On the quantum CP^N-1 model

We clarify several subtle points in the CP ^_N-1 model in 2 + 1 dimensions. It is shown that as a consequence of local gauge invariance the n_A particles, belonging to fundamental representation of SU(N) do not appear in the spectrum (in any number of dimensions) regardless of the nature of the interaction between them. In 2 + 1 dimensions we analyse the phase structure of the theory from the point of view of realization of the relevant symmetries: SU(N)/Z_N ⊗ U_Φ(1) ⊗ U_E(1). The symmetries U_Φ(1) and U_E(1) are gauge invariant analogs of the magnetic flux and electric charge symmetries in scalar QED. The charge of U_E(1) is not related to the global part of the local U(1) transformations (which counts the difference between the number of n and the number of n^* excitations). At the phase transition point the flavour SU(N) Z_N is broken down to U(N-1) while the mode of implementation of the U_Φ(1) is changed from Kosterlitz-Thouless to Wigner-Weyl. We provide a gauge invariant order parameter for the flavour symmetry breaking and identify the massless "photon" with the Kosterlitz-Thouless zero mode of U_Φ(1).