Lattice formulation of the CP^n-1 non-linear σ models

We set up a general lattice version of non-linear σ models defined on homogeneous spaces. We then apply this to the CP^n-1 models which are the correct extension of the SU(2) σ model to SU(N). We exhibit their "confinement" property: the elementary multiplets Z_α used to describe the system do not appear as physical particles but only as bound states Z_αλ_αβ^aZ_β. The method enable us to examine the "θ vacua" in the strong coupling limit by using a "dilute loop" approximation. We discuss the effect of the low activation energy for instantons which means that on a lattice, topological number is not conserved.