We examine the question concerning the relevance of classical instanton configurations in two-dimensional $\mathrm{C}{\mathrm{P}}^N$ $\sigma$ models. We show that in contrast to recent claims by Witten instantons do not disappear and are not eliminated by quantum fluctuations. This conclusion is based on the observation that at the quantum effective-action level instantons instead of being stationary points now appear in the form of poles. In addition we demonstrate that the $\frac{1}{N}$ expansion and the semiclassical method correspond to two alternative contour integrations of the functional integral.

• Received 13 July 1979

DOI:https://doi.org/10.1103/PhysRevD.20.3331