Critical points in the CP N-1model

We use scale invariant scattering theory to obtain the exact equations determining the renormalization group fixed points of the two-dimensional CP ( N-1) model, for N real. Also due to special degeneracies at N = 2 and 3, the space of solutions for N > 2 reduces to that of the O(N (2) - 1) model, and accounts for a zero temperature critical point. For N < 2 the space of solutions becomes larger than that of the O(N (2) - 1) model, with the appearance of new branches of fixed points relevant for criticality in gases of intersecting loops.