Theta, time reversal and temperature

SU(N) gauge theory is time reversal invariant at theta = 0 and theta = pi. We show that at theta = pi there is a discrete 't Hooft anomaly involving time reversal and the center symmetry. This anomaly leads to constraints on the vacua of the theory. It follows that at theta = pi the vacuum cannot be a trivial non-degenerate gapped state. (By contrast, the vacuum at theta = 0 is gapped, non-degenerate, and trivial.) Due to the anomaly, the theory admits nontrivial domain walls supporting lower-dimensional theories. Depending on the nature of the vacuum at theta = pi, several phase diagrams are possible. Assuming area law for space-like loops, one arrives at an inequality involving the temperatures at which CP and the center symmetry are restored. We also analyze alternative scenarios for SU(2) gauge theory. The underlying symmetry at theta = pi is the dihedral group of 8 elements. If deconfined loops are allowed, one can have two O(2)-symmetric fixed points. It may also be that the four-dimensional theory around theta = pi is gapless, e.g. a Coulomb phase could match the underlying anomalies.