between spin and vector variables . Below a negative value of the four-spin coupling the transition is always continuous.