  The dependence on the topological theta angle term in quantum field theory is
usually discussed in the context of instanton calculus. There the observables
are 2 pi periodic, analytic functions of theta. However, in strongly coupled
theories, the semi-classical instanton approximation can break down due to
infrared divergences. Instances are indeed known where analyticity in theta can
be lost, while the 2 pi periodicity is preserved. In this short note we exhibit
a simple two dimensional example where the 2 pi periodicity is lost. The
observables remain periodic under the transformation theta -> theta + 2 k pi
for some k >= 2. We also briefly discuss the case of four dimensional N=2
supersymmetric gauge theories.
