  In a Euclidean path integral formulation of gauge theory and quantum
mechanics, the theta-term induces a sign problem, and relatedly, a complex
phase for the fugacity of topological defects; whereas in Minkowskian
formulation, it induces a topological (geometric) phase multiplying ordinary
path-amplitudes. In an SU(2) Yang-Mills theory which admits a semi-classical
limit, we show that the complex fugacity generates interference between
Euclidean path histories, i.e., monopole-instanton events, and radically alters
the vacuum structure. At theta=0, a mass gap is due to the monopole-instanton
plasma, and the theory has a unique vacuum. At theta=pi, the monopole induced
mass gap vanishes, despite the fact that monopole density is independent of
theta, due to destructive topological interference. The theory has two options:
to remain gapless or to be gapped with a two-fold degenerate vacua. We show the
latter is realized by the magnetic bion mechanism, and the two-vacua are
realization of spontaneous CP-breaking.
  The effect of the theta-term in the circle-compactified gauge theory is a
generalization of Aharonov-Bohm effect, and the geometric (Berry) phase. As
theta varies from 0 to pi, the gauge theory interpolates between even- and
odd-integer spin quantum anti-ferromagnets on two spatial dimensional
bi-partite lattices, which have ground state degeneracies one and two,
respectively, as it is in gauge theory at theta=0 and theta=pi.
