Vacuum structure of Yang-Mills theory as a function of θ

It is believed that in SU(N) Yang-Mills theory observables are N-branched functions of the topological theta angle. This is supposed to be due to the existence of a set of locally-stable candidate vacua, which compete for global stability as a function of theta. We study the number of theta vacua, their interpretation, and their stability properties using systematic semiclassical analysis in the context of adiabatic circle compactification on R-3 x S-1. We find that while observables are indeed N-branched functions of theta, there are only approximate to N/2 locally-stable candidate vacua for any given theta. We point out that the different theta vacua are distinguished by the expectation values of certain magnetic line operators that carry non-zero GNO charge but zero 't Hooft charge. Finally, we show that in the regime of validity of our analysis YM theory has spinodal points as a function of theta, and gather evidence for the conjecture that these spinodal points are present even in the R-4 limit.