  The Lebwohl-Lasher model describes the isotropic-nematic transition in liquid
crystals. In two dimensions, where its continuous symmetry cannot break
spontaneously, it is investigated numerically since decades to verify, in
particular, the conjecture of a topological transition leading to a nematic
phase with quasi-long-range order. We use scale invariant scattering theory to
exactly determine the renormalization group fixed points in the general case of
$N$ director components ($RP^{N-1}$ model), which yields the Lebwohl-Lasher
model for $N=3$. For $N>2$ we show the absence of quasi-long-range order and
the presence of a zero temperature critical point in the universality class of
the $O(N(N+1)/2-1)$ model. For $N=2$ the fixed point equations yield the
Berezinskii-Kosterlitz-Thouless transition required by the correspondence
$RP^1\sim O(2)$.
