Phase Transitions at Unusual Values of $\theta$

We calculate the $\theta$ dependence in a cousin of QCD, where the vacuum structure can be analyzed exactly. The theory is $\mathcal{N}=2$ $SU(2)$ gauge theory with $N_F=0,1,2,3$ flavors of fundamentals, explicitly broken to $\mathcal{N}=1$ via an adjoint superpotential, and coupled to anomaly mediated supersymmetry breaking (AMSB). The hierarchy $m_{AMSB}\ll \mu_{\mathcal{N}=1}\ll \Lambda$ ensures the validity of our IR analysis. As expected from ordinary QCD, the vacuum energy is a function of $\theta$ which undergoes 1st order phase transitions between different vacua where the various dyons condense. For $N_F=0$ we find the expected phase transition at $\theta=\pi$, while for $N_F=1,2,3$ we find phase transitions at fractional values of $\pi$.