Theta dependence, sign problems, and topological interference

In a Euclidean path integral formulation of gauge theory and quantum mechanics, the θ-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 θ=0, a mass gap is due to the monopole-instanton plasma, and the theory has a unique vacuum. At θ=π, the monopole induced mass gap vanishes, despite the fact that monopole density is independent of θ, 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 θ-term in the circle-compactified gauge theory is a generalization of Aharonov-Bohm effect, and the geometric (Berry) phase. As θ varies from 0 to π, 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 θ=0 and θ=π.