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We show that parity symmetry is not spontaneously broken in the CP^N sigmamodel for any value of N when the coefficient of the $\theta$--term becomes$\theta=\pi$ (mod $2\pi$). The result follows from a non-perturbative analysisof the nodal structure of the vacuum functional $\psi_0(z)$. The dynamical roleof sphalerons turns out to be very important for the argument. The resultintroduces severe constraints on the possible critical behavior of the modelsat $\theta=\pi$ (mod $2\pi$).Comment: 8 pages, revtex, to appear in Phys. Rev. Let

physical review letters

Vacuum Structure of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msup><mml:mrow><mml:mi>CP</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="italic">N</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math>Sigma Models at<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi mathvariant="italic">θ</mml:mi><mml:mi mathvariant="italic" /><mml:mspace /><mml:mo>=</mml:mo><mml:mspace /><mml:mi mathvariant="italic">π</mml:mi></mml:…

Vacuum Structure ofCPNSigma Models atθ=π