Critical points of two-dimensional sigma models and implications for liquid crystals and gases of intersecting loops.

We use the recently introduced scale invariant scattering theory to exactly determine the renormalization group fixed points of $RP^{N-1}$ and $CP^{N-1}$ models in two dimensions, which differ from vector models for an additional local symmetry: respectively the liquid crystal head-tail symmetry and a $U(1)$ symmetry. We show that, also due to subtle degeneracies at specific values of $N$, above a threshold value $N_c$ there is only a zero temperature critical point of $O(N(N+1)/2-1)$ type for $RP^{N-1}$ and of $O(N^2-1)$ type for $CP^{N-1}$. Below $N_c$ new branches of fixed points emerge which are relevant for criticality in gases of loops with crossings. For liquid crystals $N_c=2.24421..$, and a topological transition of Berezinskii-Kosterlitz-Thouless type exists only for $N=2$. For $CP^{N-1}$ $N_c=2$ and no topological transition occurs for $N$ integer.