VACUUM-STRUCTURE OF CPN SIGMA-MODELS AT THETA=PI

We show that parity symmetry is not spontaneously broken in the CPN si gma model for any value of N when the coefficient of the theta term be comes theta = pi (mod 2 pi). The result follows from a nonperturbative analysis of the nodal structure of the vacuum functional psi(0)(z). T he dynamical role of sphalerons turns out to be very important for the argument. The result introduces severe constraints on the possible cr itical behavior of the models at theta = pi (mod 2 pi).