Theta Dependence in the LargeNLimit of Four-Dimensional Gauge Theories

The $\theta$ dependence of pure gauge theories in four dimensions can be studied using a duality of large $N$ gauge theories with string theory on a certain spacetime. Via this duality, one can argue that for every $\theta$, there are infinitely many vacua that are stable in the large $N$ limit. The true vacuum, found by minimizing the energy in this family, is a smooth function of $\theta$ except at $\theta =\pi$ where it jumps. This jump is associated with spontaneous breaking of $\mathrm{CP}$ symmetry. Domain walls separating adjacent vacua are described in terms of wrapped six-branes.