Can θ / N Dependence for Gluodynamics be Compatible with 2 π Periodicity in θ ?

In a number of field theoretical models the vacuum angle θ enters physics in the combination θ/N , where N stands generically for the number of colors or flavors, in an apparent contradiction with the expected 2π periodicity in θ. We argue that a resolution of this puzzle is related to the existence of a number of different θ dependent sectors in a finite volume formulation, which can not be seen in the naive thermodynamic limit V → ∞. It is shown that, when the limit V → ∞ is properly defined, physics is always 2π periodic in θ for any integer, and even rational, values of N, with vacuum doubling at θ = π (mod 2π). We demonstrate this phenomenon in both the multi-flavor Schwinger model with the bosonization technique, and four-dimensional gluodynamics with the effective Lagrangian method. The proposed mechanism works for an arbitrary gauge group.