  We consider various sufficiently nonlinear sigma models for nematic liquid
crystal ordering of RP^{N-1} type and of lattice gauge type with continous
symmetries. We rigorously show that they exhibit a first-order transition in
the temperature. The result holds in dimension 2 or more for the RP^{N-1}
models and in dimension 3 or more for the lattice gauge models. In the
two-dimensional case our results clarify and solve a recent controversy about
the possibility of such transitions. For lattice gauge models our methods
provide the first proof of a first-order transition in a model with a
continuous gauge symmetry.
