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It has recently been shown that a center-twisted compactification of thefour-dimensional pure $SU(N)$ Yang-Mills theory on a three-torus gives rise tothe two-dimensional $\mathbb{CP}^{N-1}$-model on a circle with a flavor-twistedboundary condition. We verify the consistency of this statementnon-perturbatively at theta angle $\theta=\pi$, in terms of the mixed 't Hooftanomalies for flavor symmetries and the time-reversal symmetry. This providesfurther support for the approach to the confinement of four-dimensionalYang-Mills theory from the two-dimensional $\mathbb{CP}^{N-1}$-model.

Relating 't Hooft Anomalies of 4d Pure Yang-Mills and 2d $\mathbb{CP}^{N-1}$ Model