C-P-T anomaly matching in bosonic quantum field theory and spin chains

Understand

We consider the $O(3)$ nonlinear sigma model with the $\theta$-term and its linear counterpart in 1+1D.

• The model has discrete time-reflection and space-reflection symmetries at any $\theta$, and enjoys the periodicity in $\theta\rightarrow \theta+2\pi$.
• At $\theta=0,\pi$ it also has a charge-conjugation $C$-symmetry.
• Gauging the discrete space-time reflection symmetries is interpreted as putting the theory on the nonorientable $\mathbb RP^2$ manifold, after which the $2\pi$ periodicity of $\theta$ and the $C$ symmetry at $\theta=\pi$ are lost.