O(N) and RP(N-1) models in two dimensions

I provide evidence that the 2D RP(N-1) model for N greater than or equal to 3 is equivalent to the O(N)-invariant nonlinear sigma model in the continuum limit. To this end, I mainly study particular versions of the models, to be called constraint models. I prove that the constraint RP(N-1) and O(N) models are equivalent for sufficiently weak coupling. Numerical results for the step-scaling function of the running coupling g(-2)=m(L)L are presented. The data confirm that the constraint O(N) model is in the same universality class as the O(N) model with standard action. I show that in the weak coupling limit periodic boundary conditions for the RP(N-1) model correspond to fluctuating boundary conditions for the O(N) model. The effect of boundary conditions on finite size scaling curves is discussed. It is concluded, in contrast with Caracciolo et al., that RP(N-1) and O(N) models share a unique universality class.